Pull requests - nigel.stanger/Publications

nigel.stanger / Publications

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Commits on 2017-05-30

Nigel Stanger	• Moved .gitignore file into subdirectory.	b82bb56
Nigel Stanger	• Ensured Business School logo image isn’t ignored.	016d9ae
Nigel Stanger	• Moved files in preparation for repository merge.	0c11945
Nigel Stanger	• Removed already globally-ignored files.	7717e0d
Nigel Stanger	• Added untracked Business School logo.	4c0bc51

Commits on 2017-02-08

Nigel Stanger	• Corrected errors.	d5e19fb
Nigel Stanger	• Fixed permissions.	ee4855d
Nigel Stanger	• Ignored .tgz files.	e4c1d08
Nigel Stanger	• Added presentation files.	92eaf20

Commits on 2016-11-07

Nigel Stanger	• Added header comments about required version of PGF and use of TeX Gyre fonts.	315b9c2
Nigel Stanger	• Added header comments about use of colour. • Minor textual tweaks for final version.	7217cd9

Commits on 2016-11-04

Nigel Stanger	• Added indicators to transformation figures of what’s actually being transformed.	812e72a
Nigel Stanger	• Updated mentions of non-relational contexts. • Forced final two SIG figures to appear at top of page, to create more space for text. • Updated figure reference that no longer crosses a page. • Added several more references. • Numerous minor rephrasings to shorten paragraphs. • Added new final paragraph about newer work on information capacity.	3ed6c9a
Nigel Stanger	• Added some references for non-relational and deductive view updating.	d71f200
Nigel Stanger	• Slightly rewrote first three sections to create more space.	5877037
Nigel Stanger	• Inserted ACM-provided copyright, ISBN, and DOI.	746a588

Commits on 2016-11-03

Nigel Stanger

• Added descriptive comments at top of file, and expanded some other comments. • Commented out \todo macro. • Updated copyright; added DOI and ISBN placeholders. • Moved line break in title. • Added CCS categories, and removed “relational model” from keywords. • Removed commented out material for camera-ready version. • Re-enabled column balancing.

8331577

Commits on 2016-08-22

Nigel Stanger

• Final updates and changes prior to final re-submission.

fe5bc7d

Commits on 2016-08-19

Nigel Stanger

• Updated SIG edge annotations to match latest knowledge. • Rewrote part about trivial projection edges to refer instead to unique projection edges. • Changed \T to \Type to avoid potential future macros clashes. • Corrected definition of key edges. • Added footnote about Miller’s mention of inclusion dependencies. • Added more edges that lose annotations in S3 → S1 transformation. • Removed conclusion paragraph about moving edges across bijective edges.

548e5f4

Commits on 2016-08-08

Nigel Stanger	• Added keywords, and as a consequence: – Rewrote and reformatted several paragraphs to resolve awkward page and column breaks. – Rearranged positioning of SIG figures. • Fixed a couple of standalone operator expressions that should have been using a macro. • Other minor phrasing tweaks.	3e5a3f6
Nigel Stanger	• Minor phrasing, grammatical, and spelling tweaks.	c5dabfb
Nigel Stanger	• Minor grammatical and phrasing tweaks. • Clarified removal of totality annotations.	ad84313
Nigel Stanger	• Added a Date citation. • Minor phrasing and grammatical tweaks.	fdc4522
Nigel Stanger	• Minor phrasing and grammatical tweaks. • Added “constraints” to list of isomorphic items. • Fixed format of subfigure captions. • Flipped Figure 1 horizontally so that the functional path is more intuitive.	a7e325a
Nigel Stanger	• Minor formatting, phrasing, and grammatical tweaks.	63213b0
Nigel Stanger	• Minor phrasing tweaks. • Updated discussion of Date’s pragmatic approach.	72443bf
Nigel Stanger	• Minor phrasing tweaks. • Rewrote sentence about update side effects.	515c73e
Nigel Stanger	• Corrected publication year of Date’s book.	13fbb41

Commits on 2016-08-07

Nigel Stanger

• Minor tweak to conclusion.

c154a43

Commits on 2016-08-05

Nigel Stanger	• Rewrote conclusion. • Other minor phrasing tweaks.	30be921
Nigel Stanger	• Moved positions of Table 2 and Figure 5 to produce better layout.	9f815d9
Nigel Stanger	• Various phrasing tweaks.	8b0d3ea
Nigel Stanger	• Removed bold math in figure captions.	424509a

Commits on 2016-08-04

Nigel Stanger	• Finished table of implied edges and rewrote description of S1 SIG construction to match.	10c3b72
Nigel Stanger	• Shifted details of implied edges into a table.	e8bc291
Nigel Stanger	• Rolled rewritten functional dependency constraints into the lists with the other rewritten constraints.	1a16c66
Nigel Stanger	• Made consistent use of math mode for \LS and \NLS.	e0f7c12
Nigel Stanger	• Switched to newtxtext and newtxmath packages from txfonts. • Removed no longer needed \!, as the spacing is now better.	ef52ad1
Nigel Stanger	• Switched to subfig package from subfigure. • Switched to align environment instead of eqnarray.	be6ac8b
Nigel Stanger	• Fixed a couple of unconverted - to \setminus.	853ce9f
Nigel Stanger	• Re-ordered and re-numbered rules. • Added note about likelihood of S3 having different information capacity, due to not being able to map all the constraints from S0.	214e5b0
Nigel Stanger	• Added macros to generate constraint numbers and constraint lists in standard format. • Removed unnecessary \makeat* macros. • Changed notation for propagated constraints. • Minor tweaks to phrasing.	815fbdf
Nigel Stanger	• Rephrased definition of key constraint edges.	7d45d99
Nigel Stanger	• Rearranged position of some figures.	267c69f

Commits on 2016-08-03

Nigel Stanger	• Fixed horrible line breaking in math formula. • Corrected annotations of copied projection edge between \TT{NLS} and \TSSNL. • Added note about whether copied projection edges actually still count as projection edges.	c2be516
Nigel Stanger	• Switched to \setminus for set differences. • Made proper use of \TrivialSelection and \TrivialProjection. • Deleted sentence explaining interpretation of edge annotations with respect to reading direction. • Deleted paragraph containing Batini et al's transformation terminology; not important. • Clarified what isomorphism means. • Rewrote transformation discussion to make it clear that the SIGs are being transformed rather than the schemas (even though we are effectively transforming the schemas). • Clarified composition transformations. • Various minor phrasing and grammatical changes.	0c00b2f
Nigel Stanger	• Removed confusing “clarification” of edge annotations. • Removed footnote about nulls, as it didn’t seem to make sense and added nothing useful. • Edited discussion of selection and projection edges.	b95ada7

Commits on 2016-08-02

Nigel Stanger

• Various changes to phrasing up to Section 4.1.

6aff650

Commits on 2016-07-28

Nigel Stanger

• Corrected publication year of Date’s view updating book.

92dee8c

Commits on 2016-07-26

Nigel Stanger	• Updated Section 7.	c5add16
Nigel Stanger	• Updated Section 6.	87d0682
Nigel Stanger	• Updated section 5.	37313b5
Nigel Stanger	• Various edits to Sections 1–4. • Oriented focus more towards modelling.	ff5a009
Nigel Stanger	• Updated to use ACM style.	7ff56bb
Nigel Stanger	• Added missing ACM copyright style file.	a73c079
Nigel Stanger	• Renamed ACM class file back to original name.	fd2fb02

Patch

Showing 15 changed files

Show notes View .gitignore 100644 → 0
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Show notes View APCCM2017_Stanger.bib 100644 → 0
@string{acj = {Australian Computer Journal}} @string{acmcsur = {ACM Computing Surveys}} @string{acmq = {ACM Queue}} @string{acmtocs = {ACM Transactions on Computing Systems}} @string{acmtods = {ACM Transactions on Database Systems}} @string{acmtois = {ACM Transactions on Information Systems}} @string{acmtoit = {ACM Transactions on Internet Technology}} @string{acmtosem = {ACM Transactions on Software Engineering and Methodology}} @string{acmtosn = {ACM Transactions on Sensor Networks}} @string{acmtweb = {ACM Transactions on the Web}} @string{adt = {Application Development Trends}} @string{ai = {Artificial Intelligence}} @string{ajis = {Australasian Journal of Information Systems}} @string{basist = {Bulletin of the American Society for Information Science and Technology}} @string{bit = {Behaviour {\&} Information Technology}} @string{byte = {B{YTE}}} @string{cacm = {Communications of the ACM}} @string{ccis = {Communications in Computer and Information Science}} @string{cj = {The Computer Journal}} @string{crpit = {Conferences in Research and Practice in Information Techology}} @string{database = {ACM SIGMIS Database}} @string{dbj = {Database Journal}} @string{dbms = {DBMS Magazine}} @string{dbpd = {Database Programming {\&} Design}} @string{ddj = {Dr. Dobb's Journal}} @string{develop = {{d}evelop, The Apple Technical Journal}} @string{directions = {Apple Directions}} @string{dke = {Data {\&} Knowledge Engineering}} @string{dlib = {D-Lib Magazine}} @string{ejis = {European Journal of Information Systems}} @string{fm = {First Monday}} @string{hbr = {Harvard Business Review}} @string{ibmjrd = {IBM Journal of Research and Development}} @string{ibmsj = {IBM Systems Journal}} @string{idt = {Internet Development Trends}} @string{ieeeahc = {IEEE Annals of the History of Computing}} @string{ieeec = {IEEE Computer}} @string{ieeedeb = {IEEE Data Engineering Bulletin}} @string{ieeeic = {IEEE Internet Computing}} @string{ieeeis = {IEEE Intelligent Systems}} @string{ieeepc = {IEEE Pervasive Computing}} @string{ieees = {IEEE Software}} @string{ieeesp = {IEEE Spectrum}} @string{ieeetnn = {IEEE Transactions on Neural Networks}} @string{ieeetse = {IEEE Transactions on Software Engineering}} @string{ijast = {International Journal of Applied Software Technology}} @string{ijcatr = {International Journal of Computer Applications Technology and Research}} @string{ijgis = {International Journal of Geographical Information Science}} @string{ijgisold = {International Journal of Geographical Information Systems}} @string{ijhcs = {International Journal of Human-Computer Studies}} @string{ijmms = {International Journal of Man-Machine Studies}} @string{ijseke = {International Journal of Software Engineering and Knowledge Engineering}} @string{ijswis = {International Journal on Semantic Web and Information Systems}} @string{ijwet = {International Journal of Web Engineering and Technology}} @string{ipm = {Information Processing and Management}} @string{is = {Information Systems}} @string{isedj = {Information Systems Education Journal}} @string{isj = {Information Systems Journal}} @string{ist = {Information and Software Technology}} @string{jacm = {Journal of the ACM}} @string{jasist = {Journal of the American Society for Information Science and Technology}} @string{jdim = {Journal of Digital Information Management}} @string{jdmm = {Journal of Digital Media Management}} @string{jiis = {Journal of Intelligent Information Systems}} @string{jlp = {Journal of Logic Programming}} @string{jodi = {Journal of Digital Information}} @string{jot = {Journal of Object Technology}} @string{jrpit = {Journal of Research and Practice in Information Technology}} @string{jss = {The Journal of Systems and Software}} @string{lncs = {Lecture Notes in Computer Science}} @string{misq = {MIS Quarterly}} @string{nzjc = {New Zealand Journal of Computing}} @string{nzjis = {New Zealand Journal of Information Systems}} @string{nzlimj = {New Zealand Library {\&} Information Management Journal}} @string{oracle = {Oracle Magazine}} @string{oss = {OCLC Systems {\&} Services: International Digital Library Perspectives}} @string{pcbi = {PLoS Computational Biology}} @string{pnas = {Proceedings of the National Academy of Sciences of the United States of America}} @string{pvldb = {Proceedings of the VLDB Endowment}} @string{sciam = {Scientific American}} @string{sej = {Software Engineering Journal}} @string{sigmetrics = {ACM SIGMETRICS Performance Evaluation Review}} @string{sigmod = {ACM SIGMOD Record}} @string{sigplan = {ACM SIGPLAN Notices}} @string{sigsoft = {ACM SIGSOFT Software Engineering Notes}} @string{spe = {Software---Practice and Experience}} @string{swj = {Semantic Web}} @string{tkde = {IEEE Transactions on Knowledge and Data Engineering}} @string{vldb = {The VLDB Journal}} @article{Bancilhon.F-1981a-Update, Author = {Fran{\c c}ois Bancilhon and Nicolas Spyratos}, Doi = {10.1145/319628.319634}, Journal = acmtods, Month = dec, Number = {4}, Pages = {557--575}, Title = {Update semantics of relational views}, Volume = {6}, Year = {1981}} @book{Batini.C-1992b-Conceptual, Address = {Redwood City, California, USA}, Booktitle = {Conceptual Database Design: An Entity-Relationship Approach}, Editor = {Carlo Batini and Stefano Ceri and Shamkant B. Navathe}, Isbn = {978-0-8053-0244-1}, Publisher = {Benjamin/Cummings}, Title = {Conceptual Database Design: An Entity-Relationship Approach}, Year = {1992}} @article{Chen.H-2011a-Survey, Author = {Haitao Chen and Husheng Liao}, Doi = {10.7763/IJCTE.2011.V3.278}, Journal = {International Journal of Computer Theory and Engineering}, Month = feb, Number = {1}, Pages = {1793--8201}, Title = {A survey to view update problem}, Url = {http://www.ijcte.org/papers/278-D038.pdf}, Volume = {3}, Year = {2011}} @article{Codd.E-1970a-Relational, Author = {Edgar F. Codd}, Doi = {10.1145/362384.362685}, Journal = cacm, Month = jun, Number = {6}, Pages = {377--387}, Title = {A relational model of data for large shared data banks}, Volume = {13}, Year = {1970}} @incollection{Codd.E-1972a-Completeness, Author = {Edgar F. Codd}, Crossref = {Rustin.R-1972a-Data}, Pages = {65--98}, Title = {Relational completeness of data base sublanguages}} @inproceedings{Codd.E-1974a-Recent, Author = {Edgar F. Codd}, Crossref = {Rosenfeld.J-1974a-IFIP}, Pages = {1017--1021}, Title = {Recent investigations in relational data base systems}} @article{Cosmadakis.S-1984a-Updates, Author = {Stavros S. Cosmadakis and Christos H. Papadimitriou}, Doi = {10.1145/1634.1887}, Journal = jacm, Month = sep, Number = {4}, Pages = {742--760}, Title = {Updates of relational views}, Volume = {31}, Year = {1984}} @book{Date.C-2013a-View, Author = {Chris J. Date}, Booktitle = {View Updating \{\&} Relational Theory: Solving the View Update Problem}, Date-Added = {2016-01-25 02:14:14 +0000}, Date-Modified = {2016-01-25 02:15:43 +0000}, Isbn = {9781449357849}, Publisher = {O'Reilly}, Series = {Theory in Practice}, Title = {View Updating {\&} Relational Theory: Solving the View Update Problem}, Year = {2013}} @book{Date.C-2012a-SQL-and-Relational, Author = {Chris J. Date}, Booktitle = {SQL and Relational Theory: How to Write Accurate SQL Code}, Date-Added = {2016-01-25 02:12:09 +0000}, Date-Modified = {2016-01-25 02:16:01 +0000}, Edition = {second}, Isbn = {9781449316402}, Publisher = {O'Reilly}, Series = {Theory in Practice}, Title = {SQL and Relational Theory: How to Write Accurate SQL Code}, Year = {2012}} @article{Dayal.U-1982a-Correct, Author = {Umeshwar Dayal and Philip A. Bernstein}, Doi = {10.1145/319732.319740}, Journal = acmtods, Month = sep, Number = {3}, Pages = {381--416}, Title = {On the correct translation of update operations on relational views}, Volume = {7}, Year = {1982}} @article{Gottlob.G-1988a-Properties, Author = {Georg Gottlob and Paolo Paolini and Roberto Zicari}, Doi = {10.1145/49346.50068}, Journal = acmtods, Month = dec, Number = {4}, Pages = {486--524}, Title = {Properties and update semantics of consistent views}, Volume = {13}, Year = {1988}} @article{Hegner.S-2004a-Order-based, Author = {Stephen J. Hegner}, Doi = {10.1023/A:1026158013113}, Journal = {Annals of Mathematics and Artificial Intelligence}, Month = jan, Number = {1--2}, Pages = {63--125}, Title = {An order-based theory of updates for closed database views}, Volume = {40}, Year = {2004}} @article{Hull.R-1986a-Relative, Author = {Richard Hull}, Journal = {SIAM Journal on Computing}, Month = aug, Number = {3}, Pages = {856--886}, Title = {Relative information capacity of simple relational database schemata}, Volume = {15}, Year = {1986}} @inproceedings{Keller.A-1985a-Algorithms, Author = {Arthur Michael Keller}, Crossref = {PODS-1985a-Proceedings}, Doi = {10.1145/325405.325423}, Pages = {154--163}, Title = {Algorithms for translating view updates to database updates for views involving selections, projections, and joins}} @article{Keller.A-1986a-Role, Author = {Arthur Michael Keller}, Doi = {10.1109/MC.1986.1663034}, Journal = ieeec, Month = jan, Number = {1}, Pages = {63--73}, Title = {The role of semantics in translating view updates}, Volume = {19}, Year = {1986}} @article{Keller.A-1987a-Comment, Author = {Arthur Michael Keller}, Doi = {10.1145/27629.214296}, Journal = acmtods, Month = sep, Number = {3}, Pages = {521--523}, Title = {Comments on {B}ancilhon and {S}pyratos' ``{U}pdate semantics and relational views''}, Volume = {12}, Year = {1987}} @inproceedings{Kotidis.Y-2006a-Updates, Annote = {Article 2.}, Author = {Yannis Kotidis and Divesh Srivastava and Yannis Velegrakis}, Crossref = {Liu.L-2006a-ICDE}, Doi = {10.1109/ICDE.2006.167}, Pages = {2}, Read = {1}, Title = {Updates through views: {A} new hope}} @article{Laurent.D-2001a-Monotonic, Author = {Dominique Laurent and Jens Lechtenb{\"o}rger and Nicolas Spyratos and Gottfried Vossen}, Doi = {10.1007/s007780100055}, Journal = vldb, Month = dec, Number = {4}, Pages = {295--315}, Title = {Monotonic complements for independent data warehouses}, Volume = {10}, Year = {2001}} @inproceedings{Lechtenborger.J-2003a-Impact, Author = {Jens Lechtenb{\"o}rger}, Crossref = {Neven.F-2003a-PODS}, Doi = {10.1145/773153.773159}, Pages = {49--55}, Title = {The impact of the constant complement approach towards view updating}} @article{Lechtenborger.J-2003a-Computation, Author = {Jens Lechtenb{\"o}rger and Gottfried Vossen}, Doi = {10.1145/777943.777946}, Journal = acmtods, Month = jun, Number = {2}, Pages = {175--208}, Title = {On the computation of relational view complements}, Volume = {28}, Year = {2003}} @inproceedings{Masunaga.Y-1984a-Relational, Author = {Yoshifumi Masunaga}, Crossref = {Dayal.U-1984a-VLDB}, Pages = {309--320}, Title = {A relational database view update translation mechanism}, Url = {http://www.vldb.org/conf/1984/P309.PDF}} @inproceedings{Miller.R-1993a-Information, Author = {Ren{\'e}e Jean Miller and Y. E. Ioannidis and Raghu Ramakrishnan}, Crossref = {Agrawal.R-1993a-VLDB}, Pages = {120--133}, Title = {The use of information capacity in schema integration and translation}, Url = {http://www.vldb.org/conf/1993/P120.PDF}} @phdthesis{Miller.R-1994a-PhD, Address = {Madison, Wisconsin, USA}, Author = {Miller, Ren{\'e}e Jean}, School = {University of Wisconsin-Madison}, Title = {Managing Structural Heterogeneity}, Type = {{PhD} thesis}, Url = {http://www.cs.toronto.edu/~miller/papers/dissertation.ps}, Year = {1994}} @techreport{Miller.R-1994a-SIG, Address = {Madison, Wisconsin, USA}, Author = {Ren{\'e}e Jean Miller and Y. E. Ioannidis and Raghu Ramakrishnan}, Institution = {Department of Computer Sciences, University of Wisconsin-Madison}, Month = jan, Number = {CS-TR-94-1185}, Title = {Schema intension graphs: {A} formal model for the study of schema equivalence}, Type = {Technical report}, Url = {http://www.cs.toronto.edu/~miller/papers/MIR93c.ps}, Year = {1994}} @article{Miller.R-1994a-Schema, Author = {Ren{\'e}e Jean Miller and Y. E. Ioannidis and Raghu Ramakrishnan}, Doi = {10.1016/0306-4379(94)90024-8}, Journal = is, Month = jan, Number = {1}, Pages = {3--31}, Title = {Schema equivalence in heterogeneous systems: {B}ridging theory and practice}, Url = {http://www.cs.toronto.edu/~miller/papers/MIR94b.ps}, Volume = {19}, Year = {1994}} @article{Shu.H-2000a-Using, Author = {Hua Shu}, Doi = {10.1023/A:1008726123320}, Journal = jiis, Month = sep, Number = {2}, Pages = {147--173}, Title = {Using constraint satisfaction for view update}, Volume = {15}, Year = {2000}} @techreport{Qian.X-1995a-Correct, Address = {Menlo Park, California, USA}, Author = {Xiaolei Qian}, Institution = {Computer Science Laboratory, SRI International}, Month = jul, Number = {SRI-CSL-95-08}, Title = {Correct schema transformations}, Type = {Technical report}, Url = {http://www.csl.sri.com/reports/postscript/sri-csl-95-08.ps.gz}, Year = {1995}} %% crossrefs @proceedings{Agrawal.R-1993a-VLDB, Address = {Dublin, Ireland}, Booktitle = {Proceedings of the Nineteenth International Conference on Very Large Data Bases (VLDB)}, Editor = {Rakesh Agrawal and Se{\'a}n Baker and David A. Bell}, Isbn = {978-1-55860-152-X}, Month = aug # {~24--27}, Publisher = {Morgan Kaufmann}, Title = {Proceedings of the Nineteenth International Conference on Very Large Data Bases (VLDB)}, Year = {1993}} @proceedings{Dayal.U-1984a-VLDB, Address = {Singapore}, Booktitle = {Proceedings of the Tenth International Conference on Very Large Data Bases (VLDB 1984)}, Editor = {Umeshwar Dayal and Gunter Schlageter and Lim Huat Seng}, Isbn = {0-934613-16-8}, Month = aug # {~27--31}, Publisher = {Morgan Kaufmann}, Title = {Proceedings of the Tenth International Conference on Very Large Data Bases (VLDB 1984)}, Year = {1984}} @proceedings{Liu.L-2006a-ICDE, Address = {Atlanta, Georgia, USA}, Booktitle = {Proceedings of the 22nd International Conference on Data Engineering (ICDE 2006)}, Editor = {Ling Liu and Andreas Reuter and Kyu-Young Whang and Jianjun Zhang}, Isbn = {0-7695-2570-9}, Month = {3--8~} # apr, Publisher = {IEEE Computer Society}, Title = {Proceedings of the 22nd International Conference on Data Engineering (ICDE 2006)}, Year = {2006}} @proceedings{Neven.F-2003a-PODS, Address = {San Diego, California, USA}, Booktitle = {Proceedings of the Twenty-Second ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS 2003)}, Editor = {Frank Neven and Catriel Beeri and Tova Milo}, Isbn = {1-58113-670-6}, Month = jun # {~9--12}, Organization = {ACM}, Title = {Proceedings of the Twenty-Second ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS 2003)}, Year = {2003}} @proceedings{PODS-1985a-Proceedings, Address = {Portland, Oregon, USA}, Booktitle = {Proceedings of the Fourth ACM SIGACT-SIGMOD Symposium on Principles of Database Systems (PODS '85)}, Isbn = {0-89791-153-9}, Month = mar # {~25--27}, Publisher = {ACM}, Title = {Proceedings of the Fourth ACM SIGACT-SIGMOD Symposium on Principles of Database Systems (PODS '85)}, Year = {1985}} @proceedings{Rosenfeld.J-1974a-IFIP, Address = {Stockholm, Sweden}, Booktitle = {Proceedings of the IFIP Congress '74 (Information Processing '74)}, Editor = {Jack L. Rosenfeld}, Isbn = {978-0-7204-2803-3}, Month = aug # {~5--10}, Publisher = {North-Holland}, Title = {Proceedings of the IFIP Congress '74 (Information Processing '74)}, Year = {1974}} @book{Rustin.R-1972a-Data, Address = {Englewood Cliffs, New Jersey, USA}, Booktitle = {Data Base Systems}, Editor = {Randall Rustin}, Publisher = {Prentice-Hall}, Series = {Courant Computer Science Symposia Series 6}, Title = {Data Base Systems}, Year = {1972}}

Show notes View

APCCM2017_Stanger.bib 100644 → 0

@string{acj = {Australian Computer Journal}}
@string{acmcsur = {ACM Computing Surveys}}
@string{acmq = {ACM Queue}}
@string{acmtocs = {ACM Transactions on Computing Systems}}
@string{acmtods = {ACM Transactions on Database Systems}}
@string{acmtois = {ACM Transactions on Information Systems}}
@string{acmtoit = {ACM Transactions on Internet Technology}}
@string{acmtosem = {ACM Transactions on Software Engineering and Methodology}}
@string{acmtosn = {ACM Transactions on Sensor Networks}}
@string{acmtweb = {ACM Transactions on the Web}}
@string{adt = {Application Development Trends}}
@string{ai = {Artificial Intelligence}}
@string{ajis = {Australasian Journal of Information Systems}}
@string{basist = {Bulletin of the American Society for Information Science and Technology}}
@string{bit = {Behaviour {\&} Information Technology}}
@string{byte = {B{YTE}}}
@string{cacm = {Communications of the ACM}}
@string{ccis = {Communications in Computer and Information Science}}
@string{cj = {The Computer Journal}}
@string{crpit = {Conferences in Research and Practice in Information Techology}}
@string{database = {ACM SIGMIS Database}}
@string{dbj = {Database Journal}}
@string{dbms = {DBMS Magazine}}
@string{dbpd = {Database Programming {\&} Design}}
@string{ddj = {Dr. Dobb's Journal}}
@string{develop = {{d}evelop, The Apple Technical Journal}}
@string{directions = {Apple Directions}}
@string{dke = {Data {\&} Knowledge Engineering}}
@string{dlib = {D-Lib Magazine}}
@string{ejis = {European Journal of Information Systems}}
@string{fm = {First Monday}}
@string{hbr = {Harvard Business Review}}
@string{ibmjrd = {IBM Journal of Research and Development}}
@string{ibmsj = {IBM Systems Journal}}
@string{idt = {Internet Development Trends}}
@string{ieeeahc = {IEEE Annals of the History of Computing}}
@string{ieeec = {IEEE Computer}}
@string{ieeedeb = {IEEE Data Engineering Bulletin}}
@string{ieeeic = {IEEE Internet Computing}}
@string{ieeeis = {IEEE Intelligent Systems}}
@string{ieeepc = {IEEE Pervasive Computing}}
@string{ieees = {IEEE Software}}
@string{ieeesp = {IEEE Spectrum}}
@string{ieeetnn = {IEEE Transactions on Neural Networks}}
@string{ieeetse = {IEEE Transactions on Software Engineering}}
@string{ijast = {International Journal of Applied Software Technology}}
@string{ijcatr = {International Journal of Computer Applications Technology and Research}}
@string{ijgis = {International Journal of Geographical Information Science}}
@string{ijgisold = {International Journal of Geographical Information Systems}}
@string{ijhcs = {International Journal of Human-Computer Studies}}
@string{ijmms = {International Journal of Man-Machine Studies}}
@string{ijseke = {International Journal of Software Engineering and Knowledge Engineering}}
@string{ijswis = {International Journal on Semantic Web and Information Systems}}
@string{ijwet = {International Journal of Web Engineering and Technology}}
@string{ipm = {Information Processing and Management}}
@string{is = {Information Systems}}
@string{isedj = {Information Systems Education Journal}}
@string{isj = {Information Systems Journal}}
@string{ist = {Information and Software Technology}}
@string{jacm = {Journal of the ACM}}
@string{jasist = {Journal of the American Society for Information Science and Technology}}
@string{jdim = {Journal of Digital Information Management}}
@string{jdmm = {Journal of Digital Media Management}}
@string{jiis = {Journal of Intelligent Information Systems}}
@string{jlp = {Journal of Logic Programming}}
@string{jodi = {Journal of Digital Information}}
@string{jot = {Journal of Object Technology}}
@string{jrpit = {Journal of Research and Practice in Information Technology}}
@string{jss = {The Journal of Systems and Software}}
@string{lncs = {Lecture Notes in Computer Science}}
@string{misq = {MIS Quarterly}}
@string{nzjc = {New Zealand Journal of Computing}}
@string{nzjis = {New Zealand Journal of Information Systems}}
@string{nzlimj = {New Zealand Library {\&} Information Management Journal}}
@string{oracle = {Oracle Magazine}}
@string{oss = {OCLC Systems {\&} Services: International Digital Library Perspectives}}
@string{pcbi = {PLoS Computational Biology}}
@string{pnas = {Proceedings of the National Academy of Sciences of the United States of America}}
@string{pvldb = {Proceedings of the VLDB Endowment}}
@string{sciam = {Scientific American}}
@string{sej = {Software Engineering Journal}}
@string{sigmetrics = {ACM SIGMETRICS Performance Evaluation Review}}
@string{sigmod = {ACM SIGMOD Record}}
@string{sigplan = {ACM SIGPLAN Notices}}
@string{sigsoft = {ACM SIGSOFT Software Engineering Notes}}
@string{spe = {Software---Practice and Experience}}
@string{swj = {Semantic Web}}
@string{tkde = {IEEE Transactions on Knowledge and Data Engineering}}
@string{vldb = {The VLDB Journal}}

@article{Bancilhon.F-1981a-Update,
	Author = {Fran{\c c}ois Bancilhon and Nicolas Spyratos},
	Doi = {10.1145/319628.319634},
	Journal = acmtods,
	Month = dec,
	Number = {4},
	Pages = {557--575},
	Title = {Update semantics of relational views},
	Volume = {6},
	Year = {1981}}

@book{Batini.C-1992b-Conceptual,
	Address = {Redwood City, California, USA},
	Booktitle = {Conceptual Database Design: An Entity-Relationship Approach},
	Editor = {Carlo Batini and Stefano Ceri and Shamkant B. Navathe},
	Isbn = {978-0-8053-0244-1},
	Publisher = {Benjamin/Cummings},
	Title = {Conceptual Database Design: An Entity-Relationship Approach},
	Year = {1992}}

@article{Chen.H-2011a-Survey,
	Author = {Haitao Chen and Husheng Liao},
	Doi = {10.7763/IJCTE.2011.V3.278},
	Journal = {International Journal of Computer Theory and Engineering},
	Month = feb,
	Number = {1},
	Pages = {1793--8201},
	Title = {A survey to view update problem},
	Url = {http://www.ijcte.org/papers/278-D038.pdf},
	Volume = {3},
	Year = {2011}}

@article{Codd.E-1970a-Relational,
	Author = {Edgar F. Codd},
	Doi = {10.1145/362384.362685},
	Journal = cacm,
	Month = jun,
	Number = {6},
	Pages = {377--387},
	Title = {A relational model of data for large shared data banks},
	Volume = {13},
	Year = {1970}}

@incollection{Codd.E-1972a-Completeness,
	Author = {Edgar F. Codd},
	Crossref = {Rustin.R-1972a-Data},
	Pages = {65--98},
	Title = {Relational completeness of data base sublanguages}}

@inproceedings{Codd.E-1974a-Recent,
	Author = {Edgar F. Codd},
	Crossref = {Rosenfeld.J-1974a-IFIP},
	Pages = {1017--1021},
	Title = {Recent investigations in relational data base systems}}

@article{Cosmadakis.S-1984a-Updates,
	Author = {Stavros S. Cosmadakis and Christos H. Papadimitriou},
	Doi = {10.1145/1634.1887},
	Journal = jacm,
	Month = sep,
	Number = {4},
	Pages = {742--760},
	Title = {Updates of relational views},
	Volume = {31},
	Year = {1984}}

@book{Date.C-2013a-View,
	Author = {Chris J. Date},
	Booktitle = {View Updating \{\&} Relational Theory: Solving the View Update Problem},
	Date-Added = {2016-01-25 02:14:14 +0000},
	Date-Modified = {2016-01-25 02:15:43 +0000},
	Isbn = {9781449357849},
	Publisher = {O'Reilly},
	Series = {Theory in Practice},
	Title = {View Updating {\&} Relational Theory: Solving the View Update Problem},
	Year = {2013}}

@book{Date.C-2012a-SQL-and-Relational,
	Author = {Chris J. Date},
	Booktitle = {SQL and Relational Theory: How to Write Accurate SQL Code},
	Date-Added = {2016-01-25 02:12:09 +0000},
	Date-Modified = {2016-01-25 02:16:01 +0000},
	Edition = {second},
	Isbn = {9781449316402},
	Publisher = {O'Reilly},
	Series = {Theory in Practice},
	Title = {SQL and Relational Theory: How to Write Accurate SQL Code},
	Year = {2012}}

@article{Dayal.U-1982a-Correct,
	Author = {Umeshwar Dayal and Philip A. Bernstein},
	Doi = {10.1145/319732.319740},
	Journal = acmtods,
	Month = sep,
	Number = {3},
	Pages = {381--416},
	Title = {On the correct translation of update operations on relational views},
	Volume = {7},
	Year = {1982}}

@article{Gottlob.G-1988a-Properties,
	Author = {Georg Gottlob and Paolo Paolini and Roberto Zicari},
	Doi = {10.1145/49346.50068},
	Journal = acmtods,
	Month = dec,
	Number = {4},
	Pages = {486--524},
	Title = {Properties and update semantics of consistent views},
	Volume = {13},
	Year = {1988}}

@article{Hegner.S-2004a-Order-based,
	Author = {Stephen J. Hegner},
	Doi = {10.1023/A:1026158013113},
	Journal = {Annals of Mathematics and Artificial Intelligence},
	Month = jan,
	Number = {1--2},
	Pages = {63--125},
	Title = {An order-based theory of updates for closed database views},
	Volume = {40},
	Year = {2004}}

@article{Hull.R-1986a-Relative,
	Author = {Richard Hull},
	Journal = {SIAM Journal on Computing},
	Month = aug,
	Number = {3},
	Pages = {856--886},
	Title = {Relative information capacity of simple relational database schemata},
	Volume = {15},
	Year = {1986}}

@inproceedings{Keller.A-1985a-Algorithms,
	Author = {Arthur Michael Keller},
	Crossref = {PODS-1985a-Proceedings},
	Doi = {10.1145/325405.325423},
	Pages = {154--163},
	Title = {Algorithms for translating view updates to database updates for views involving selections, projections, and joins}}

@article{Keller.A-1986a-Role,
	Author = {Arthur Michael Keller},
	Doi = {10.1109/MC.1986.1663034},
	Journal = ieeec,
	Month = jan,
	Number = {1},
	Pages = {63--73},
	Title = {The role of semantics in translating view updates},
	Volume = {19},
	Year = {1986}}

@article{Keller.A-1987a-Comment,
	Author = {Arthur Michael Keller},
	Doi = {10.1145/27629.214296},
	Journal = acmtods,
	Month = sep,
	Number = {3},
	Pages = {521--523},
	Title = {Comments on {B}ancilhon and {S}pyratos' ``{U}pdate semantics and relational views''},
	Volume = {12},
	Year = {1987}}

@inproceedings{Kotidis.Y-2006a-Updates,
	Annote = {Article 2.},
	Author = {Yannis Kotidis and Divesh Srivastava and Yannis Velegrakis},
	Crossref = {Liu.L-2006a-ICDE},
	Doi = {10.1109/ICDE.2006.167},
	Pages = {2},
	Read = {1},
	Title = {Updates through views: {A} new hope}}

@article{Laurent.D-2001a-Monotonic,
	Author = {Dominique Laurent and Jens Lechtenb{\"o}rger and Nicolas Spyratos and Gottfried Vossen},
	Doi = {10.1007/s007780100055},
	Journal = vldb,
	Month = dec,
	Number = {4},
	Pages = {295--315},
	Title = {Monotonic complements for independent data warehouses},
	Volume = {10},
	Year = {2001}}

@inproceedings{Lechtenborger.J-2003a-Impact,
	Author = {Jens Lechtenb{\"o}rger},
	Crossref = {Neven.F-2003a-PODS},
	Doi = {10.1145/773153.773159},
	Pages = {49--55},
	Title = {The impact of the constant complement approach towards view updating}}

@article{Lechtenborger.J-2003a-Computation,
	Author = {Jens Lechtenb{\"o}rger and Gottfried Vossen},
	Doi = {10.1145/777943.777946},
	Journal = acmtods,
	Month = jun,
	Number = {2},
	Pages = {175--208},
	Title = {On the computation of relational view complements},
	Volume = {28},
	Year = {2003}}

@inproceedings{Masunaga.Y-1984a-Relational,
	Author = {Yoshifumi Masunaga},
	Crossref = {Dayal.U-1984a-VLDB},
	Pages = {309--320},
	Title = {A relational database view update translation mechanism},
	Url = {http://www.vldb.org/conf/1984/P309.PDF}}

@inproceedings{Miller.R-1993a-Information,
	Author = {Ren{\'e}e Jean Miller and Y. E. Ioannidis and Raghu Ramakrishnan},
	Crossref = {Agrawal.R-1993a-VLDB},
	Pages = {120--133},
	Title = {The use of information capacity in schema integration and translation},
	Url = {http://www.vldb.org/conf/1993/P120.PDF}}

@phdthesis{Miller.R-1994a-PhD,
	Address = {Madison, Wisconsin, USA},
	Author = {Miller, Ren{\'e}e Jean},
	School = {University of Wisconsin-Madison},
	Title = {Managing Structural Heterogeneity},
	Type = {{PhD} thesis},
	Url = {http://www.cs.toronto.edu/~miller/papers/dissertation.ps},
	Year = {1994}}

@techreport{Miller.R-1994a-SIG,
	Address = {Madison, Wisconsin, USA},
	Author = {Ren{\'e}e Jean Miller and Y. E. Ioannidis and Raghu Ramakrishnan},
	Institution = {Department of Computer Sciences, University of Wisconsin-Madison},
	Month = jan,
	Number = {CS-TR-94-1185},
	Title = {Schema intension graphs: {A} formal model for the study of schema equivalence},
	Type = {Technical report},
	Url = {http://www.cs.toronto.edu/~miller/papers/MIR93c.ps},
	Year = {1994}}

@article{Miller.R-1994a-Schema,
	Author = {Ren{\'e}e Jean Miller and Y. E. Ioannidis and Raghu Ramakrishnan},
	Doi = {10.1016/0306-4379(94)90024-8},
	Journal = is,
	Month = jan,
	Number = {1},
	Pages = {3--31},
	Title = {Schema equivalence in heterogeneous systems: {B}ridging theory and practice},
	Url = {http://www.cs.toronto.edu/~miller/papers/MIR94b.ps},
	Volume = {19},
	Year = {1994}}

@article{Shu.H-2000a-Using,
	Author = {Hua Shu},
	Doi = {10.1023/A:1008726123320},
	Journal = jiis,
	Month = sep,
	Number = {2},
	Pages = {147--173},
	Title = {Using constraint satisfaction for view update},
	Volume = {15},
	Year = {2000}}

@techreport{Qian.X-1995a-Correct,
	Address = {Menlo Park, California, USA},
	Author = {Xiaolei Qian},
	Institution = {Computer Science Laboratory, SRI International},
	Month = jul,
	Number = {SRI-CSL-95-08},
	Title = {Correct schema transformations},
	Type = {Technical report},
	Url = {http://www.csl.sri.com/reports/postscript/sri-csl-95-08.ps.gz},
	Year = {1995}}

%% crossrefs

@proceedings{Agrawal.R-1993a-VLDB,
	Address = {Dublin, Ireland},
	Booktitle = {Proceedings of the Nineteenth International Conference on Very Large Data Bases (VLDB)},
	Editor = {Rakesh Agrawal and Se{\'a}n Baker and David A. Bell},
	Isbn = {978-1-55860-152-X},
	Month = aug # {~24--27},
	Publisher = {Morgan Kaufmann},
	Title = {Proceedings of the Nineteenth International Conference on Very Large Data Bases (VLDB)},
	Year = {1993}}

@proceedings{Dayal.U-1984a-VLDB,
	Address = {Singapore},
	Booktitle = {Proceedings of the Tenth International Conference on Very Large Data Bases (VLDB 1984)},
	Editor = {Umeshwar Dayal and Gunter Schlageter and Lim Huat Seng},
	Isbn = {0-934613-16-8},
	Month = aug # {~27--31},
	Publisher = {Morgan Kaufmann},
	Title = {Proceedings of the Tenth International Conference on Very Large Data Bases (VLDB 1984)},
	Year = {1984}}

@proceedings{Liu.L-2006a-ICDE,
	Address = {Atlanta, Georgia, USA},
	Booktitle = {Proceedings of the 22nd International Conference on Data Engineering (ICDE 2006)},
	Editor = {Ling Liu and Andreas Reuter and Kyu-Young Whang and Jianjun Zhang},
	Isbn = {0-7695-2570-9},
	Month = {3--8~} # apr,
	Publisher = {IEEE Computer Society},
	Title = {Proceedings of the 22nd International Conference on Data Engineering (ICDE 2006)},
	Year = {2006}}

@proceedings{Neven.F-2003a-PODS,
	Address = {San Diego, California, USA},
	Booktitle = {Proceedings of the Twenty-Second ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS 2003)},
	Editor = {Frank Neven and Catriel Beeri and Tova Milo},
	Isbn = {1-58113-670-6},
	Month = jun # {~9--12},
	Organization = {ACM},
	Title = {Proceedings of the Twenty-Second ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS 2003)},
	Year = {2003}}

@proceedings{PODS-1985a-Proceedings,
	Address = {Portland, Oregon, USA},
	Booktitle = {Proceedings of the Fourth ACM SIGACT-SIGMOD Symposium on Principles of Database Systems (PODS '85)},
	Isbn = {0-89791-153-9},
	Month = mar # {~25--27},
	Publisher = {ACM},
	Title = {Proceedings of the Fourth ACM SIGACT-SIGMOD Symposium on Principles of Database Systems (PODS '85)},
	Year = {1985}}

@proceedings{Rosenfeld.J-1974a-IFIP,
	Address = {Stockholm, Sweden},
	Booktitle = {Proceedings of the IFIP Congress '74 (Information Processing '74)},
	Editor = {Jack L. Rosenfeld},
	Isbn = {978-0-7204-2803-3},
	Month = aug # {~5--10},
	Publisher = {North-Holland},
	Title = {Proceedings of the IFIP Congress '74 (Information Processing '74)},
	Year = {1974}}

@book{Rustin.R-1972a-Data,
	Address = {Englewood Cliffs, New Jersey, USA},
	Booktitle = {Data Base Systems},
	Editor = {Randall Rustin},
	Publisher = {Prentice-Hall},
	Series = {Courant Computer Science Symposia Series 6},
	Title = {Data Base Systems},
	Year = {1972}}

Show notes View APCCM2017_Stanger.tex 100644 → 0
%% %% MAX 12 PAGES (all material) %% \documentclass{vldb} \usepackage{balance} \usepackage{relalg} \usepackage{txfonts} \usepackage{bm} \usepackage{subfigure} \usepackage{booktabs} \usepackage{tikz} \usetikzlibrary{positioning} \usetikzlibrary{graphs} \usetikzlibrary{decorations.pathreplacing} \usetikzlibrary{calc} \usetikzlibrary{arrows} % "empty" arrow tip \pgfarrowsdeclare{:}{:}{}{} % custom bar arrow tip, offset from end of line (use "empty" tip at line ends if no >) \tikzset{crossbar/.tip={\|[scale width=1.75,sep=0.25em]}} % various edge styles for TikZ \tikzset{ function/.style={arrows={->}}, injection/.style={arrows={<-}}, total/.style={arrows={:{crossbar}-}}, surjection/.style={arrows={-{crossbar}:}}, bijection/.style={arrows={<{crossbar}-{crossbar}>}}, projection/.style={arrows={:{crossbar}-{crossbar}>}}, projection left/.style={arrows={:{crossbar}-{crossbar}>},edge label={\scriptsize\(\RelProject\)}}, projection right/.style={arrows={:{crossbar}-{crossbar}>},edge label'={\scriptsize\(\RelProject\)}}, selection left/.style={arrows={<-{crossbar}>},edge label={\scriptsize\(\RelRestrict\)}}, selection right/.style={arrows={<-{crossbar}>},edge label'={\scriptsize\(\RelRestrict\)}}, funcdep left/.style={arrows={->},edge node={node[sloped,midway,above] {\scriptsize\emph{key}}}}, funcdep right/.style={arrows={->},edge node={node[sloped,midway,below] {\scriptsize\emph{key}}}}, surtotal/.style={arrows={:{crossbar}-{crossbar}:}}, input keep/.style={blue,thick}, input delete/.style={blue!40,thick,dashed}, output/.style={red,thick}, output temp/.style={red,thick,dashed}, path 1/.style={green!60!black,thick}, path 2/.style={orange,thick} } % projection and selection edge annotations for TikZ \makeatletter \newcommand{\ProjectionAnnotation}[3][]{% \path (#2) to node[above,#1] {\scriptsize\(\RelProject\)} (#3);% } \newcommand{\SelectionAnnotation}[3][]{% \path (#2) to node[above,#1] {\scriptsize\(\RelRestrict\)} (#3);% } \makeatother \newcounter{constraint} % misc \newcommand{\todo}[1]{\textbf{!!TODO!!} {[#1]}} % commonly used elements \newcommand{\LS}{\ensuremath{\mathit{LS}}} \newcommand{\NLS}{\ensuremath{\mathit{NLS}}} \newcommand{\LSsub}{\ensuremath{\mathit{LS}_{\mathit{sub}}}} \newcommand{\NLSsub}{\ensuremath{\mathit{NLS}_{\mathit{sub}}}} \newcommand{\Sno}{\ensuremath{\mathit{Sno}}} \newcommand{\Sname}{\ensuremath{\mathit{Sname}}} \newcommand{\Status}{\ensuremath{\mathit{Status}}} \newcommand{\City}{\ensuremath{\mathit{City}}} \newcommand{\T}[1]{\ensuremath{T_{#1}}} \newcommand{\TT}[1]{\ensuremath{T_{\{#1\}}}} \newcommand{\CityLondon}{\ensuremath{\{\City\colon\mathit{'London'}\}}} \newcommand{\TCityMinusLondon}{\ensuremath{\T{\City} - \CityLondon}} \newcommand{\TSSC}{\ensuremath{\T{\Sname} \times \T{\Status} \times \T{\City}}} \newcommand{\TSSL}{\ensuremath{\T{\Sname} \times \T{\Status} \times \CityLondon}} \newcommand{\TSSNL}{\ensuremath{\T{\Sname} \times \T{\Status} \times (\TCityMinusLondon)}} \newcommand{\TLSPlusNLS}{\ensuremath{\T{\LS} + \T{\NLS}}} \newcommand{\TTLSPlusNLS}{\ensuremath{\TT{\LS} + \TT{\NLS}}} \newcommand{\TLSPlusNLSsub}{\ensuremath{\T{\LSsub} + \T{\NLSsub}}} \newcommand{\TTLSPlusNLSsub}{\ensuremath{\TT{\LSsub} + \TT{\NLSsub}}} \newcommand{\StackTLSPlusNLS}{\ensuremath{\begin{array}{c}\T{\LS}\,+ \\ \T{\NLS}\end{array}}} \newcommand{\StackTTLSPlusNLS}{\ensuremath{\begin{array}{c}\TT{\LS}\,+ \\ \TT{\NLS}\end{array}}} \newcommand{\StackTLSPlusNLSsub}{\ensuremath{\begin{array}{c}\T{\LSsub}\,+ \\ \T{\NLSsub}\end{array}}} \newcommand{\StackTTLSPlusNLSsub}{\ensuremath{\begin{array}{c}\TT{\LSsub}\,+ \\ \TT{\NLSsub}\end{array}}} \newcommand{\StackTSSC}{\ensuremath{\begin{array}{c}\T{\Sname}\,\times \\ \T{\Status} \times \T{\City}\end{array}}} \newcommand{\StackTSSL}{\ensuremath{\begin{array}{c}\T{\Sname} \times \T{\Status}\,\times \\ \CityLondon\end{array}}} \newcommand{\StackTSSNL}{\ensuremath{\begin{array}{c}\T{\Sname} \times \T{\Status}\,\times \\ (\TCityMinusLondon)\end{array}}} \newcommand{\StackTCityMinusLondon}{\ensuremath{\begin{array}{c}\T{\City}\,- \\ \CityLondon\end{array}}} \newcommand{\SC}[1]{\ensuremath{\mathcal{S}_{#1}}} % dominates \newcommand{\Dominates}[2]{\ensuremath{#2 \preceq #1}} \newcommand{\Equivalent}[2]{\ensuremath{#1 \equiv #2}} % SIG notation (in text) \newcommand{\Sedge}[1]{\ensuremath{\sigma_{\textrm{#1}}}} \newcommand{\SedgeP}[1]{\ensuremath{\sigma_{\textrm{#1}}^{'}}} \newcommand{\medmid}{\raise.125ex\hbox{\scalebox{1}[0.75]{$\mid$}}} % General SIG edges for use in formulas. % Adapted from: http://tex.stackexchange.com/questions/96330/adding-symbols-at-the-ends-of-a-horizontal-line \makeatletter \newlength{\@annotskipleft} \newlength{\@annotskipright} % #1 = left edge component % #2 = right edge component % #3 = left bar annotation % #4 = right bar annotation \newcommand\@sig@edge[4]{% \let\@middle\joinrel% \ifx#1\relbar% \@annotskipleft=.3em% % scrunch up the bare line so it's similar length to \long...arrow \ifx#2\relbar\def\@middle{\joinrel\joinrel\relbar\joinrel\joinrel}\fi% \else% % arrows need a little more clearance \@annotskipleft=.4em% \fi% \ifx#2\relbar\@annotskipright=.3em\else\@annotskipright=.4em\fi% \mathrel{\ooalign{$#1\@middle#2$\cr\hskip\@annotskipleft$#3$\hfil$#4$\hskip\@annotskipright\cr}}% } % 0 = nothing % 1 = bar % 2 = arrowhead % 3 = both \def\@sigedge#1#2{% \ifcase\numexpr#14+#2\relax% \@sig@edge{\relbar}{\relbar}{}{}\or % 00 = ----- \@sig@edge{\relbar}{\relbar}{}{\medmid}\or % 01 = ---+- \@sig@edge{\relbar}{\rightarrow}{}{}\or % 02 = ----> \@sig@edge{\relbar}{\rightarrow}{}{\medmid}\or % 03 = ---+> \@sig@edge{\relbar}{\relbar}{\medmid}{}\or % 10 = -+--- \@sig@edge{\relbar}{\relbar}{\medmid}{\medmid}\or % 11 = -+-+- \@sig@edge{\relbar}{\rightarrow}{\medmid}{}\or % 12 = -+--> \@sig@edge{\relbar}{\rightarrow}{\medmid}{\medmid}\or % 13 = -+-+> \@sig@edge{\leftarrow}{\relbar}{}{}\or % 20 = <---- \@sig@edge{\leftarrow}{\relbar}{}{\medmid}\or % 21 = <--+- \@sig@edge{\leftarrow}{\rightarrow}{}{}\or % 22 = <---> \@sig@edge{\leftarrow}{\rightarrow}{}{\medmid}\or % 23 = <--+> \@sig@edge{\leftarrow}{\relbar}{\medmid}{}\or % 30 = <+--- \@sig@edge{\leftarrow}{\relbar}{\medmid}{\medmid}\or % 31 = <+-+- \@sig@edge{\leftarrow}{\rightarrow}{\medmid}{}\or % 32 = <+--> \@sig@edge{\leftarrow}{\rightarrow}{\medmid}{\medmid} % 33 = <+-+> \fi% } \newcommand{\sigedge}[1]{\ensuremath{\@sigedge#1}} \makeatother \newcommand{\Edge}{\sigedge{00}} \newcommand{\Total}{\sigedge{10}} \newcommand{\Surjective}{\sigedge{01}} \newcommand{\SurTotal}{\sigedge{11}} \newcommand{\Functional}{\sigedge{02}} \newcommand{\Injective}{\sigedge{20}} \newcommand{\Bijective}{\sigedge{33}} % Hack to get the overset symbols to appear at the right height. % \smash removes the spacing around the operator, hence \mathop. \newcommand{\LabelledEdge}[2]{\mathop{\overset{\raisebox{0.3em}{\scriptsize\ensuremath{#2}}}{\smash[t]{#1}}}} \newcommand{\ProjectionEdge}{\LabelledEdge{\sigedge{13}}{\RelProject}} \newcommand{\SelectionEdge}{\LabelledEdge{\sigedge{23}}{\RelRestrict}} \newcommand{\TrivialProjection}{\ensuremath{\LabelledEdge{\Bijective}{\RelProject}}} \newcommand{\TrivialSelection}{\ensuremath{\LabelledEdge{\Bijective}{\RelRestrict}}} \newcommand{\KeyEdge}{\ensuremath{\LabelledEdge{\Functional}{\mathit{key}}}} \hyphenation{co-pied sche-ma} \pagestyle{empty} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{Characterising relational view updates using relative information capacity} \numberofauthors{1} \author{ \alignauthor Nigel Stanger \\ \affaddr{Department of Information Science}\\ \affaddr{University of Otago}\\ \affaddr{Dunedin, New Zealand}\\ \email{nigel.stanger@otago.ac.nz} } \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} In 2012, Chris Date published a book about the view update problem in the context of the Relational Model. He presented several detailed examples of different varieties of view updates and characterised their behaviour, in the process deriving a set of principles for updating views based around the notion of \emph{information equivalence} of view schemas. In this paper we examine how this notion can be more formally characterised using Hull's concept of \emph{relative information capacity}. Specifically, we use Miller's \emph{schema intension graph} formalism to characterise the information equivalence of Date's restriction view example, and show that this supports his operational definition of information equivalence. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \noindent The view update problem is a well known and long standing problem of great practical significance in the database field. Indeed, Date \cite{Date.C-2013a-View} likens it to significant unsolved problems in other fields such as the Riemann Hypothesis in mathematics or ``\(P = \mathit{NP}\)?'' in computational complexity theory. In his 2012 book \cite{Date.C-2013a-View} Date developed a comprehensive set of principles for updating relational views, based on the notion of \emph{information equivalence}. However, that notion was defined more operationally than by a formal theoretical definition. View updating can be considered a special case of schema translation: we effectively have one schema comprising a set of views and another schema comprising a set of base relations, and need to translate updates from one schema to the other.\footnote{These schemas may be actually separate, or more likely may be sub-schemas of a larger database schema, with effective separation achieved via the access privilege mechanism.} It therefore makes sense that formalisms developed to assist with characterising sche\-ma translations would also be applicable to view updates. To that end, we propose to use Hull's notion of \emph{relative information capacity} \cite{Hull.R-1986a-Relative} as a means to characterise the equivalence of the schemas involved in a given view configuration. In particular, we will use Miller's \emph{schema intension graph} (SIG) formalism, which facilitates the comparison of relative information capacity across schemas. In this paper, we show how SIGs can be used to characterise the equivalence of schemas within a view configuration. To facilitate this, we extend the SIG formalism with additional concepts, including relation and tuple types, key constraints and functional dependencies. We also show how the formalism can be practically used to characterise the information equivalence of a view configuration, by applying it to Date's example of disjoint restriction views \cite{Date.C-2013a-View}. The results our formal analysis are shown to be consistent with Date's operational analysis of information equivalence in this example. In Section~\ref{sec-view-update-problem}, we briefly review the view update problem and relevant research, and in Section~\ref{sec-relations-types} we introduce some core concepts that are used throughout the remainder of the paper. In Section~\ref{sec-info-capacity} we provide a detailed explanation of relative information capacity and the SIG formalism, then explain in Section~\ref{sec-relational-sigs} how to construct a SIG that represents a relational schema. In Section~\ref{sec-characterising} we analyse Date's restriction view example \cite{Date.C-2013a-View} in detail, and characterise the information capacity of the various sub-schemas involved. We discuss conclusions and future work in Section~\ref{sec-conclusion}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The view update problem} \label{sec-view-update-problem} \noindent The view update problem was first discussed by Codd in 1974 \cite{Codd.E-1974a-Recent}. When an update operation is applied to a view, that update needs to be translated into corresponding updates on the base relations from which the view is derived \cite{Keller.A-1985a-Algorithms}. Sometimes this translation is clear, but sometimes there are multiple such translations that all produce the desired view update, but produce different---possibly incompatible---changes in the base relations. Worse, there may be translations that have side effects (e.g., for some views based on joins), which produce a different result in the view than what we would expect if we could apply the update to the view directly \cite{Keller.A-1985a-Algorithms}. The problem then is how to choose the most appropriate translation. As yet there has been no solution that completely and automatically resolves this ambiguity. There has been considerable research into this problem over the last four decades, and numerous different approaches have been suggested. We shall now briefly review some key developments in the relational context. % Paolini? Banchilhon and Spyratos proposed the concept of \emph{constant complement} \cite{Bancilhon.F-1981a-Update} as a way to resolve view update issues. The \emph{complement} of a view describes all the information from the original database that is not in the view, so composing a view with its complement provides enough information to reconstruct the original database. Keeping the complement invariant under view update facilitates translation construction. This is a useful approach that has been developed further by other authors \cite{Hegner.S-2004a-Order-based,Lechtenborger.J-2003a-Impact}, but it has been shown that reasonable translations exist that cannot be carried out under constant complement \cite{Keller.A-1987a-Comment}. Computing the complement of a given view is also difficult, and several authors have worked on computing complements more efficiently \cite{Cosmadakis.S-1984a-Updates,Laurent.D-2001a-Monotonic,Lechtenborger.J-2003a-Computation}. Finally, this approach is purely structural in nature and does not consider the semantics of the schema, which is necessary to properly resolve translation ambiguity \cite{Keller.A-1986a-Role}. Gottlob et al.\ generalised the idea of constant complement into the class of \emph{consistent views} \cite{Gottlob.G-1988a-Properties}. They define consistent views as having the property that ``if the effect of a view update on a view state is determined, then the effect of the corresponding database update is unambiguously determined'' \cite{Gottlob.G-1988a-Properties}. However, there are still reasonable views that are not consistent under this definition, and this approach again does not consider schema semantics. % not sure this is really needed? %Keller nicely illustrated the view update problem in terms of mappings between database states \cite{Keller.A-1985a-Algorithms}, as shown in Figure~\ref{fig-equivalence}. A view \(V(\mathit{DB})\) is derived via a view mapping \(V\) from the original database \(\mathit{DB}\). When an update \(U\) is applied to \(V(\mathit{DB})\), a view translator \(T\) translates it into an update \(T(U)\) on \(\mathit{DB}\), resulting in a new database state \(\mathit{DB}'\). The new view state \(V(\mathit{DB}')\) is then derived. Problems arise if \(V(\mathit{DB}')\) does not match what we would expect if \(U\) were applied directly to \(V(\mathit{DB})\), because of side effects introduced by \(T\). Ideally, we would be able to find a side effect free translator \(T\) for any given view, but there are some views that cannot be translated without side effects (e.g. certain joins) \cite{Keller.A-1985a-Algorithms}. % Date \cite{Date.C-2013a-View} notes that this arises from a lack of \emph{information equivalence} between \(\mathit{DB}\) and \(V(\mathit{DB})\). %%%%%%%%%%%%%%%%%%%% %\newcommand{\qeq}{\ensuremath{\mathop{\smash[t]{\overset{?}{=}}}}} % %\begin{figure} % \centering % \begin{tikzpicture}[every edge/.style={draw,semithick}] % \node (vdb) {\(V(\mathit{DB})\)}; % \node (db) [below=11mm of vdb] {\(\mathit{DB}\vphantom{(')}\)}; % % \node (vdbp) [right=3cm of vdb] {\(V(\mathit{DB}')\)}; % \node (uvdb) [left=-1mm of vdbp] {\(U(V(\mathit{DB})) \qeq\)}; % % \node (dbp) at (db -\| vdbp) {\(\mathit{DB}\smash[t]{'}\vphantom{()}\)}; % \node (tudb) [left=-1mm of dbp] {\(T(U)(\mathit{DB}) =\)}; % % \graph{ % { (db), (dbp) } -> [edge label={\(V\)},swap] { (vdb), (vdbp) }; % % (vdb) -> [edge node={node (u) [above=-0.6mm] {\(U\)}}] (uvdb); % (db) -> [edge node={node (tu) [above=-0.6mm] {\(T(U)\)}}] (tudb); % % (u) -> [double equal sign distance=2pt,-implies] (tu); % }; % \end{tikzpicture} % \caption{Equivalence of view updates \protect\cite{Keller.A-1985a-Algorithms}.} % \label{fig-equivalence} %\end{figure} %%%%%%%%%%%%%%%%%%%% Keller noted that semantic information is essential in order to choose an appropriate translation \cite{Keller.A-1986a-Role}, and suggested this could be collected at view definition time from the database administrator (DBA) \cite{Keller.A-1985a-Algorithms}. Masunaga argued, however, that this would be insufficient to completely resolve ambiguity, and suggested collecting further semantic information from end users at update time \cite{Masunaga.Y-1984a-Relational}. Shu \cite{Shu.H-2000a-Using} discussed an interesting approach to this kind of information gathering, by treating the problem of generating suitable view translations as a constraint satisfaction problem. This enables constraints to be added incrementally at view update time. However, while collecting such information from the DBA seems reasonable, collecting it from the end user at update time could be seen as too burdensome, and could impact the usability of query interfaces. It is also unclear which information needs to be captured, and how best to capture it. Kotidis et al.\ \cite{Kotidis.Y-2006a-Updates} discussed a different approach to avoiding side effects in view translations. They proposed creating a materialised copy of the view that could be updated directly, rather than mapping updates from the (virtual) view to the base tables. Clearly this approach enables the view to updated without translation side effects, but does imply that the physical manifestation of the view may no longer match the original view query (i.e., its extension may not match its intension). Finally, Date has published several books on the theoretical underpinnings of the Relational Model over the last decade, including one specifically targeting the view update problem \cite{Date.C-2013a-View} (indeed, the book is subtitled ``Solving the View Update Problem''). He developed a comprehensive set of rules for updating views in a relational context, supported by a wide array of worked examples. Date advocated explicitly stating all relevant constraints at view creation time, which is similar to the information gathering approaches discussed above \cite{Keller.A-1985a-Algorithms,Masunaga.Y-1984a-Relational,Shu.H-2000a-Using}. He argued that if a complete set of constraints was available, then ambiguity would be diminished or eliminated, and it would be possible to automatically generate a set of appropriate \emph{compensatory rules} to automatically manage the view update process. Compensatory rules are similar in principle to database triggers, except that they are intended to be declarative so that their definitions can made explicitly visible to end users. Date's work is interesting and compelling, and raises several questions worthy of further investigation. However, there are still some situations where the correct view translation may be ambiguous, e.g., views based on certain types of join, as previously noted by Keller \cite{Keller.A-1985a-Algorithms}. Somewhat disappointingly, Date took a pragmatic approach in these cases, effectively saying ``this feels like the correct solution''. His justification was well argued and logical, but it still falls somewhat short of the ideal of a fully automated solution. As noted previously, Date characterises the ability to define an appropriate and effective view translation using the notion of \emph{information equivalence}. Two schemas \(\SC{1}\) and \(\SC{2}\) are said to be information equivalent ``if and only if the constraints that apply to [\(\SC{1}\)] and [\(\SC{2}\)] are such that every proposition that can be represented by [\(\SC{1}\)] can be represented by [\(\SC{2}\)] and vice versa'' \cite{Date.C-2013a-View}. While information equivalence is clearly a formal concept, the definition and characterisation in \cite{Date.C-2013a-View} are more operational and pragmatic in nature. This is what we explore further in this paper. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Relations and types} \label{sec-relations-types} \noindent We generally follow Date's \cite{Date.C-2012a-SQL-and-Relational} notation and terminology regarding the Relational Model. The main exception is that we do not use Tutorial D notation Date uses in his examples, preferring instead a more classical algebraic style notation. In this paper, \(A\) represents an attribute (comprising a name:type pair), \(R\) represents a relation variable (\emph{relvar}), and \(v\) represents a value (of an attribute, tuple, relation, etc.). In the following discussion it will generally be clear which attribute a value is associated with, but where necessary, the value will be prefixed with the attribute name, i.e., \(\mathit{name}\colon\mathit{value}\). Date defines a \emph{type} \(\T{i}\) as ``essentially a \emph{named, finite set of values}'' \cite{Date.C-2012a-SQL-and-Relational}, representing every possible value of some specific kind, e.g., every possible ASCII character string of length 5, every possible city name, every possible relation value with a particular heading, etc. That is: \begin{equation}\label{eqn-type} \T{i} = \{v_{1}, v_{2}, \dotsc, v_{m}\}, 0 < m < k \end{equation} where \(k\) is some finite upper bound. Thus, the type of an attribute \(A_{i}\) (denoted \(\T{A_{i}}\)) is by definition the set of all possible values of \(A_{i}\). Consider a relvar \(R\) with heading \(\{A_{1}, A_{2}, \dotsc, A_{n}\}\). Following from equation~\ref{eqn-type}, the (relation) type of \(R\), denoted here as \(\T{R}\), is the set of all possible relation values with \(R\)'s heading. Similarly, the (tuple) type of \(R\)'s tuples, denoted here as \TT{R}, is the set of all possible tuple values with \(R\)'s heading, which equates to the cartesian product of \(R\)'s attribute types, i.e.: \begin{equation}\label{eqn-tuple-type} \TT{R} = \T{A_{1}} \times \T{A_{2}} \times\dotsb\times \T{A_{n}}\text{.} \end{equation} By definition any given value of \(R\) must comprise a subset (possibly empty) of tuples \(\{t_{1}, t_{2}, \dotsc, t_{m}\} \subseteq \TT{R}\). \(\T{R}\) therefore represents every possible \(k\)-combination of the elements of \TT{R}, for \(0 \le k \le \left\|\TT{R}\right\|\). For example, Date's usual suppliers-and-parts schema \cite{Date.C-2013a-View} includes a relvar \(S\!\) with heading \(\{\Sno, \Sname, \Status, \City\}\), which contains details of suppliers. As per equation~\ref{eqn-tuple-type}, the tuples of \(S\!\) have type \(\TT{S} = \T{\Sno} \times \TSSC\). This notion of type as a set of possible values is synonymous with Codd's original definition of domains in the Relational Model \cite{Codd.E-1970a-Relational,Date.C-2012a-SQL-and-Relational}, and in particular with the notion of domains as ``sets of values'' in schema intension graphs \cite{Miller.R-1994a-Schema,Miller.R-1994a-SIG}. The implications of the latter point will be explored further in Section~\ref{sec-relational-sigs}, after we have discussed the concepts of relative information capacity and schema intension graphs in Section~\ref{sec-info-capacity}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Nulls} \noindent We also follow Date in assuming that nulls are not permitted in relation values. Philosophical objections aside, however, this makes little practical difference to the approach discussed in this paper, as we are dealing exclusively with types rather than attributes. That is, an attribute can be null, but it makes little sense to say that a \emph{type} can be ``null''. It could be claimed that a type ``includes'' nulls, but it is unclear what this would even mean in practice, and runs the risk of treating nulls like ``normal'' values rather than the special markers that they actually are. In any case, this would effectively be no different from the situation without nulls, so we omit further consideration of nulls in the interest of clarity. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{Relative information capacity} \label{sec-info-capacity} \noindent A schema conveys information about the phenomenon that it models. Hull \cite{Hull.R-1986a-Relative} defined the concept of \emph{relative information capacity} to describe the information content of different schemas. The relative information capacity of a schema determines the set of all valid instances of that schema \cite{Hull.R-1986a-Relative,Miller.R-1994a-PhD,Miller.R-1993a-Information}. Relative information capacity, as its name implies, is not a quantitative measure; rather it is an indicator of the relative information content of different schemas. Relative information capacity can be used as a basis for identifying the differences between schema instances, and can therefore be used as an aid to schema integration and translation \cite{Miller.R-1994a-PhD}. As noted earlier, view updating is a special case of this, so it follows that information capacity can also be used to characterise view updates, or more precisely, to characterise the information content of views compared to the base relvar(s) from which they are derived. Miller et al.\ \cite{Miller.R-1993a-Information} defined relative information capacity in terms of two types of mapping. An \emph{information (capacity) preserving mapping} from schema \(\SC{1}\) to schema \(\SC{2}\) maps every element of \(\SC{1}\) to an element of \(\SC{2}\), but only some elements of \(\SC{2}\) to elements of \(\SC{1}\). \(\SC{2}\) has an information capacity greater than \(\SC{1}\), and is said to \emph{dominate} \(\SC{1}\), denoted \Dominates{\SC{2}}{\SC{1}}. Compare this to an \emph{equivalence preserving mapping} from \(\SC{1}\) to \(\SC{3}\), which maps every element of \(\SC{1}\) to an element \(\SC{3}\) and vice versa (i.e., both the mapping and its inverse are information preserving). The information capacities of \(\SC{1}\) and \(\SC{3}\) are identical, and the schemas are said to be \emph{equivalent}, denoted \Equivalent{\SC{1}}{\SC{3}}. (Note that Miller et al.\ \cite{Miller.R-1994a-Schema} use stricter definitions of equivalence and dominance than those used here; strictly speaking, the discussion of equivalence and dominance in this section should properly refer to \emph{absolute} dominance and equivalence, whereas the following discussion on SIG transformations should properly refer to \emph{internal} dominance and equivalence \cite{Miller.R-1994a-Schema}. The distinction, however, is not crucial to the discussion in this paper.) Since relative information capacity is a relative measure, it can only be effectively used in a comparative manner. To this end, Miller et al.\ \cite{Miller.R-1994a-SIG} defined the \emph{schema intension graph} formalism to provide a tool for comparing the information capacity of schemas. A schema intension graph (SIG) describes the associations between domains in a schema, and provides a way to ``reason about constraints on \emph{collections of entities} in an instance of the entity set, rather than about the internal structure of a single entity'' \cite{Miller.R-1994a-SIG}. In particular, SIGs provide a useful abstract formalism within which to perform schema transformations, as transforming the SIG for a schema is effectively the same as transforming the schema itself. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Constructing SIGs} \label{sec-sig-build} \noindent A SIG is a graph \(G = (N, E)\) \cite{Miller.R-1994a-Schema}. The nodes \(n_{i} \in N\) represent typed domains (i.e., sets of values). Nodes may be \emph{simple} (atomic) or \emph{constructed} (composite). Simple nodes are denoted by the name of the domain that they represent. Constructed nodes represent a collection of domains and are built using the \(\times\) (cartesian product) and \(+\) (union) operators. The edges \(e_{i} \in E\) represent binary relations (in the mathematical sense) between domains. Edges are denoted by plain lines with an optional label, and may be composed to form a \emph{path} between two nodes (the notation \(e_{2} \circ e_{1}\) means ``edge \(e_{1}\) followed by edge \(e_{2}\)''). Some edges may be designated as \emph{selection} edges, labelled \(\RelRestrict\), or \emph{projection} edges, labelled \(\RelProject\). These correspond to the relational algebra operators of the same name \cite{Codd.E-1972a-Completeness}. Edges may also be annotated with up to four properties denoting simple constraints on the binary relation represented by the edge: totality, surjectivity, functionality and injectivity. These properties are defined as follows \cite{Miller.R-1994a-Schema}: \begin{quotation} \noindent Let \(A\) and \(B\) be two sets. A binary relation \(r : A - B\) is \emph{total} (denoted \(e : A \Total B\)) if it is defined for all elements of \(A\); \emph{surjective} (\(e : A \Surjective B\)) if it is defined for all elements of \(B\); \emph{functional} (\(e : A \Functional B\)) if an element of \(A\) determines at most one element of \(B\); and \emph{injective} (\(e : A \Injective B\)) if an element of \(B\) determines at most one element of \(A\). \end{quotation} Note that the correct usage of these properties is determined by the direction in which the binary relation is read. That is, the binary relation \(A \Functional B\) is functional if read from left to right, but injective if read from right to left. A \emph{bijection} is a binary relation that has all four properties \cite{Miller.R-1994a-SIG}. Essentially, \(A \Total B\)) means that every element of \(A\) (conversely, \(B\) for \(A \Surjective B\)) must participate in the binary relation, while \(A \Functional B\) means that a given element of \(A\) (conversely, \(B\) for \(A \Injective B\)) may appear in at most one instance of the binary relation. Note that use of the term ``relation'' here, while mathematically correct, clearly has potential to cause confusion in any discussion that also involves the Relational Model. From now on, we shall therefore refer to the binary relations represented by SIG edges as ``associations''. A selection edge \(e : A \LabelledEdge{\sigedge{00}}{\RelRestrict} B\) indicates that the elements of \(B\) comprise a subset (or specialisation) of the elements of \(A\). If \(B = A\), which we term a \emph{trivial} selection, every element of B must appear in A and vice versa. Similarly, every element of \(A\) can be associated with at most one element of \(B\), and vice versa. Trivial selection edges are therefore bijective, i.e., \(A \LabelledEdge{\Bijective}{\RelRestrict} B\). If \(B \subset A\), which we term a \emph{non-trivial} selection, every element of \(B\) must appear in \(A\), but the converse no longer holds. The mapping remains one-to-one, as before. Non-trivial selection edges are therefore surjective (from the perspective of the supertype), functional, and injective, i.e., \(A \SelectionEdge B\). A projection edge \(e : A \LabelledEdge{\sigedge{00}}{\RelProject} B\), indicates that the elements of \(B\) are a projection of \(A\). This implies that \(A\) is a constructed node of the form \(B \times C \times \dotsb\). Trivial projection edges (\(B = A\)) are bijective, i.e., \(A \LabelledEdge{\Bijective}{\RelProject} B\), for the same reasons as trivial selection edges. For non-trivial projections (i.e., \(B \ne A\)), every element of \(A\) must have a corresponding element in \(B\), and vice versa. Similarly, every element of \(A\) can be associated with at most one element of \(B\), but \emph{not} vice versa. Thus, non-trivial projection edges are total, surjective, and functional, i.e., \(A \ProjectionEdge B\).\footnote{Allowing nulls could potentially alter which of these properties apply, but all non-trivial projection edges would still share the same properties regardless.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Comparing SIGs} \label{sec-sig-compare} \noindent It is possible to determine the relationship between the relative information capacities of two schemas by comparing SIGs for the two schemas. Consider two schemas \(\SC{1}\) and \(\SC{2}\), with corresponding SIGs \(G_{\SC{1}}\) and \(G_{\SC{2}}\). Intuitively, if \(G_{\SC{1}}\) and \(G_{\SC{2}}\) are isomorphic, then \(\SC{1}\) and \(\SC{2}\) are equivalent \cite{Miller.R-1994a-SIG}. An isomorphism occurs between two SIGs when the graphs match both structurally and semantically. That is, \(G_{\SC{1}}\) and \(G_{\SC{2}}\) must be more than just structurally identical---the semantics of the SIG nodes and edges must also be compatible. Miller et al.\ \cite{Miller.R-1994a-SIG} identify three possible results from comparing two SIGs \(G_{\SC{1}}\) and \(G_{\SC{2}}\): \begin{enumerate} \item \(G_{\SC{1}}\) and \(G_{\SC{2}}\) are either immediately isomorphic, or one can be manipulated by a series of equivalence preserving transformations (described below) until it is isomorphic to the other. The two schemas are equivalent (i.e., \(\SC{1} \equiv \SC{2}\)). \item \(G_{\SC{1}}\) and \(G_{\SC{2}}\) are not immediately isomorphic, but one can be manipulated by a series of information capacity augmenting transformations (described below) until it is isomorphic to the other. The transformed schema is dominated by the other (i.e., either \(\Dominates{\SC{2}}{\SC{1}}\) or \(\Dominates{\SC{1}}{\SC{2}}\)). \item \(G_{\SC{1}}\) and \(G_{\SC{2}}\) are not immediately isomorphic and cannot be transformed to be isomorphic. The information capacities of \(\SC{1}\) and \(\SC{2}\) cannot be compared. \end{enumerate} Batini et al.\ \cite{Batini.C-1992b-Conceptual} follow a similar classification scheme with respect to schema transformations: they refer to the first as \emph{information-preserving} transformations, the second as \emph{augmenting} transformations and the third as \emph{noncomparable} transformations. Miller et al.\ \cite{Miller.R-1994a-Schema} describe the following equivalence preserving and information capacity augmenting transformations on SIGs, which are summarised in Table~\ref{Tab.Viewpoints.SIGTransformations}: \begin{description} \item[Annotation transformations] allow the removal of annotations, or projection and selection constraints from edges (e.g., \(n_{1} \sigedge{23} n_{2} \Rightarrow n_{1} \sigedge{02} n_{2}\)). If we can transform \(\SC{1}\) into \(\SC{2}\) by such a transformation, then the information capacity of \(\SC{2}\) dominates that of \(\SC{1}\) (i.e., \Dominates{\SC{2}}{\SC{1}}). \item[Composition transformations] allow edges to be moved and copied around the graph. Consider an edge \(e\) between nodes \(n_{1}\) and \(n_{2}\), as shown in Figure~\ref{fig-sig-composition-original}. If there is a path \(P = e_{m} \circ \dotsb \circ e_{k}\) of functional edges from some other node \(n_{k}\) to \(n_{2}\) (i.e., \(n_{k} \Functional \dotsb \Functional n_{2}\)), \(e\) may be copied to a new edge \(e'\) between nodes \(n_{1}\) and \(n_{k}\) as shown in Figure~\ref{fig-sig-composition-copy}. The original \(e\) can then be deleted if desired, effectively moving \(e\) to \(e'\) as shown in Figure~\ref{fig-sig-composition-move}. The annotation of \(e'\) is determined by the composition of the edges of \(P\!\). If an edge with a particular annotation (such as surjectivity) is composed with an edge without that annotation, the resulting edge does not have the annotation. For example, the composition of the edges \(\Functional\) and \(\Surjective\) would be an edge with no annotations (\(\Edge\)), whereas the composition of \(\Functional\) and \(\sigedge{03}\) would be \(\Functional\). If we can transform \(\SC{1}\) into \(\SC{2}\) by such a transformation, then \Dominates{\SC{2}}{\SC{1}}. %%%%%%%%%%%%%%%%%%%% \begin{table}[tb] \centering \small \begin{tabular}{lcc} \toprule & & \textbf{Effect on infor-} \\ \textbf{Transformation} & \textbf{Type} & \textbf{mation capacity} \\ \midrule Delete annotation or \(\RelProject\)/\(\RelRestrict\) constraint & annotation & augmented \\ Move/copy edge & composition & augmented \\ Create/delete duplicate node & selection & no change \\ Create trivial selection edge & selection & no change \\ Delete trivial selection edge & selection & augmented \\ \bottomrule \end{tabular} \caption{Summary of SIG transformations.} \label{Tab.Viewpoints.SIGTransformations} \end{table} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] \centering \subfigure[Original state\label{fig-sig-composition-original}]{ \begin{tikzpicture} \node (n1) {\(n_{1}\)}; \node (n2) [below=1cm of n1] {\(n_{2}\)}; \node (n3) [right=1cm of n2] {\(n_{3}\)}; \node (ee) [right=0.75cm of n3] {\ldots}; \node (nk) [right=0.75cm of ee] {\(n_{k}\)}; \graph{ (n1) -- [edge label={\(e\)},swap] (n2); (nk) -> [edge label={\(e_{k}\)}] (ee) -> [edge label={\(e_{m+1}\)}] (n3) -> [edge label={\(e_{m}\)}] (n2); }; \node (Pleft) [above=2mm of n2.north east] {}; \node (Pright) [above=2mm of nk.north west] {}; \node (P) at ($(Pleft)!0.5!(Pright)$) {\(P\)}; \draw[decorate,decoration=brace] (n2.north east) -- (nk.north west); \end{tikzpicture} } \subfigure[Copy edge \(e\) to \(e'\)\label{fig-sig-composition-copy}]{ \begin{tikzpicture} \node (n1) {\(n_{1}\)}; \node (n2) [below=1cm of n1] {\(n_{2}\)}; \node (n3) [right=1cm of n2] {\(n_{3}\)}; \node (ee) [right=0.75cm of n3] {\ldots}; \node (nk) [right=0.75cm of ee] {\(n_{k}\)}; \graph{ (n1) -- [edge label={\(e\)},swap] (n2); (n1) -- [edge label={\(e'\)}] (nk); (nk) -> [edge label={\(e_{k}\)}] (ee) -> [edge label={\(e_{m+1}\)}] (n3) -> [edge label={\(e_{m}\)}] (n2); }; \end{tikzpicture} } \subfigure[Move edge \(e\) to \(e'\)\label{fig-sig-composition-move}]{ \begin{tikzpicture} \node (n1) {\(n_{1}\)}; \node (n2) [below=1cm of n1] {\(n_{2}\)}; \node (n3) [right=1cm of n2] {\(n_{3}\)}; \node (ee) [right=0.75cm of n3] {\ldots}; \node (nk) [right=0.75cm of ee] {\(n_{k}\)}; \graph{ (n1) -- [edge label={\(e'\)}] (nk); (nk) -> [edge label={\(e_{k}\)}] (ee) -> [edge label={\(e_{m+1}\)}] (n3) -> [edge label={\(e_{m}\)}] (n2); }; \end{tikzpicture} } \caption{Composition transformations.} \label{fig-sig-composition} \end{figure} %%%%%%%%%%%%%%%%%%%% \item[Selection transformations] allow the creation and deletion of new nodes and edges as follows: \begin{enumerate} \item An existing node \(n\) may be duplicated to create a new node \(n'\!\). A trivial selection edge \(e:n \LabelledEdge{\Bijective}{\RelRestrict} n'\) is added to enforce the restriction that \(n'\) is a duplicate of \(n\). \item A node \(n_{1}\) may be deleted if it is attached to exactly one other node \(n_{2}\) by a single trivial selection edge \(e:n_{1} \LabelledEdge{\Bijective}{\RelRestrict} n_{2}\) (implying \(n_{1}\) is a duplicate of \(n_{2}\)). This also implies deleting the attached trivial selection edge. \item A trivial selection edge \(e:n \LabelledEdge{\Bijective}{\RelRestrict} n\) may be created that connects a node \(n\) to itself. \item A trivial selection edge \(e:n_{1} \LabelledEdge{\Bijective}{\RelRestrict} n_{2}\) may be deleted, removing the association between \(n_{1}\) and \(n_{2}\). \end{enumerate} If we can transform \(\SC{1}\) into \(\SC{2}\) using any of the first three selection transformations, then \Equivalent{\SC{2}}{\SC{1}}. If we can transform \(\SC{1}\) into \(\SC{2}\) using the fourth selection transformation, then \Dominates{\SC{2}}{\SC{1}}. The latter seems somewhat counter-intuitive \cite{Qian.X-1995a-Correct}, but is nonetheless correct \cite{Miller.R-1994a-Schema}. This is because a trivial selection edge \(e:n_{1} \LabelledEdge{\Bijective}{\RelRestrict} n_{2}\) denotes that \(n_{1}\) and \(n_{2}\) correspond to the same domain, so removing the edge also removes this constraint. \end{description} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Relational schemas as SIGs} \label{sec-relational-sigs} \noindent The direct correspondence between relational types and SIG domains (as noted in Section~\ref{sec-relations-types}) enables us to use SIGs to represent the structure and constraints of relational schemas. Equating tuple types to the cartesian product of their attributes (equation \ref{eqn-tuple-type}) means that these can also be mapped directly to a constructed SIG node. Let us again consider the relvar \(S\{\Sno, \Sname, \Status, \City\}\) from Date's suppliers-and-parts schema. In addition to type constraints on the attributes, tuples, and the relvar as a whole, \(S\!\) is subject to a (primary) key constraint on \Sno. Taken together with the collection of types \(\T{\Sno}\), \(\T{\Sname}\), \(\T{\Status}\), \(\T{\City}\), \(\TT{S}\), and \(\T{S}\), we can construct the corresponding SIG shown in Figure~\ref{fig-sig-s-alone} as explained below (assuming that \(S\!\) is the only relvar in the schema). As noted in Section~\ref{sec-relations-types}, any given value of \(S\!\) must comprise a subset of \(\TT{S}\), so the association between \(\T{S}\) and \(\TT{S}\) is total. Similarly, every tuple in \(\TT{S}\) must appear in at least one value of \(S\!\), so the association is also surjective. The association is neither functional nor injective, however, as a value of \(S\!\) in general may contain multiple tuples, and each tuple in \(\TT{S}\) appears in multiple values of \(S\!\). This gives us \(\T{S}\SurTotal\TT{S}\). The key constraint on \Sno\ implies that the functional dependency \(\{\Sno\} \rightarrow \{\Sname, \Status, \City\}\) holds in \(S\!\). From equation \ref{eqn-tuple-type}, we can infer the type of \(\{\Sname, \Status, \City\}\) as \(\TSSC\). The association is functional and not injective by definition. As not every supplier number may appear in any given value of \(S\!\), there may be elements of \(\T{\Sno}\) that are not associated with an element of \(\TSSC\), and vice versa. This implies that the association is neither total nor surjective. This gives us \(\T{\Sno} \KeyEdge \TSSC\) (we introduce the label \(\mathit{key}\) to indicate that this edge represents a key constraint). Both \(\T{\Sno}\) and \(\TSSC\) are projections of \(\TT{S}\), so we can add (non-trivial) projection edges \(\TT{S} \ProjectionEdge \T{\Sname} \times \T{\Status} \times T_{\City}\) and \(\TT{S} \ProjectionEdge \T{\Sno}\). Similarly, \(\T{\Sname}\), \(\T{\Status}\), and \(\T{\City}\) are each projections of \(\TSSC\), so we can add further projection edges to these nodes, as shown in Figure~\ref{fig-sig-s-alone}. %%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] \centering \begin{tikzpicture}[level distance=12mm] \node (TS) {\(\T{S}\)}; \node (TTS) [right=1cm of TS] {\(\TT{S}\)} [sibling distance=3cm] child[arrows={:{crossbar}-{crossbar}>}] {node (TSno) {\(\T{\Sno}\)}} child[arrows={:{crossbar}-{crossbar}>}] {node (TSSC) {\(\TSSC\)} [sibling distance=15mm] child[arrows={:{crossbar}-{crossbar}>}] {node (TSname) {\(\T{\Sname}\)}} child[arrows={:{crossbar}-{crossbar}>}] {node (TStatus) {\(\T{\Status}\)}} child[arrows={:{crossbar}-{crossbar}>}] {node (TCity) {\(\T{\City}\)}}}; % TS to T{S} \draw[arrows={:{crossbar}-{crossbar}:}] (TS) to (TTS); % FD \draw[arrows={->}] (TSno) to node[below=-2pt] {\scriptsize\emph{key}} (TSSC); % projection edges \ProjectionAnnotation[above left=-2pt and -4pt]{TTS}{TSno} \ProjectionAnnotation[above right=-2pt and -4pt]{TTS}{TSSC} \ProjectionAnnotation[above left=-2pt and -4pt]{TSSC}{TSname} \ProjectionAnnotation[right=-2pt]{TSSC}{TStatus} \ProjectionAnnotation[above right=-2pt and -4pt]{TSSC}{TCity} \end{tikzpicture} \caption{SIG for relvar \(\bm{S\{\Sno, \Sname, \Status, \City\}}\).} \label{fig-sig-s-alone} \end{figure} %%%%%%%%%%%%%%%%%%%% Looking at Figure~\ref{fig-sig-s-alone}, it could be argued that the \(T_{\dots}\) type notation is redundant and unnecessarily complicates the diagram. Why not just use, e.g., \(S\!\) instead of \(\T{S}\), \(\Sno\) instead of \(\T{\Sno}\), and \(\Sno \times \Sname \times \Status \times \City\) instead of \(\TT{S}\)? It is important to remember, however, that SIGs like that shown in Figure~\ref{fig-sig-s-alone} include multiple elements of quite different kinds, including relations, tuples, and various combinations of attributes. Discarding the type notation might be reasonable in informal situations, but could lead to confusion between the different kinds of elements represented. The explicit type notation protects the semantics of the various elements of the schema and makes it clear what is being considered. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Characterising view updates} \label{sec-characterising} \noindent In this section we show how SIGs can be used to characterise the information equivalence of view updates, using Date's motivating example of restriction views \cite{Date.C-2013a-View}. Views and the base relvars they are derived from normally exist within the same ``base'' database schema (although this is not strictly required), which we shall refer to as \(\SC{0}\). However, an end user may be granted access only to the views, and thus not be aware of the existence of the base relvar(s), or vice versa. This effectively partitions the schema into sub-schemas \(\SC{1}, \SC{2}, \dotsc, \SC{n}\). There are explicit and implicit constraints in \(\SC{0}\) that should also apply to the sub-schemas. Ignoring these constraints in a sub-schema \(\SC{k}\) will affect its information capacity, and could negatively impact the information equivalence of view updates involving \(\SC{k}\). In the next section we will show how such constraints can be propagated into sub-schemas by rewriting them in terms of the sub-schema alone, which enables us to explicitly incorporate into the sub-schema the type information implicit in \(\SC{0}\) that would otherwise be omitted. We will then discuss the construction of SIGs for Date's restriction view example (Section~\ref{sec-date-sigs}), and show how to transform and compare these (Section~\ref{sec-transforming}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Constraint derivation and propagation} \label{sec-constraints} \noindent Date's motivating example in Chapter 4 of \cite{Date.C-2013a-View} explores what is arguably the simplest case of view updating: disjoint restriction views. The example includes a base relvar \(S\!\) and two views \(\LS\) and \(\NLS\) derived\footnote{Although as Date notes \cite{Date.C-2013a-View}, we could just as easily define \(\LS\) and \(\NLS\) as base relvars and then derive \(S = \LS \RelUnion \NLS\), or even define all three as base relvars. The constraints will remain the same regardless.} from \(S\!\): \begin{eqnarray} \LS & = & \RelRestrict_{\City = \mathit{'London'}}S \label{eqn-ls} \\ \NLS & = & \RelRestrict_{\City \ne \mathit{'London'}}S \label{eqn-nls} \end{eqnarray} Informally, \LS\ contains details of London suppliers, while \NLS\ contains details of non-London suppliers. \(\LS\) and \(\NLS\) represent disjoint restrictions of \(S\!\) with the same heading. They have the following tuple types: \begin{eqnarray} \TT{\LS} & = & \T{\Sno} \times \T{\Sname} \times \T{\LS} \times \CityLondon \nonumber \\ \TT{\NLS} & = & \T{\Sno} \times \T{\Sname} \times \T{\Status} \times (\TCityMinusLondon) \nonumber \end{eqnarray} The base schema for this example is \(\SC{0} = \{S, \LS, \NLS\}\). This is subject to type constraints as specified by the various types noted previously, and the following additional constraints: \begin{list}{\textbf{C\(\bm{_{\arabic{constraint}}}\):}}{\usecounter{constraint}} \item\label{constraint-key} \Sno\ is a key for each of \(S\!\), \LS, and \NLS, therefore the functional dependency \(\{\Sno\} \rightarrow \{\Sname, \Status, \City\}\) also holds in each of (a) \(S\!\), (b) \(\LS\), and (c) \(\NLS\). % not sure how relevant the FK constraints are? % how to represent? FK is a subset of the original key, so selection edge between S.Sno and each of LS.Sno and NLS.Sno % or perhaps a selection edge from Sno to itself? (labelled FK) % Notation: \T{\Sno}, \T{\LS.\Sno}, \T{\NLS.\Sno}? \item\label{constraint-lsfk} \(\RelProject_{\{\Sno\}}\LS \subseteq \RelProject_{\{\Sno\}}S\) [foreign key from \(\LS\) to \(S\!\)] \item\label{constraint-nlsfk} \(\RelProject_{\{\Sno\}}\NLS \subseteq \RelProject_{\{\Sno\}}S\) [foreign key from \(\NLS\) to \(S\!\)] \item\label{constraint-union} \(S = \LS \RelUnion \NLS\) [from (\ref{eqn-ls}) and (\ref{eqn-nls}) above] \item\label{constraint-disjoint} \(\LS \RelIntersect \NLS = \emptyset\) [\LS\ and \NLS\ are disjoint] \item\label{constraint-notlondon} \(\RelRestrict_{\City \ne \mathit{'London'}}\LS = \emptyset\) [from (\ref{eqn-ls}) above] \item\label{constraint-london} \(\RelRestrict_{\City = \mathit{'London'}}\NLS = \emptyset\) [from (\ref{eqn-nls}) above] \item\label{constraint-relation-type-union} (a) \(\T{S} = \T{\LS} \RelUnion \T{\NLS}\); (b) \(\TT{S} = \TT{\LS} \RelUnion \TT{\NLS}\) [from C\(_{\ref{constraint-union}}\)] \item\label{constraint-relation-types} (a) \(\T{\LS} \subset \T{S}\); (b) \(\TT{\LS} \subset \TT{S}\) [from (\ref{eqn-ls}) above] \item\label{constraint-tuple-types} (a) \(\T{\NLS} \subset \T{S}\); (b) \(\TT{\NLS} \subset \TT{S}\) [from (\ref{eqn-nls}) above] \item\label{constraint-implied-types-london} \(\CityLondon \subset \T{\City}\) \item\label{constraint-implied-types-nonlondon} \((\TCityMinusLondon) \subset \T{\City}\) \item\label{constraint-implied-types-TSSL} \(\TSSL \subset \TSSC\) \item\label{constraint-implied-types-TSSNL} \(\TSSNL \subset \TSSC\) \end{list} Constraints C\(_{\ref{constraint-union}}\)--C\(_{\ref{constraint-london}}\) enforce that \(\LS\) and \(\NLS\) are disjoint restrictions of \(S\!\), while C\(_{\ref{constraint-relation-type-union}}\)--C\(_{\ref{constraint-implied-types-TSSNL}}\) are additional type constraints that specify specialisation associations implied by the subtypes and supertypes occurring in \(\SC{0}\). In most typical database schemas, only constraints C\(_{\ref{constraint-key}}\), C\(_{\ref{constraint-lsfk}}\), and C\(_{\ref{constraint-nlsfk}}\) will be explicitly specified. Some database schemas may also explicitly specify constraints C\(_{\ref{constraint-union}}\) and C\(_{\ref{constraint-disjoint}}\), but it is more likely that these and all the remaining constraints will be implicit in the definition of \(\SC{0}\). It is important for the purposes of this analysis to clearly identify all such implicit constraints, so that nothing is missed when constructing corresponding SIGs. Let us partition \(\SC{0}\) into three sub-schemas: \(\SC{1} = \{S\}\), \(\SC{2} = \{\LS, \NLS\}\), and \(\SC{3} = \{LS\}\). We wish to compare the relative information capacities of the schema pairs \((\SC{1}, \SC{2})\) and \((\SC{1}, \SC{3})\), in order to determine whether the schemas in each pair are information equivalent. This will in turn characterise the view update compatibility of the two schemas. Note that in the following analysis, we omit the foreign key constraints C\(_{\ref{constraint-lsfk}}\) and C\(_{\ref{constraint-nlsfk}}\), as we have yet to determine the best way to represent inclusion dependencies of this nature in a SIG, or indeed whether such dependencies are significant. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Schema \(\SC{1}\)} \label{sec-constraints-s-i} \noindent A user granted access only to \(\SC{1}\) sees only relvar \(S\!\). Constraint C\(_{\ref{constraint-key}}\)(a) can be propagated directly into \(\SC{1}\) because it is expressed only in terms of \(S\!\). Constraints C\(_{\ref{constraint-implied-types-london}}\)--C\(_{\ref{constraint-implied-types-TSSNL}}\) do not mention any relvars and can thus also be directly propagated into \(\SC{1}\). Constraints C\(_{\ref{constraint-union}}\)--C\(_{\ref{constraint-tuple-types}}\) can be propagated to \(\SC{1}\) by rewriting them only in terms of \(S\!\). We can do this by substituting the definitions of \(\LS\) (equation~\ref{eqn-ls}) and \(\NLS\) (equation~\ref{eqn-nls}) into the constraints. For brevity, we shall denote these substitutions as \(\LSsub\) and \(\NLSsub\), so as to distinguish them from the actual \(\LS\) and \(\NLS\) relvars in \(\SC{0}\), \(\SC{2}\), and \(\SC{3}\). The superscript ``1'' in the following denotes that these constraints have been rewritten in \(\SC{1}\): \begin{list}{\textbf{C\(\bm{_{\arabic{constraint}}^{1}}\):}}{\usecounter{constraint}\setcounter{constraint}{3}} \item \(S = \LSsub \RelUnion \NLSsub\) (or even just \(S = S\)) \item \(\LSsub \RelIntersect \NLSsub = \emptyset\) \item \(\RelRestrict_{\City \ne \mathit{'London'}}\LSsub = \emptyset\) \item \(\RelRestrict_{\City = \mathit{'London'}}\NLSsub = \emptyset\) \item (a) \(\T{S} = \T{\LSsub} \RelUnion \T{\NLSsub}\); (b) \(\TT{S} = \TT{\LSsub} \RelUnion \TT{\NLSsub}\) (or even just \(\T{S} = \T{S}\) and \(\TT{S} = \TT{S}\)) \item (a) \(\T{\LSsub} \subset \T{S}\); (b) \(\TT{\LSsub} \subset \TT{S}\) [from (\ref{eqn-ls}) above] \item (a) \(\T{\NLSsub} \subset \T{S}\); (b) \(\TT{\NLSsub} \subset \TT{S}\) [from (\ref{eqn-nls}) above] \end{list} We can also rewrite C\(_{\ref{constraint-key}}\)(b) and C\(_{\ref{constraint-key}}\)(c) to C\(_{\ref{constraint-key}}^{1}\)(b) and C\(_{\ref{constraint-key}}^{1}\)(c), so that the functional dependency applies to \(\LSsub\) and \(\NLSsub\) instead of \(\LS\) and \(\NLS\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Schema \(\SC{2}\)} \label{sec-constraints-s-ii} \noindent A user granted access only to \(\SC{2}\) sees only relvars \(\LS\) and \(\NLS\). Constraints C\(_{\ref{constraint-key}}\)(b), C\(_{\ref{constraint-key}}\)(c), and C\(_{\ref{constraint-disjoint}}\)--C\(_{\ref{constraint-london}}\) can be directly propagated into \(\SC{2}\) as they are defined only in terms of \(\LS\) and \(\NLS\). C\(_{\ref{constraint-implied-types-london}}\)--C\(_{\ref{constraint-implied-types-TSSNL}}\) can be directly propagated into \(\SC{2}\) as they do not mention any relvars. C\(_{\ref{constraint-union}}\), C\(_{\ref{constraint-relation-type-union}}\), C\(_{\ref{constraint-relation-types}}\), and C\(_{\ref{constraint-tuple-types}}\) can be rewritten only in terms of \(\LS\) and \(\NLS\): \begin{list}{\textbf{C\(\bm{_{\arabic{constraint}}^{2}}\):}}{\usecounter{constraint}\setcounter{constraint}{3}} \item \(\LS \RelUnion \NLS = \LS \RelUnion \NLS\)\setcounter{constraint}{7} \item (a) \(\T{\LS} \RelUnion \T{\NLS} = \T{\LS} \RelUnion \T{\NLS}\); (b) \(\TT{\LS} \RelUnion \TT{\NLS} = \TT{\LS} \RelUnion \TT{\NLS}\) \item (a) \(\T{\LS} \subset (\T{\LS} \RelUnion \T{\NLS})\); (b) \(\TT{\LS} \subset (\TT{\LS} \RelUnion \TT{\NLS})\) \item (a) \(\T{\NLS} \subset \T{S}\); (b) \(\TT{\NLS} \subset \TT{S}\) \end{list} We can also rewrite C\(_{\ref{constraint-key}}\)(a) to C\(_{\ref{constraint-key}}^{2}\)(a), so that the functional dependency applies to \(\LS \RelUnion \NLS\) instead of \(S\!\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Schema \(\SC{3}\)} \label{sec-constraints-s-iii} \noindent A user granted access only to \(\SC{3}\) sees only relvar \(\LS\). Constraints C\(_{\ref{constraint-key}}\)(b) and C\(_{\ref{constraint-notlondon}}\) can be directly propagated into \(\SC{3}\) as they are defined only in terms of \(\LS\). C\(_{\ref{constraint-implied-types-london}}\)--C\(_{\ref{constraint-implied-types-TSSNL}}\) can also be directly propagated as previously discussed. The remaining constraints cannot be rewritten only in terms of \(\LS\), and therefore cannot be propagated to \(\SC{3}\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \subsection{Constructing the SIGs} \label{sec-date-sigs} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Schema \(\SC{1}\)} \label{sec-sigs-s-i} \noindent The SIG for \(\SC{1}\) will not just be that for relvar \(S\!\) (see Figure~\ref{fig-sig-s-alone}), as that schema did not incorporate the constraints implicit in \(\SC{0}\). The process for constructing the SIG will be similar to that for the earlier example, however. The constraints appearing in \(\SC{1}\) are C\(_{\ref{constraint-key}}^{1}\), C\(_{\ref{constraint-union}}^{1}\)--C\(_{\ref{constraint-tuple-types}}^{1}\), and C\(_{\ref{constraint-implied-types-london}}\)--C\(_{\ref{constraint-implied-types-TSSNL}}\). The SIG will include all of the edges that appear in Figure~\ref{fig-sig-s-alone}. \(\T{S} \SurTotal \TT{S}\) implies \(\T{\LSsub} \SurTotal \TT{\LSsub}\) and \(\T{\NLSsub} \SurTotal \TT{\NLSsub}\). Constraints C\(_{\ref{constraint-implied-types-london}}\)--C\(_{\ref{constraint-implied-types-TSSNL}}\) imply the existence of several subtype nodes, plus corresponding selection and projection edges: \begin{itemize} \item \(\TT{\LSsub} \ProjectionEdge \T{\Sno}\) and \(\TT{\NLSsub} \ProjectionEdge \T{\Sno}\) \item \(\TT{\LSsub} \ProjectionEdge \TSSL\) \item \(\TT{\NLSsub} \ProjectionEdge \TSSNL\) \item \(\TSSL \ProjectionEdge \T{\Sname}\) \item \(\TSSL \ProjectionEdge \T{\Status}\) \item \(\TSSL \ProjectionEdge\,\CityLondon\) \item \(\TSSNL \ProjectionEdge \T{\Sname}\) \item \(\TSSNL \ProjectionEdge \T{\Status}\) \item \(\TSSNL \ProjectionEdge \T{\City} -\) \newline \(\CityLondon\) \end{itemize} C\(_{\ref{constraint-key}}^{1}\)(b) and C\(_{\ref{constraint-key}}^{1}\)(c) imply two additional key edges: \begin{itemize} \item \(\T{\Sno} \KeyEdge \TSSL\) \item \(\T{\Sno} \KeyEdge \TSSNL\) \end{itemize} C\(_{\ref{constraint-relation-type-union}}^{1}\)(a) implies node \(\TLSPlusNLSsub\) to represent \(\T{\LSsub} \RelUnion \T{\NLSsub}\). However, this node would be effectively identical to \(\T{S}\!\), so we can ignore C\(_{\ref{constraint-relation-type-union}}^{1}\)(a) and just continue to use \(\T{S}\!\). A similar argument applies for C\(_{\ref{constraint-relation-type-union}}^{1}\)(b) and \(\TT{S}\). C\(_{\ref{constraint-relation-types}}^{1}\)--C\(_{\ref{constraint-implied-types-TSSNL}}^{1}\) imply the following selection edges: \begin{itemize} \item \(\T{S} \SelectionEdge \T{\LSsub}\) and \(\TT{S} \SelectionEdge \TT{\LSsub}\) \item \(\T{S} \SelectionEdge \T{\NLSsub}\) and \(\TT{S} \SelectionEdge \TT{\NLSsub}\) \item \(\T{\City} \SelectionEdge\,\CityLondon\) \item \(\T{\City} \SelectionEdge\,(\TCityMinusLondon)\) \item \(\TSSC \SelectionEdge\) \(\TSSL\) \item \(\TSSC \SelectionEdge \T{\Sname} \times \T{\Status} \times \T{\City} -\) \newline \(\CityLondon\) \end{itemize} The complete SIG for \(\SC{1}\) is shown in Figure~\ref{fig-sig-s}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node (TLS) {\(\T{\LSsub}\)}; &[7mm] \node (TTLS) {\(\TT{\LSsub}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\T{\Sname}\)}; \\ \node (TLSPlusNLS) {\(\T{S}\)}; & \node (TTLSPlusNLS) {\(\TT{S}\)}; & \node (TSno) {\(\T{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\T{City}\)}; \\ & & & & & \node (TStatus) {\(\T{\Status}\)}; \\ \node (TNLS) {\(\T{\NLSsub}\)}; & \node (TTNLS) {\(\TT{\NLSsub}\)}; & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) }, }; % FDs (TSno) -- [funcdep left,bend left] (TSSL); (TSno) -- [funcdep right] (TSSNL); % projection edges { [edges=projection left] (TTLSPlusNLS) -- (TSno), (TTLS) -- { (TSno), (TSSL) }, (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TTNLS) -- { (TSno), (TSSNL) }, (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (TStatus), (NLSCity), (TSname) }, }; % selection edges { [edges=selection left] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) }, (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, (TCity) -- (LSCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right] (TSSC), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,bend right] (TSSC), }; \end{tikzpicture} \caption{SIG for schema \(\bm{\SC{1} = \{S\}}\).} \label{fig-sig-s} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Schema \(\SC{2}\)} \label{sec-sigs-s-ii} \noindent The constraints appearing in \(\SC{2}\) are C\(_{\ref{constraint-key}}^{2}\), C\(_{\ref{constraint-union}}^{2}\), C\(_{\ref{constraint-disjoint}}\)--C\(_{\ref{constraint-london}}\), C\(_{\ref{constraint-relation-type-union}}^{2}\)--C\(_{\ref{constraint-tuple-types}}^{2}\), and C\(_{\ref{constraint-implied-types-london}}\)--C\(_{\ref{constraint-implied-types-TSSNL}}\). This is the same set of constraints, albeit with different rewritings, as \(\SC{1}\). This means that the SIG for \(\SC{2}\) can be built in essentially the same manner as for \(\SC{1}\), with some nodes differently named. In particular, C\(_{\ref{constraint-relation-type-union}}\) implies that there will be nodes \(\TLSPlusNLS\) and \(\TTLSPlusNLS\) instead of \(\T{S}\) and \(\TT{S}\). There will also be \(\T{\LS}\) instead of \(\T{\LSsub}\), etc. The complete SIG for \(\SC{2}\) is shown in Figure~\ref{fig-sig-ls-nls}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node (TLS) {\(\T{\LS}\)}; &[7mm] \node (TTLS) {\(\TT{\LS}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\T{\Sname}\)}; \\ \node (TLSPlusNLS) {\(\StackTLSPlusNLS\)}; & \node (TTLSPlusNLS) {\(\StackTTLSPlusNLS\)}; & \node (TSno) {\(\T{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\T{City}\)}; \\ & & & & & \node (TStatus) {\(\T{\Status}\)}; \\ \node (TNLS) {\(\T{\NLS}\)}; & \node (TTNLS) {\(\TT{\NLS}\)}; & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) }, }; % FDs (TSno) -- [funcdep left,bend left] (TSSL); (TSno) -- [funcdep right] (TSSNL); % projection edges { [edges=projection left] (TTLSPlusNLS) -- (TSno), (TTLS) -- { (TSno), (TSSL) }, (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TTNLS) -- { (TSno), (TSSNL) }, (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (TStatus), (NLSCity), (TSname) }, }; % selection edges { [edges=selection left] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) }, (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, (TCity) -- (LSCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right] (TSSC), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,bend right] (TSSC), }; \end{tikzpicture} \caption{SIG for schema \(\bm{\SC{2} = \{\LS, \NLS\}}\).} \label{fig-sig-ls-nls} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Schema \(\SC{3}\)} \label{sec-sigs-s-iii} \noindent The constraints appearing in \(\SC{3}\) are C\(_{\ref{constraint-key}}\)(b), C\(_{\ref{constraint-notlondon}}\), and C\(_{\ref{constraint-implied-types-london}}\)--C\(_{\ref{constraint-implied-types-TSSNL}}\). The SIG will only include edges and nodes that relate either directly to \(\LS\) or are independent of any relvar, such as the subtype node \(\TSSNL\). The complete SIG for \(\SC{3}\) is shown in Figure~\ref{fig-sig-ls}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node (TLS) {\(\T{\LS}\)}; &[7.5mm] \node (TTLS) {\(\TT{\LS}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\T{\Sname}\)}; \\ & & \node (TSno) {\(\T{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\T{City}\)}; \\ & & & & & \node (TStatus) {\(\T{\Status}\)}; \\ & & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TLS) -- (TTLS), }; % projection edges { [edges=projection left] (TTLS) -- { (TSno), (TSSL) }, (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (TStatus), (NLSCity), (TSname) }, }; % selection edges { [edges=selection left] (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] (TCity) -- (LSCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); }; \end{tikzpicture} \caption{SIG for schema \(\bm{\SC{3} = \{\LS\}}\).} \label{fig-sig-ls} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Transforming the SIGs} \label{sec-transforming} \noindent Now that we have constructed SIGs for all three schemas, we can compare the relative information capacities of the schema pairs \((\SC{1}, \SC{2})\) and \((\SC{1}, \SC{3})\). We do this by attempting to transform the SIG for one of the schemas until it is isomorphic with the other. Looking at the SIGs for \(\SC{1}\) and \(\SC{2}\), we can see immediately that both SIGs have the same node and edge structure; indeed the only difference is in the names of some of the nodes. However, we already know from the discussion in Sections~\ref{sec-constraints-s-i} and~\ref{sec-constraints-s-ii} that these differently-named nodes represent the same domains. We therefore have an immediate isomorphism between the two SIGs and can conclude that the information capacities of \(\SC{1}\) and \(\SC{2}\) are identical, i.e., \(\Equivalent{\SC{1}}{\SC{2}}\). The comparison between \(\SC{1}\) and \(\SC{3}\) is more interesting. There are some shared structures, but the two SIGs are clearly different. Since the SIG transformations discussed in Section~\ref{sec-sig-compare} only ever preserve or enhance information capacity, intuitively we should attempt to transform the ``smaller'' SIG for \(\SC{3}\) into the ``larger'' SIG for \(\SC{1}\). SIG transformations can generally be carried out in two broad phases (using the current transformation as an example): \begin{description} \item[Duplicate nodes] in \(\SC{3}\) (producing \(\SC{3}'\)) that correspond to supertypes in \(\SC{1}\), to produce nodes corresponding to the extra nodes that \(\SC{1}\) has over \(\SC{3}\). The goal is to produce a node structure similar to that of \(\SC{1}\). The duplicated nodes will be connected to their original nodes by trivial selection edges. These transformations (if applicable) will not alter the information capacity of \(\SC{3}'\) (i.e., \(\equiv\)). \item[Copy and move edges] across the bijective paths connecting the new duplicated nodes in \(\SC{3}'\). The goal is to produce an edge structure similar to that of \(\SC{1}\). The paths that the edges are copied or moved across are bijective, so the annotations of the affected edges will remain unchanged. These transformations (if applicable) will \emph{increase} the information capacity of \(\SC{3}'\) (i.e., \(\preceq\)). \end{description} These two phases generally occur in order, but may be interspersed throughout with the following: \begin{description} \item[Delete duplicate nodes] in \(\SC{3}'\) that are no longer needed, especially those that do not exist in \(\SC{1}\). These transformations (if applicable) will not alter the information capacity of \(\SC{3}'\) (\(\equiv\)). \item[Remove totality annotations] from trivial selection edges in \(\SC{3}'\) to change them into non-trivial selection edges. This decouples subtypes and supertypes to ensure that they represent different domains. These transformations (if applicable) will \emph{increase} the information capacity of \(\SC{3}'\) (\(\preceq\)). \end{description} For example, we might duplicate some nodes, remove some totality annotations, move or copy some edges, delete some no longer needed nodes, move or copy some more edges, then remove some more totality annotations. Following this process, we first duplicate node \(\T{\LS}\) to \(\T{S}\) and node \(\TT{\LS}\) to \(\TT{S}\). To obtain a node structure similar to that of \(\SC{1}\), we also need to further duplicate \(\T{S}\) to \(\T{\NLS}\) and \(\TT{S}\) to \(\TT{\NLS}\). The result of these duplications is shown in Figure~\ref{fig-transform-all-duplicates}. The transformed SIG now has essentially the same node structure as that for \(\SC{1}\), albeit with different node semantics. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node [input keep] (TLS) {\(\T{\LS}\)}; &[7mm] \node [input keep] (TTLS) {\(\TT{\LS}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\T{\Sname}\)}; \\ \node [output] (TLSPlusNLS) {\(\T{S}\)}; & \node [output] (TTLSPlusNLS) {\(\TT{S}\)}; & \node (TSno) {\(\T{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\T{City}\)}; \\ & & & & & \node (TStatus) {\(\T{\Status}\)}; \\ \node [output] (TNLS) {\(\T{\NLS}\)}; & \node [output] (TTNLS) {\(\TT{\NLS}\)}; & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TLS) -- (TTLS), }; % projection edges { [edges=projection left] (TTLS) -- { (TSno), (TSSL) }, (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (TStatus), (NLSCity), (TSname) }, }; % selection edges { [edges=selection left] (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] (TCity) -- (LSCity), }; { [edges={bijection,output},edge label={\scriptsize\(\RelRestrict\)}] { (TLS), (TTLS) } -- { (TLSPlusNLS), (TTLSPlusNLS) }, { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); }; \end{tikzpicture} \caption{SIG for modified schema \(\bm{S_{3}'}\) after duplicating various nodes to produce analogues of the additional nodes in \(\bm{S_{1}}\). Inputs to the transformation are coloured \textcolor{blue}{blue}, outputs are coloured \textcolor{red}{red}. [\(\equiv\)]} \label{fig-transform-all-duplicates} \end{figure} %%%%%%%%%%%%%%%%%%%% The next phase is to copy and move edges. To produce the edge \(\T{S} \SurTotal \TT{S}\), we first \emph{copy} \(\T{\LS} \SurTotal \TT{\LS}\) across the path \(\TT{S} \TrivialSelection \TT{\LS}\) (giving \(\T{\LS} \SurTotal \TT{S}\)), then \emph{move} it across \(\T{S} \TrivialSelection \T{\LS}\). We can carry out a similar series of transformations to copy and move \(\T{S} \SurTotal \TT{S}\) to \(\T{\NLS} \SurTotal \TT{\NLS}\). Next, we copy and move the projection edge \(\TT{\LS} \ProjectionEdge \T{\Sno}\) to produce the edges \(\TT{S} \ProjectionEdge \T{\Sno}\) and \(\TT{\NLS} \ProjectionEdge \T{\Sno}\). Similarly, we copy and move \(\TT{\LS} \ProjectionEdge \TSSL\) to produce \(\TT{S} \LabelledEdge{\sigedge{03}}{\RelProject} \TSSC\) and \(\TT{\NLS} \LabelledEdge{\sigedge{02}}{\RelProject} \TSSNL\). Note that the first edge in the latter pair of transformations loses the totality annotation, and the second edge the surjectivity annotation, as a result of composition with the pre-existing non-bijective selection edges in the middle of the SIG. Finally, we carry out similar copy and move transformations to copy the key edge \(\T{\Sno} \KeyEdge \TSSC\) to \(\T{\Sno} \KeyEdge \TSSL\) and \(\T{\Sno} \KeyEdge \TSSNL\). The annotations for these edges are unaffected. The result of all these edge transformations is shown in Figure~\ref{fig-transform-edge-moves}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node (TLS) {\(\T{\LS}\)}; &[7mm] \node (TTLS) {\(\TT{\LS}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\T{\Sname}\)}; \\ \node (TLSPlusNLS) {\(\T{S}\)}; & \node (TTLSPlusNLS) {\(\TT{S}\)}; & \node (TSno) {\(\T{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\T{City}\)}; \\ & & & & & \node (TStatus) {\(\T{\Status}\)}; \\ \node (TNLS) {\(\T{\NLS}\)}; & \node (TTNLS) {\(\TT{\NLS}\)}; & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TLS) -- [input keep] (TTLS), (TLSPlusNLS) -- [output] (TTLSPlusNLS), (TNLS) -- [output] (TTNLS), }; % FDs (TSno) -- [funcdep left,bend left,output] (TSSL); (TSno) -- [funcdep right,output] (TSSNL); % projection edges { [edges=projection left] (TTLS) -- [input keep] { (TSno), (TSSL) }, (TSSL) -- { (TStatus), (LSCity) }, (TTLSPlusNLS) -- [output] (TSno), }; { [edges=projection right] (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (TStatus), (NLSCity), (TSname) }, (TTNLS) -- [output] (TSno), (TTNLS) -- [arrows={:->},output] (TSSNL), }; % selection edges { [edges=selection left] (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] (TCity) -- (LSCity), }; { [edges=bijection,edge label={\scriptsize\(\RelRestrict\)}] { (TLS), (TTLS) } -- { (TLSPlusNLS), (TTLSPlusNLS) }, { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right] (TSSC), }; (TSno) -- [funcdep right,input keep] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,arrows={:-{crossbar}>},bend right,output] (TSSC), }; \end{tikzpicture} \caption{SIG for modified schema \(\bm{S_{3}'}\) after copying and moving edges. [\(\preceq\)]} \label{fig-transform-edge-moves} \end{figure*} %%%%%%%%%%%%%%%%%%%% If we compare Figure~\ref{fig-sig-s} with Figure~\ref{fig-transform-edge-moves}, we can see that the transformed SIG for \(\SC{3}'\) now has the same node and edge structure as the SIG for \(\SC{1}\). We still need to remove the totality annotations from the four remaining trivial selection edges, but even when this is done, we do not achieve the desired isomorphism, as the projection edges \(\TT{S} \LabelledEdge{\sigedge{03}}{\RelProject} \TSSC\) and \(\TT{\NLS} \LabelledEdge{\sigedge{02}}{\RelProject} \TSSNL\) have different annotations from the corresponding edges in the SIG for \(\SC{1}\). Technically this means that the information capacities of \(\SC{1}\) and \(\SC{3}\) are incomparable, but the structural differences are so small that it is probably safe to say that the information capacity of \(\SC{3}\) is less than that of \(\SC{1}\) (i.e., \Dominates{\SC{1}}{\SC{3}}). Either way, it is clear that view updates based on this pair of schemas will be problematic. Incidentally, if we created a fourth sub-schema \(\SC{4} = \{\NLS\}\) and compared this with \(\SC{1}\), the result would be similar to \(\SC{3}\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusion \& future work} \label{sec-conclusion} \noindent In this paper we have shown how to use the concept of relative information capacity and the schema intension graph (SIG) formalism to characterise information equivalence with respect to view updating. The results of our analysis of Date's restriction view example \cite{Date.C-2013a-View} are consistent with his findings. The information capacities of \(\SC{1} = \{S\}\) and \(\SC{2} = \{\LS,\NLS\}\) are equivalent, so it should be possible to effectively propagate view updates between \(\SC{1}\) and \(\SC{2}\), regardless of the configuration of views and base relvars. Conversely, the information capacity of \(\SC{3} = \{\LS\}\) is at best less than that of \(\SC{1}\), implying that there are updates that cannot be propagated between \(\SC{1}\) and \(\SC{3}\). Our analysis shows the importance of clearly identifying all explicit and implicit constraints that can be propagated from the base schema \(\SC{0}\) to its sub-schemas. Omission of constraints could lead to a different SIG structure that might not be comparable, or might lead to an erroneous conclusion about a schema's information capacity relative to another. There are several avenues for future work. While the discussion in this paper focused on the relational context, relative information capacity and the SIG formalism are data model agnostic, and thus should be applicable in non-relational view contexts (e.g., XML). It is unclear at present whether the omission of inclusion dependencies such as foreign keys is significant. If such dependencies are significant, the question then is how best to represent them in the SIG formalism. It is also unclear at present whether copying or moving an edge across a bijective path should necessarily increase information capacity, as no annotations are lost in such a transformation. It may be that the structural change to the SIG implies an increase in information capacity regardless. Further investigation is required. Finally, we have only examined disjoint restriction views in this paper, which are arguably the simplest case of view updating. Further work needs to be done to characterise other types of view configuration, such as projection views, join views, and union views. This will enable us to further refine the application of SIGs in this context. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \balance %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibliographystyle{abbrv} \bibliography{VLDB2016_Stanger} \end{document}

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APCCM2017_Stanger.tex 100644 → 0

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\documentclass{vldb}

\usepackage{balance}
\usepackage{relalg}
\usepackage{txfonts}
\usepackage{bm}
\usepackage{subfigure}
\usepackage{booktabs}

\usepackage{tikz}
\usetikzlibrary{positioning}
\usetikzlibrary{graphs}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{calc}
\usetikzlibrary{arrows}

% "empty" arrow tip
\pgfarrowsdeclare{:}{:}{}{}
% custom bar arrow tip, offset from end of line (use "empty" tip at line ends if no >)
\tikzset{crossbar/.tip={|[scale width=1.75,sep=0.25em]}}
% various edge styles for TikZ
\tikzset{
    function/.style={arrows={->}},
    injection/.style={arrows={<-}},
    total/.style={arrows={:{crossbar}-}},
    surjection/.style={arrows={-{crossbar}:}},
    bijection/.style={arrows={<{crossbar}-{crossbar}>}},
    projection/.style={arrows={:{crossbar}-{crossbar}>}},
    projection left/.style={arrows={:{crossbar}-{crossbar}>},edge label={\scriptsize$\RelProject$}},
    projection right/.style={arrows={:{crossbar}-{crossbar}>},edge label'={\scriptsize$\RelProject$}},
    selection left/.style={arrows={<-{crossbar}>},edge label={\scriptsize$\RelRestrict$}},
    selection right/.style={arrows={<-{crossbar}>},edge label'={\scriptsize$\RelRestrict$}},
    funcdep left/.style={arrows={->},edge node={node[sloped,midway,above] {\scriptsize\emph{key}}}},
    funcdep right/.style={arrows={->},edge node={node[sloped,midway,below] {\scriptsize\emph{key}}}},
    surtotal/.style={arrows={:{crossbar}-{crossbar}:}},
    input keep/.style={blue,thick},
    input delete/.style={blue!40,thick,dashed},
    output/.style={red,thick},
    output temp/.style={red,thick,dashed},
    path 1/.style={green!60!black,thick},
    path 2/.style={orange,thick}
}
        
% projection and selection edge annotations for TikZ
\makeatletter
\newcommand{\ProjectionAnnotation}[3][]{%
    \path (#2) to node[above,#1] {\scriptsize$\RelProject$} (#3);%
}
\newcommand{\SelectionAnnotation}[3][]{%
    \path (#2) to node[above,#1] {\scriptsize$\RelRestrict$} (#3);%
}
\makeatother

\newcounter{constraint}

% misc
\newcommand{\todo}[1]{\textbf{!!TODO!!} {[#1]}}

% commonly used elements
\newcommand{\LS}{\ensuremath{\mathit{LS}}}
\newcommand{\NLS}{\ensuremath{\mathit{NLS}}}
\newcommand{\LSsub}{\ensuremath{\mathit{LS}_{\mathit{sub}}}}
\newcommand{\NLSsub}{\ensuremath{\mathit{NLS}_{\mathit{sub}}}}
\newcommand{\Sno}{\ensuremath{\mathit{Sno}}}
\newcommand{\Sname}{\ensuremath{\mathit{Sname}}}
\newcommand{\Status}{\ensuremath{\mathit{Status}}}
\newcommand{\City}{\ensuremath{\mathit{City}}}

\newcommand{\T}[1]{\ensuremath{T_{#1}}}
\newcommand{\TT}[1]{\ensuremath{T_{\{#1\}}}}

\newcommand{\CityLondon}{\ensuremath{\{\City\colon\mathit{'London'}\}}}
\newcommand{\TCityMinusLondon}{\ensuremath{\T{\City} - \CityLondon}}
\newcommand{\TSSC}{\ensuremath{\T{\Sname} \times \T{\Status} \times \T{\City}}}
\newcommand{\TSSL}{\ensuremath{\T{\Sname} \times \T{\Status} \times \CityLondon}}
\newcommand{\TSSNL}{\ensuremath{\T{\Sname} \times \T{\Status} \times (\TCityMinusLondon)}}

\newcommand{\TLSPlusNLS}{\ensuremath{\T{\LS} + \T{\NLS}}}
\newcommand{\TTLSPlusNLS}{\ensuremath{\TT{\LS} + \TT{\NLS}}}
\newcommand{\TLSPlusNLSsub}{\ensuremath{\T{\LSsub} + \T{\NLSsub}}}
\newcommand{\TTLSPlusNLSsub}{\ensuremath{\TT{\LSsub} + \TT{\NLSsub}}}

\newcommand{\StackTLSPlusNLS}{\ensuremath{\begin{array}{c}\T{\LS}\,+ \\ \T{\NLS}\end{array}}}
\newcommand{\StackTTLSPlusNLS}{\ensuremath{\begin{array}{c}\TT{\LS}\,+ \\ \TT{\NLS}\end{array}}}
\newcommand{\StackTLSPlusNLSsub}{\ensuremath{\begin{array}{c}\T{\LSsub}\,+ \\ \T{\NLSsub}\end{array}}}
\newcommand{\StackTTLSPlusNLSsub}{\ensuremath{\begin{array}{c}\TT{\LSsub}\,+ \\ \TT{\NLSsub}\end{array}}}
\newcommand{\StackTSSC}{\ensuremath{\begin{array}{c}\T{\Sname}\,\times \\ \T{\Status} \times \T{\City}\end{array}}}
\newcommand{\StackTSSL}{\ensuremath{\begin{array}{c}\T{\Sname} \times \T{\Status}\,\times \\ \CityLondon\end{array}}}
\newcommand{\StackTSSNL}{\ensuremath{\begin{array}{c}\T{\Sname} \times \T{\Status}\,\times \\ (\TCityMinusLondon)\end{array}}}
\newcommand{\StackTCityMinusLondon}{\ensuremath{\begin{array}{c}\T{\City}\,- \\ \CityLondon\end{array}}}

\newcommand{\SC}[1]{\ensuremath{\mathcal{S}_{#1}}}

% dominates
\newcommand{\Dominates}[2]{\ensuremath{#2 \preceq #1}}
\newcommand{\Equivalent}[2]{\ensuremath{#1 \equiv #2}}

% SIG notation (in text)
\newcommand{\Sedge}[1]{\ensuremath{\sigma_{\textrm{#1}}}}
\newcommand{\SedgeP}[1]{\ensuremath{\sigma_{\textrm{#1}}^{'}}}

\newcommand{\medmid}{\raise.125ex\hbox{\scalebox{1}[0.75]{$\mid$}}}

% General SIG edges for use in formulas.
% Adapted from: http://tex.stackexchange.com/questions/96330/adding-symbols-at-the-ends-of-a-horizontal-line
\makeatletter
\newlength{\@annotskipleft}
\newlength{\@annotskipright}
% #1 = left edge component
% #2 = right edge component
% #3 = left bar annotation
% #4 = right bar annotation
\newcommand\@sig@edge[4]{%
    \let\@middle\joinrel%
    \ifx#1\relbar%
        \@annotskipleft=.3em%
        % scrunch up the bare line so it's similar length to \long...arrow
        \ifx#2\relbar\def\@middle{\joinrel\joinrel\relbar\joinrel\joinrel}\fi%
    \else% 
        % arrows need a little more clearance
        \@annotskipleft=.4em%
    \fi%
    \ifx#2\relbar\@annotskipright=.3em\else\@annotskipright=.4em\fi%
    \mathrel{\ooalign{$#1\@middle#2$\cr\hskip\@annotskipleft$#3$\hfil$#4$\hskip\@annotskipright\cr}}%
}

% 0 = nothing
% 1 = bar
% 2 = arrowhead
% 3 = both
\def\@sigedge#1#2{%
    \ifcase\numexpr#1*4+#2\relax%
        \@sig@edge{\relbar}{\relbar}{}{}\or                     % 00 = -----
        \@sig@edge{\relbar}{\relbar}{}{\medmid}\or              % 01 = ---+-
        \@sig@edge{\relbar}{\rightarrow}{}{}\or                 % 02 = ---->
        \@sig@edge{\relbar}{\rightarrow}{}{\medmid}\or          % 03 = ---+>
        \@sig@edge{\relbar}{\relbar}{\medmid}{}\or              % 10 = -+---
        \@sig@edge{\relbar}{\relbar}{\medmid}{\medmid}\or       % 11 = -+-+-
        \@sig@edge{\relbar}{\rightarrow}{\medmid}{}\or          % 12 = -+-->
        \@sig@edge{\relbar}{\rightarrow}{\medmid}{\medmid}\or   % 13 = -+-+>
        \@sig@edge{\leftarrow}{\relbar}{}{}\or                  % 20 = <----
        \@sig@edge{\leftarrow}{\relbar}{}{\medmid}\or           % 21 = <--+-
        \@sig@edge{\leftarrow}{\rightarrow}{}{}\or              % 22 = <--->
        \@sig@edge{\leftarrow}{\rightarrow}{}{\medmid}\or       % 23 = <--+>
        \@sig@edge{\leftarrow}{\relbar}{\medmid}{}\or           % 30 = <+---
        \@sig@edge{\leftarrow}{\relbar}{\medmid}{\medmid}\or    % 31 = <+-+-
        \@sig@edge{\leftarrow}{\rightarrow}{\medmid}{}\or       % 32 = <+-->
        \@sig@edge{\leftarrow}{\rightarrow}{\medmid}{\medmid}   % 33 = <+-+>
    \fi%
}

\newcommand{\sigedge}[1]{\ensuremath{\@sigedge#1}}
\makeatother

\newcommand{\Edge}{\sigedge{00}}
\newcommand{\Total}{\sigedge{10}}
\newcommand{\Surjective}{\sigedge{01}}
\newcommand{\SurTotal}{\sigedge{11}}
\newcommand{\Functional}{\sigedge{02}}
\newcommand{\Injective}{\sigedge{20}}
\newcommand{\Bijective}{\sigedge{33}}

% Hack to get the overset symbols to appear at the right height.
% \smash removes the spacing around the operator, hence \mathop.
\newcommand{\LabelledEdge}[2]{\mathop{\overset{\raisebox{0.3em}{\scriptsize\ensuremath{#2}}}{\smash[t]{#1}}}}
\newcommand{\ProjectionEdge}{\LabelledEdge{\sigedge{13}}{\RelProject}}
\newcommand{\SelectionEdge}{\LabelledEdge{\sigedge{23}}{\RelRestrict}}
\newcommand{\TrivialProjection}{\ensuremath{\LabelledEdge{\Bijective}{\RelProject}}}
\newcommand{\TrivialSelection}{\ensuremath{\LabelledEdge{\Bijective}{\RelRestrict}}}
\newcommand{\KeyEdge}{\ensuremath{\LabelledEdge{\Functional}{\mathit{key}}}}

\hyphenation{co-pied sche-ma}

\pagestyle{empty}

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\begin{document}

\title{Characterising relational view updates using relative information capacity}

\numberofauthors{1}
\author{
\alignauthor Nigel Stanger \\
    \affaddr{Department of Information Science}\\
    \affaddr{University of Otago}\\
    \affaddr{Dunedin, New Zealand}\\
    \email{nigel.stanger@otago.ac.nz}
}

\maketitle

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\begin{abstract}
In 2012, Chris Date published a book about the view update problem in the context of the Relational Model. He presented several detailed examples of different varieties of view updates and characterised their behaviour, in the process deriving a set of principles for updating views based around the notion of \emph{information equivalence} of view schemas. In this paper we examine how this notion can be more formally characterised using Hull's concept of \emph{relative information capacity}. Specifically, we use Miller's \emph{schema intension graph} formalism to characterise the information equivalence of Date's restriction view example, and show that this supports his operational definition of information equivalence.
\end{abstract}

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\section{Introduction}

\noindent The view update problem is a well known and long standing problem of great practical significance in the database field. Indeed, Date \cite{Date.C-2013a-View} likens it to significant unsolved problems in other fields such as the Riemann Hypothesis in mathematics or ``$P = \mathit{NP}$?'' in computational complexity theory. In his 2012 book \cite{Date.C-2013a-View} Date developed a comprehensive set of principles for updating relational views, based on the notion of \emph{information equivalence}. However, that notion was defined more operationally than by a formal theoretical definition.

View updating can be considered a special case of schema translation: we effectively have one schema comprising a set of views and another schema comprising a set of base relations, and need to translate updates from one schema to the other.\footnote{These schemas may be actually separate, or more likely may be sub-schemas of a larger database schema, with effective separation achieved via the access privilege mechanism.} It therefore makes sense that formalisms developed to assist with characterising sche\-ma translations would also be applicable to view updates. To that end, we propose to use Hull's notion of \emph{relative information capacity} \cite{Hull.R-1986a-Relative} as a means to characterise the equivalence of the schemas involved in a given view configuration.

In particular, we will use Miller's \emph{schema intension graph} (SIG) formalism, which facilitates the comparison of relative information capacity across schemas. In this paper, we show how SIGs can be used to characterise the equivalence of schemas within a view configuration. To facilitate this, we extend the SIG formalism with additional concepts, including relation and tuple types, key constraints and functional dependencies. We also show how the formalism can be practically used to characterise the information equivalence of a view configuration, by applying it to Date's example of disjoint restriction views \cite{Date.C-2013a-View}. The results our formal analysis are shown to be consistent with Date's operational analysis of information equivalence in this example.

In Section~\ref{sec-view-update-problem}, we briefly review the view update problem and relevant research, and in Section~\ref{sec-relations-types} we introduce some core concepts that are used throughout the remainder of the paper. In Section~\ref{sec-info-capacity} we provide a detailed explanation of relative information capacity and the SIG formalism, then explain in Section~\ref{sec-relational-sigs} how to construct a SIG that represents a relational schema. In Section~\ref{sec-characterising} we analyse Date's restriction view example \cite{Date.C-2013a-View} in detail, and characterise the information capacity of the various sub-schemas involved. We discuss conclusions and future work in Section~\ref{sec-conclusion}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The view update problem}
\label{sec-view-update-problem}

\noindent The view update problem was first discussed by Codd in 1974 \cite{Codd.E-1974a-Recent}. When an update operation is applied to a view, that update needs to be translated into corresponding updates on the base relations from which the view is derived \cite{Keller.A-1985a-Algorithms}. Sometimes this translation is clear, but sometimes there are multiple such translations that all produce the desired view update, but produce different---possibly incompatible---changes in the base relations. Worse, there may be translations that have side effects (e.g., for some views based on joins), which produce a different result in the view than what we would expect if we could apply the update to the view directly \cite{Keller.A-1985a-Algorithms}. The problem then is how to choose the most appropriate translation. As yet there has been no solution that completely and automatically resolves this ambiguity.

There has been considerable research into this problem over the last four decades, and numerous different approaches have been suggested. We shall now briefly review some key developments in the relational context.

% Paolini?

Banchilhon and Spyratos proposed the concept of \emph{constant complement} \cite{Bancilhon.F-1981a-Update} as a way to resolve view update issues. The \emph{complement} of a view describes all the information from the original database that is not in the view, so composing a view with its complement provides enough information to reconstruct the original database. Keeping the complement invariant under view update facilitates translation construction. This is a useful approach that has been developed further by other authors \cite{Hegner.S-2004a-Order-based,Lechtenborger.J-2003a-Impact}, but it has been shown that reasonable translations exist that cannot be carried out under constant complement \cite{Keller.A-1987a-Comment}. Computing the complement of a given view is also difficult, and several authors have worked on computing complements more efficiently \cite{Cosmadakis.S-1984a-Updates,Laurent.D-2001a-Monotonic,Lechtenborger.J-2003a-Computation}. Finally, this approach is purely structural in nature and does not consider the semantics of the schema, which is necessary to properly resolve translation ambiguity \cite{Keller.A-1986a-Role}.

Gottlob et al.\ generalised the idea of constant complement into the class of \emph{consistent views} \cite{Gottlob.G-1988a-Properties}. They define consistent views as having the property that ``if the effect of a view update on a view state is determined, then the effect of the corresponding database update is unambiguously determined'' \cite{Gottlob.G-1988a-Properties}. However, there are still reasonable views that are not consistent under this definition, and this approach again does not consider schema semantics.

% not sure this is really needed?
%Keller nicely illustrated the view update problem in terms of mappings between database states \cite{Keller.A-1985a-Algorithms}, as shown in Figure~\ref{fig-equivalence}. A view $V(\mathit{DB})$ is derived via a view mapping $V$ from the original database $\mathit{DB}$. When an update $U$ is applied to $V(\mathit{DB})$, a view translator $T$ translates it into an update $T(U)$ on $\mathit{DB}$, resulting in a new database state $\mathit{DB}'$. The new view state $V(\mathit{DB}')$ is then derived. Problems arise if $V(\mathit{DB}')$ does not match what we would expect if $U$ were applied directly to $V(\mathit{DB})$, because of side effects introduced by $T$. Ideally, we would be able to find a side effect free translator $T$ for any given view, but there are some views that cannot be translated without side effects (e.g. certain joins) \cite{Keller.A-1985a-Algorithms}.

% Date \cite{Date.C-2013a-View} notes that this arises from a lack of \emph{information equivalence} between $\mathit{DB}$ and $V(\mathit{DB})$.

%%%%%%%%%%%%%%%%%%%%

%\newcommand{\qeq}{\ensuremath{\mathop{\smash[t]{\overset{?}{=}}}}}
%
%\begin{figure}
%    \centering
%    \begin{tikzpicture}[every edge/.style={draw,semithick}]
%        \node (vdb)                         {$V(\mathit{DB})$};
%        \node (db)   [below=11mm of vdb]     {$\mathit{DB}\vphantom{(')}$};
%    
%        \node (vdbp) [right=3cm of vdb]     {$V(\mathit{DB}')$};
%        \node (uvdb) [left=-1mm of vdbp]    {$U(V(\mathit{DB})) \qeq$};
%    
%        \node (dbp)  at (db -| vdbp)        {$\mathit{DB}\smash[t]{'}\vphantom{()}$};
%        \node (tudb) [left=-1mm of dbp]     {$T(U)(\mathit{DB}) =$};
%        
%        \graph{
%            { (db), (dbp) } -> [edge label={$V$},swap] { (vdb), (vdbp) };
%            
%            (vdb) -> [edge node={node (u)  [above=-0.6mm] {$U$}}]    (uvdb);
%            (db)  -> [edge node={node (tu) [above=-0.6mm] {$T(U)$}}] (tudb);
%            
%            (u) -> [double equal sign distance=2pt,-implies] (tu);
%        };
%    \end{tikzpicture}
%    \caption{Equivalence of view updates \protect\cite{Keller.A-1985a-Algorithms}.}
%    \label{fig-equivalence}
%\end{figure}

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Keller noted that semantic information is essential in order to choose an appropriate translation \cite{Keller.A-1986a-Role}, and suggested this could be collected at view definition time from the database administrator (DBA)  \cite{Keller.A-1985a-Algorithms}. Masunaga argued, however, that this would be insufficient to completely resolve ambiguity, and suggested collecting further semantic information from end users at update time \cite{Masunaga.Y-1984a-Relational}. Shu \cite{Shu.H-2000a-Using} discussed an interesting approach to this kind of information gathering, by treating the problem of generating suitable view translations as a constraint satisfaction problem. This enables constraints to be added incrementally at view update time. However, while collecting such information from the DBA seems reasonable, collecting it from the end user at update time could be seen as too burdensome, and could impact the usability of query interfaces. It is also unclear which information needs to be captured, and how best to capture it.

Kotidis et al.\ \cite{Kotidis.Y-2006a-Updates} discussed a different approach to avoiding side effects in view translations. They proposed creating a materialised copy of the view that could be updated directly, rather than mapping updates from the (virtual) view to the base tables. Clearly this approach enables the view to updated without translation side effects, but does imply that the physical manifestation of the view may no longer match the original view query (i.e., its extension may not match its intension).

Finally, Date has published several books on the theoretical underpinnings of the Relational Model over the last decade, including one specifically targeting the view update problem \cite{Date.C-2013a-View} (indeed, the book is subtitled ``Solving the View Update Problem''). He developed a comprehensive set of rules for updating views in a relational context, supported by a wide array of worked examples. Date advocated explicitly stating all relevant constraints at view creation time, which is similar to the information gathering approaches discussed above \cite{Keller.A-1985a-Algorithms,Masunaga.Y-1984a-Relational,Shu.H-2000a-Using}. He argued that if a complete set of constraints was available, then ambiguity would be diminished or eliminated, and it would be possible to automatically generate a set of appropriate \emph{compensatory rules} to automatically manage the view update process. Compensatory rules are similar in principle to database triggers, except that they are intended to be declarative so that their definitions can made explicitly visible to end users.

Date's work is interesting and compelling, and raises several questions worthy of further investigation. However, there are still some situations where the correct view translation may be ambiguous, e.g., views based on certain types of join, as previously noted by Keller \cite{Keller.A-1985a-Algorithms}. Somewhat disappointingly, Date took a pragmatic approach in these cases, effectively saying ``this feels like the correct solution''. His justification was well argued and logical, but it still falls somewhat short of the ideal of a fully automated solution.

As noted previously, Date characterises the ability to define an appropriate and effective view translation using the notion of \emph{information equivalence}. Two schemas $\SC{1}$ and $\SC{2}$ are said to be information equivalent ``if and only if the constraints that apply to [$\SC{1}$] and [$\SC{2}$] are such that every proposition that can be represented by [$\SC{1}$] can be represented by [$\SC{2}$] and vice versa'' \cite{Date.C-2013a-View}. While information equivalence is clearly a formal concept, the definition and characterisation in \cite{Date.C-2013a-View} are more operational and pragmatic in nature. This is what we explore further in this paper.

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\section{Relations and types}
\label{sec-relations-types}

\noindent We generally follow Date's \cite{Date.C-2012a-SQL-and-Relational} notation and terminology regarding the Relational Model. The main exception is that we do not use Tutorial D notation Date uses in his examples, preferring instead a more classical algebraic style notation.

In this paper, $A$ represents an attribute (comprising a name:type pair), $R$ represents a relation variable (\emph{relvar}), and $v$ represents a value (of an attribute, tuple, relation, etc.). In the following discussion it will generally be clear which attribute a value is associated with, but where necessary, the value will be prefixed with the attribute name, i.e., $\mathit{name}\colon\mathit{value}$.

Date defines a \emph{type} $\T{i}$ as ``essentially a \emph{named, finite set of values}'' \cite{Date.C-2012a-SQL-and-Relational}, representing every possible value of some specific kind, e.g., every possible ASCII character string of length 5, every possible city name, every possible relation value with a particular heading, etc. That is:
\begin{equation}\label{eqn-type}
    \T{i} = \{v_{1}, v_{2}, \dotsc, v_{m}\}, 0 < m < k
\end{equation}
where $k$ is some finite upper bound. Thus, the type of an attribute $A_{i}$ (denoted $\T{A_{i}}$) is by definition the set of all possible values of $A_{i}$.

Consider a relvar $R$ with heading $\{A_{1}, A_{2}, \dotsc, A_{n}\}$. Following from equation~\ref{eqn-type}, the (relation) type of $R$, denoted here as $\T{R}$, is the set of all possible relation values with $R$'s heading. Similarly, the (tuple) type of $R$'s tuples, denoted here as \TT{R}, is the set of all possible tuple values with $R$'s heading, which equates to the cartesian product of $R$'s attribute types, i.e.:
\begin{equation}\label{eqn-tuple-type}
\TT{R} = \T{A_{1}} \times \T{A_{2}} \times\dotsb\times \T{A_{n}}\text{.}
\end{equation}
By definition any given value of $R$ must comprise a subset (possibly empty) of tuples $\{t_{1}, t_{2}, \dotsc, t_{m}\} \subseteq \TT{R}$. $\T{R}$ therefore represents every possible $k$-combination of the elements of \TT{R}, for $0 \le k \le \left|\TT{R}\right|$.

For example, Date's usual suppliers-and-parts schema \cite{Date.C-2013a-View} includes a relvar $S\!$ with heading $\{\Sno, \Sname, \Status, \City\}$, which contains details of suppliers. As per equation~\ref{eqn-tuple-type}, the tuples of $S\!$ have type $\TT{S} = \T{\Sno} \times \TSSC$.

This notion of type as a set of possible values is synonymous with Codd's original definition of domains in the Relational Model \cite{Codd.E-1970a-Relational,Date.C-2012a-SQL-and-Relational}, and in particular with the notion of domains as ``sets of values'' in schema intension graphs \cite{Miller.R-1994a-Schema,Miller.R-1994a-SIG}. The implications of the latter point will be explored further in Section~\ref{sec-relational-sigs}, after we have discussed the concepts of relative information capacity and schema intension graphs in Section~\ref{sec-info-capacity}.

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\subsection{Nulls}

\noindent We also follow Date in assuming that nulls are not permitted in relation values. Philosophical objections aside, however, this makes little practical difference to the approach discussed in this paper, as we are dealing exclusively with types rather than attributes. That is, an attribute can be null, but it makes little sense to say that a \emph{type} can be ``null''. It could be claimed that a type ``includes'' nulls, but it is unclear what this would even mean in practice, and runs the risk of treating nulls like ``normal'' values rather than the special markers that they actually are. In any case, this would effectively be no different from the situation without nulls, so we omit further consideration of nulls in the interest of clarity.

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\newpage

\section{Relative information capacity}
\label{sec-info-capacity}

\noindent A schema conveys information about the phenomenon that it models. Hull \cite{Hull.R-1986a-Relative} defined the concept of \emph{relative information capacity} to describe the information content of different schemas. The relative information capacity of a schema determines the set of all valid instances of that schema \cite{Hull.R-1986a-Relative,Miller.R-1994a-PhD,Miller.R-1993a-Information}. Relative information capacity, as its name implies, is not a quantitative measure; rather it is an indicator of the relative information content of different schemas.

Relative information capacity can be used as a basis for identifying the differences between schema instances, and can therefore be used as an aid to schema integration and translation \cite{Miller.R-1994a-PhD}. As noted earlier, view updating is a special case of this, so it follows that information capacity can also be used to characterise view updates, or more precisely, to characterise the information content of views compared to the base relvar(s) from which they are derived.

Miller et al.\ \cite{Miller.R-1993a-Information} defined relative information capacity in terms of two types of mapping. An \emph{information (capacity) preserving mapping} from schema $\SC{1}$ to schema $\SC{2}$ maps every element of $\SC{1}$ to an element of $\SC{2}$, but only some elements of $\SC{2}$ to elements of $\SC{1}$. $\SC{2}$ has an information capacity greater than $\SC{1}$, and is said to \emph{dominate} $\SC{1}$, denoted \Dominates{\SC{2}}{\SC{1}}. Compare this to an \emph{equivalence preserving mapping} from $\SC{1}$ to $\SC{3}$, which maps every element of $\SC{1}$ to an element $\SC{3}$ and vice versa (i.e., both the mapping and its inverse are information preserving). The information capacities of $\SC{1}$ and $\SC{3}$ are identical, and the schemas are said to be \emph{equivalent}, denoted \Equivalent{\SC{1}}{\SC{3}}.

(Note that Miller et al.\ \cite{Miller.R-1994a-Schema} use stricter definitions of equivalence and dominance than those used here; strictly speaking, the discussion of equivalence and dominance in this section should properly refer to \emph{absolute} dominance and equivalence, whereas the following discussion on SIG transformations should properly refer to \emph{internal} dominance and equivalence \cite{Miller.R-1994a-Schema}. The distinction, however, is not crucial to the discussion in this paper.)

Since relative information capacity is a relative measure, it can only be effectively used in a comparative manner. To this end, Miller et al.\ \cite{Miller.R-1994a-SIG} defined the \emph{schema intension graph} formalism to provide a tool for comparing the information capacity of schemas. A schema intension graph (SIG) describes the associations between domains in a schema, and provides a way to ``reason about constraints on \emph{collections of entities} in an instance of the entity set, rather than about the internal structure of a single entity'' \cite{Miller.R-1994a-SIG}. In particular, SIGs provide a useful abstract formalism within which to perform schema transformations, as transforming the SIG for a schema is effectively the same as transforming the schema itself.

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\subsection{Constructing SIGs}
\label{sec-sig-build}

\noindent A SIG is a graph $G = (N, E)$ \cite{Miller.R-1994a-Schema}. The nodes $n_{i} \in N$ represent typed domains (i.e., sets of values). Nodes may be \emph{simple} (atomic) or \emph{constructed} (composite). Simple nodes are denoted by the name of the domain that they represent. Constructed nodes represent a collection of domains and are built using the $\times$ (cartesian product) and $+$ (union) operators.

The edges $e_{i} \in E$ represent binary relations (in the mathematical sense) between domains. Edges are denoted by plain lines with an optional label, and may be composed to form a \emph{path} between two nodes (the notation $e_{2} \circ e_{1}$ means ``edge $e_{1}$ followed by edge $e_{2}$''). Some edges may be designated as \emph{selection} edges, labelled $\RelRestrict$, or \emph{projection} edges, labelled $\RelProject$. These correspond to the relational algebra operators of the same name \cite{Codd.E-1972a-Completeness}.

Edges may also be annotated with up to four properties denoting simple constraints on the binary relation represented by the edge: totality, surjectivity, functionality and injectivity. These properties are defined as follows \cite{Miller.R-1994a-Schema}:
\begin{quotation}
    \noindent Let $A$ and $B$ be two sets. A binary relation $r : A - B$ is \emph{total} (denoted $e : A \Total B$) if it is defined for all elements of $A$; \emph{surjective} ($e : A \Surjective B$) if it is defined for all elements of $B$; \emph{functional} ($e : A \Functional B$) if an element of $A$ determines at most one element of $B$; and \emph{injective} ($e : A \Injective B$) if an element of $B$ determines at most one element of $A$.
\end{quotation}
Note that the correct usage of these properties is determined by the direction in which the binary relation is read. That is, the binary relation $A \Functional B$ is functional if read from left to right, but injective if read from right to left. A \emph{bijection} is a binary relation that has all four properties \cite{Miller.R-1994a-SIG}.

Essentially, $A \Total B$) means that every element of $A$ (conversely, $B$ for $A \Surjective B$) must participate in the binary relation, while $A \Functional B$ means that a given element of $A$ (conversely, $B$ for $A \Injective B$) may appear in at most one instance of the binary relation.

Note that use of the term ``relation'' here, while mathematically correct, clearly has potential to cause confusion in any discussion that also involves the Relational Model. From now on, we shall therefore refer to the binary relations represented by SIG edges as ``associations''.

A selection edge $e : A \LabelledEdge{\sigedge{00}}{\RelRestrict} B$ indicates that the elements of $B$ comprise a subset (or specialisation) of the elements of $A$. If $B = A$, which we term a \emph{trivial} selection, every element of B must appear in A and vice versa. Similarly, every element of $A$ can be associated with at most one element of $B$, and vice versa. Trivial selection edges are therefore bijective, i.e., $A \LabelledEdge{\Bijective}{\RelRestrict} B$. If $B \subset A$, which we term a \emph{non-trivial} selection, every element of $B$ must appear in $A$, but the converse no longer holds. The mapping remains one-to-one, as before. Non-trivial selection edges are therefore surjective (from the perspective of the supertype), functional, and injective, i.e., $A \SelectionEdge B$.

A projection edge $e : A \LabelledEdge{\sigedge{00}}{\RelProject} B$, indicates that the elements of $B$ are a projection of $A$. This implies that $A$ is a constructed node of the form $B \times C \times \dotsb$. Trivial projection edges ($B = A$) are bijective, i.e., $A \LabelledEdge{\Bijective}{\RelProject} B$, for the same reasons as trivial selection edges. For non-trivial projections (i.e., $B \ne A$), every element of $A$ must have a corresponding element in $B$, and vice versa. Similarly, every element of $A$ can be associated with at most one element of $B$, but \emph{not} vice versa. Thus, non-trivial projection edges are total, surjective, and functional, i.e., $A \ProjectionEdge B$.\footnote{Allowing nulls could potentially alter which of these properties apply, but all non-trivial projection edges would still share the same properties regardless.}

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\subsection{Comparing SIGs}
\label{sec-sig-compare}

\noindent It is possible to determine the relationship between the relative information capacities of two schemas by comparing SIGs for the two schemas. Consider two schemas $\SC{1}$ and $\SC{2}$, with corresponding SIGs $G_{\SC{1}}$ and $G_{\SC{2}}$. Intuitively, if $G_{\SC{1}}$ and $G_{\SC{2}}$ are isomorphic, then $\SC{1}$ and $\SC{2}$ are equivalent \cite{Miller.R-1994a-SIG}. An isomorphism occurs between two SIGs when the graphs match both structurally and semantically. That is, $G_{\SC{1}}$ and $G_{\SC{2}}$ must be more than just structurally identical---the semantics of the SIG nodes and edges must also be compatible.

Miller et al.\ \cite{Miller.R-1994a-SIG} identify three possible results from comparing two SIGs $G_{\SC{1}}$ and $G_{\SC{2}}$:
\begin{enumerate}
    
    \item $G_{\SC{1}}$ and $G_{\SC{2}}$ are either immediately isomorphic, or one can be manipulated by a series of equivalence preserving transformations (described below) until it is isomorphic to the other. The two schemas are equivalent (i.e., $\SC{1} \equiv \SC{2}$).
    
    \item $G_{\SC{1}}$ and $G_{\SC{2}}$ are not immediately isomorphic, but one can be manipulated by a series of information capacity augmenting transformations (described below) until it is isomorphic to the other. The transformed schema is dominated by the other (i.e., either $\Dominates{\SC{2}}{\SC{1}}$ or $\Dominates{\SC{1}}{\SC{2}}$).
    
    \item $G_{\SC{1}}$ and $G_{\SC{2}}$ are not immediately isomorphic and cannot be transformed to be isomorphic. The information capacities of $\SC{1}$ and $\SC{2}$ cannot be compared.
    
\end{enumerate}
Batini et al.\ \cite{Batini.C-1992b-Conceptual} follow a similar classification scheme with respect to schema transformations: they refer to the first as \emph{information-preserving} transformations, the second as \emph{augmenting} transformations and the third as \emph{noncomparable} transformations.

Miller et al.\ \cite{Miller.R-1994a-Schema} describe the following equivalence preserving and information capacity augmenting transformations on SIGs, which are summarised in Table~\ref{Tab.Viewpoints.SIGTransformations}:
\begin{description}
    
    \item[Annotation transformations] allow the removal of annotations, or projection and selection constraints from edges (e.g., $n_{1} \sigedge{23} n_{2} \Rightarrow n_{1} \sigedge{02} n_{2}$). If we can transform $\SC{1}$ into $\SC{2}$ by such a transformation, then the information capacity of $\SC{2}$ dominates that of $\SC{1}$ (i.e., \Dominates{\SC{2}}{\SC{1}}).
    
    \item[Composition transformations] allow edges to be moved and copied around the graph. Consider an edge $e$ between nodes $n_{1}$ and $n_{2}$, as shown in Figure~\ref{fig-sig-composition-original}. If there is a path $P = e_{m} \circ \dotsb \circ e_{k}$ of functional edges from some other node $n_{k}$ to $n_{2}$ (i.e., $n_{k} \Functional \dotsb \Functional n_{2}$), $e$ may be copied to a new edge $e'$ between nodes $n_{1}$ and $n_{k}$ as shown in Figure~\ref{fig-sig-composition-copy}. The original $e$ can then be deleted if desired, effectively moving $e$ to $e'$ as shown in Figure~\ref{fig-sig-composition-move}.
    
    The annotation of $e'$ is determined by the composition of the edges of $P\!$. If an edge with a particular annotation (such as surjectivity) is composed with an edge without that annotation, the resulting edge does not have the annotation. For example, the composition of the edges $\Functional$ and $\Surjective$ would be an edge with no annotations ($\Edge$), whereas the composition of $\Functional$ and $\sigedge{03}$ would be $\Functional$. If we can transform $\SC{1}$ into $\SC{2}$ by such a transformation, then \Dominates{\SC{2}}{\SC{1}}.

%%%%%%%%%%%%%%%%%%%%

\begin{table}[tb]
    \centering
    \small
    \begin{tabular}{lcc}
        \toprule
                                                        &                   &   \textbf{Effect on infor-}   \\
        \textbf{Transformation}                         &   \textbf{Type}   &   \textbf{mation capacity}    \\
        \midrule
        Delete annotation or $\RelProject$/$\RelRestrict$ constraint    &   annotation      &   augmented   \\
        Move/copy edge                                  &   composition     &   augmented   \\
        Create/delete duplicate node                    &   selection       &   no change   \\
        Create trivial selection edge                   &   selection       &   no change   \\
        Delete trivial selection edge                   &   selection       &   augmented   \\
        \bottomrule
    \end{tabular}
    \caption{Summary of SIG transformations.}
    \label{Tab.Viewpoints.SIGTransformations}
\end{table}

%%%%%%%%%%%%%%%%%%%%

\begin{figure}[htb]
    \centering
    \subfigure[Original state\label{fig-sig-composition-original}]{
        \begin{tikzpicture}
            \node (n1)                      {$n_{1}$};
            \node (n2) [below=1cm of n1]    {$n_{2}$};
            \node (n3) [right=1cm of n2]    {$n_{3}$};
            \node (ee) [right=0.75cm of n3] {\ldots};
            \node (nk) [right=0.75cm of ee] {$n_{k}$};
        
            \graph{
                (n1) -- [edge label={$e$},swap] (n2);
                (nk) -> [edge label={$e_{k}$}] (ee) -> [edge label={$e_{m+1}$}] (n3) -> [edge label={$e_{m}$}] (n2);
            };
            
            \node (Pleft) [above=2mm of n2.north east] {};
            \node (Pright) [above=2mm of nk.north west] {};
            \node (P) at ($(Pleft)!0.5!(Pright)$) {$P$};
            \draw[decorate,decoration=brace] (n2.north east) -- (nk.north west);
        \end{tikzpicture}
    }
    
    \subfigure[Copy edge $e$ to $e'$\label{fig-sig-composition-copy}]{
        \begin{tikzpicture}
            \node (n1)                      {$n_{1}$};
            \node (n2) [below=1cm of n1]    {$n_{2}$};
            \node (n3) [right=1cm of n2]    {$n_{3}$};
            \node (ee) [right=0.75cm of n3] {\ldots};
            \node (nk) [right=0.75cm of ee] {$n_{k}$};
        
            \graph{
                (n1) -- [edge label={$e$},swap] (n2);
                (n1) -- [edge label={$e'$}]     (nk);
                (nk) -> [edge label={$e_{k}$}] (ee) -> [edge label={$e_{m+1}$}] (n3) -> [edge label={$e_{m}$}] (n2);
            };
        \end{tikzpicture}
    }
    
    \subfigure[Move edge $e$ to $e'$\label{fig-sig-composition-move}]{
        \begin{tikzpicture}
            \node (n1)                      {$n_{1}$};
            \node (n2) [below=1cm of n1]    {$n_{2}$};
            \node (n3) [right=1cm of n2]    {$n_{3}$};
            \node (ee) [right=0.75cm of n3] {\ldots};
            \node (nk) [right=0.75cm of ee] {$n_{k}$};
        
            \graph{
                (n1) -- [edge label={$e'$}] (nk);
                (nk) -> [edge label={$e_{k}$}] (ee) -> [edge label={$e_{m+1}$}] (n3) -> [edge label={$e_{m}$}] (n2);
            };
        \end{tikzpicture}
    }
    \caption{Composition transformations.}
    \label{fig-sig-composition}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

\item[Selection transformations] allow the creation and deletion of new nodes and edges as follows:
    \begin{enumerate}
        
        \item An existing node $n$ may be duplicated to create a new node $n'\!$. A trivial selection edge $e:n \LabelledEdge{\Bijective}{\RelRestrict} n'$ is added to enforce the restriction that $n'$ is a duplicate of $n$.
        
        \item A node $n_{1}$ may be deleted if it is attached to exactly one other node $n_{2}$ by a single trivial selection edge $e:n_{1} \LabelledEdge{\Bijective}{\RelRestrict} n_{2}$ (implying $n_{1}$ is a duplicate of $n_{2}$). This also implies deleting the attached trivial selection edge.
        
        \item A trivial selection edge $e:n \LabelledEdge{\Bijective}{\RelRestrict} n$ may be created that connects a node $n$ to itself.
        
        \item A trivial selection edge $e:n_{1} \LabelledEdge{\Bijective}{\RelRestrict} n_{2}$ may be deleted, removing the association between $n_{1}$ and $n_{2}$.
        
    \end{enumerate}
    
    If we can transform $\SC{1}$ into $\SC{2}$ using any of the first three selection transformations, then \Equivalent{\SC{2}}{\SC{1}}. If we can transform $\SC{1}$ into $\SC{2}$ using the fourth selection transformation, then \Dominates{\SC{2}}{\SC{1}}. The latter seems somewhat counter-intuitive \cite{Qian.X-1995a-Correct}, but is nonetheless correct \cite{Miller.R-1994a-Schema}. This is because a trivial selection edge $e:n_{1} \LabelledEdge{\Bijective}{\RelRestrict} n_{2}$ denotes that $n_{1}$ and $n_{2}$ correspond to the same domain, so removing the edge also removes this constraint.
    
\end{description}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Relational schemas as SIGs}
\label{sec-relational-sigs}

\noindent The direct correspondence between relational types and SIG domains (as noted in Section~\ref{sec-relations-types}) enables us to use SIGs to represent the structure and constraints of relational schemas. Equating tuple types to the cartesian product of their attributes (equation \ref{eqn-tuple-type}) means that these can also be mapped directly to a constructed SIG node.

Let us again consider the relvar $S\{\Sno, \Sname, \Status, \City\}$ from Date's suppliers-and-parts schema. In addition to type constraints on the attributes, tuples, and the relvar as a whole, $S\!$ is subject to a (primary) key constraint on \Sno. Taken together with the collection of types $\T{\Sno}$, $\T{\Sname}$, $\T{\Status}$, $\T{\City}$, $\TT{S}$, and $\T{S}$, we can construct the corresponding SIG shown in Figure~\ref{fig-sig-s-alone} as explained below (assuming that $S\!$ is the only relvar in the schema).

As noted in Section~\ref{sec-relations-types}, any given value of $S\!$ must comprise a subset of $\TT{S}$, so the association between $\T{S}$ and $\TT{S}$ is total. Similarly, every tuple in $\TT{S}$ must appear in at least one value of $S\!$, so the association is also surjective. The association is neither functional nor injective, however, as a value of $S\!$ in general may contain multiple tuples, and each tuple in $\TT{S}$ appears in multiple values of $S\!$. This gives us $\T{S}\SurTotal\TT{S}$.
    
The key constraint on \Sno\ implies that the functional dependency $\{\Sno\} \rightarrow \{\Sname, \Status, \City\}$ holds in $S\!$. From equation \ref{eqn-tuple-type}, we can infer the type of $\{\Sname, \Status, \City\}$ as $\TSSC$. The association is functional and not injective by definition. As not every supplier number may appear in any given value of $S\!$, there may be elements of $\T{\Sno}$ that are not associated with an element of $\TSSC$, and vice versa. This implies that the association is neither total nor surjective. This gives us $\T{\Sno} \KeyEdge \TSSC$ (we introduce the label $\mathit{key}$ to indicate that this edge represents a key constraint).
    
Both $\T{\Sno}$ and $\TSSC$ are projections of $\TT{S}$, so we can add (non-trivial) projection edges $\TT{S} \ProjectionEdge \T{\Sname} \times \T{\Status} \times T_{\City}$ and $\TT{S} \ProjectionEdge \T{\Sno}$. Similarly, $\T{\Sname}$, $\T{\Status}$, and $\T{\City}$ are each projections of $\TSSC$, so we can add further projection edges to these nodes, as shown in Figure~\ref{fig-sig-s-alone}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}[htb]
    \centering
    \begin{tikzpicture}[level distance=12mm]
        \node (TS) {$\T{S}$};
        \node (TTS) [right=1cm of TS] {$\TT{S}$}
            [sibling distance=3cm]
            child[arrows={:{crossbar}-{crossbar}>}] {node (TSno) {$\T{\Sno}$}}
            child[arrows={:{crossbar}-{crossbar}>}] {node (TSSC) {$\TSSC$}
                [sibling distance=15mm]
                child[arrows={:{crossbar}-{crossbar}>}] {node (TSname) {$\T{\Sname}$}}
                child[arrows={:{crossbar}-{crossbar}>}] {node (TStatus) {$\T{\Status}$}}
                child[arrows={:{crossbar}-{crossbar}>}] {node (TCity) {$\T{\City}$}}};
        
        % TS to T{S}
        \draw[arrows={:{crossbar}-{crossbar}:}] (TS) to (TTS);
        
        % FD
        \draw[arrows={->}] (TSno) to node[below=-2pt] {\scriptsize\emph{key}} (TSSC);
        
        % projection edges
        \ProjectionAnnotation[above left=-2pt and -4pt]{TTS}{TSno}
        \ProjectionAnnotation[above right=-2pt and -4pt]{TTS}{TSSC}

\ProjectionAnnotation[above left=-2pt and -4pt]{TSSC}{TSname}
        \ProjectionAnnotation[right=-2pt]{TSSC}{TStatus}
        \ProjectionAnnotation[above right=-2pt and -4pt]{TSSC}{TCity}
    \end{tikzpicture}
    \caption{SIG for relvar $\bm{S\{\Sno, \Sname, \Status, \City\}}$.}
    \label{fig-sig-s-alone}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

Looking at Figure~\ref{fig-sig-s-alone}, it could be argued that the $T_{\dots}$ type notation is redundant and unnecessarily complicates the diagram. Why not just use, e.g., $S\!$ instead of $\T{S}$, $\Sno$ instead of $\T{\Sno}$, and $\Sno \times \Sname \times \Status \times \City$ instead of $\TT{S}$? It is important to remember, however, that SIGs like that shown in Figure~\ref{fig-sig-s-alone} include multiple elements of quite different kinds, including relations, tuples, and various combinations of attributes. Discarding the type notation might be reasonable in informal situations, but could lead to confusion between the different kinds of elements represented. The explicit type notation protects the semantics of the various elements of the schema and makes it clear what is being considered.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Characterising view updates}
\label{sec-characterising}

\noindent In this section we show how SIGs can be used to characterise the information equivalence of view updates, using Date's motivating example of restriction views \cite{Date.C-2013a-View}.

Views and the base relvars they are derived from normally exist within the same ``base'' database schema (although this is not strictly required), which we shall refer to as $\SC{0}$. However, an end user may be granted access only to the views, and thus not be aware of the existence of the base relvar(s), or vice versa. This effectively partitions the schema into sub-schemas $\SC{1}, \SC{2}, \dotsc, \SC{n}$.

There are explicit and implicit constraints in $\SC{0}$ that should also apply to the sub-schemas. Ignoring these constraints in a sub-schema $\SC{k}$ will affect its information capacity, and could negatively impact the information equivalence of view updates involving $\SC{k}$. In the next section we will show how such constraints can be propagated into sub-schemas by rewriting them in terms of the sub-schema alone, which enables us to explicitly incorporate into the sub-schema the type information implicit in $\SC{0}$ that would otherwise be omitted. We will then discuss the construction of SIGs for Date's restriction view example (Section~\ref{sec-date-sigs}), and show how to transform and compare these (Section~\ref{sec-transforming}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Constraint derivation and propagation}
\label{sec-constraints}

\noindent Date's motivating example in Chapter 4 of \cite{Date.C-2013a-View} explores what is arguably the simplest case of view updating: disjoint restriction views. The example includes a base relvar $S\!$ and two views $\LS$ and $\NLS$ derived\footnote{Although as Date notes \cite{Date.C-2013a-View}, we could just as easily define $\LS$ and $\NLS$ as base relvars and then derive $S = \LS \RelUnion \NLS$, or even define all three as base relvars. The constraints will remain the same regardless.} from $S\!$:
\begin{eqnarray}
    \LS  & = & \RelRestrict_{\City = \mathit{'London'}}S \label{eqn-ls}   \\
    \NLS & = & \RelRestrict_{\City \ne \mathit{'London'}}S \label{eqn-nls}
\end{eqnarray}

Informally, \LS\ contains details of London suppliers, while \NLS\ contains details of non-London suppliers. $\LS$ and  $\NLS$ represent disjoint restrictions of $S\!$ with the same heading. They have the following tuple types:
\begin{eqnarray}
    \TT{\LS}  & = & \T{\Sno} \times \T{\Sname} \times \T{\LS} \times \CityLondon \nonumber    \\
    \TT{\NLS} & = & \T{\Sno} \times \T{\Sname} \times \T{\Status} \times (\TCityMinusLondon) \nonumber
\end{eqnarray}

The base schema for this example is $\SC{0} = \{S, \LS, \NLS\}$. This is subject to type constraints as specified by the various types noted previously, and the following additional constraints:
\begin{list}{\textbf{C$\bm{_{\arabic{constraint}}}$:}}{\usecounter{constraint}}

\item\label{constraint-key} \Sno\ is a key for each of $S\!$, \LS, and \NLS, therefore the functional dependency $\{\Sno\} \rightarrow \{\Sname, \Status, \City\}$ also holds in each of (a) $S\!$, (b) $\LS$, and (c) $\NLS$.
    
    % not sure how relevant the FK constraints are?
    % how to represent? FK is a subset of the original key, so selection edge between S.Sno and each of LS.Sno and NLS.Sno
    % or perhaps a selection edge from Sno to itself? (labelled FK)
    % Notation: \T{\Sno}, \T{\LS.\Sno}, \T{\NLS.\Sno}?
    \item\label{constraint-lsfk} $\RelProject_{\{\Sno\}}\LS \subseteq \RelProject_{\{\Sno\}}S$ [foreign key from $\LS$ to $S\!$]
    
    \item\label{constraint-nlsfk} $\RelProject_{\{\Sno\}}\NLS \subseteq \RelProject_{\{\Sno\}}S$ [foreign key from $\NLS$ to $S\!$]
    
    \item\label{constraint-union} $S = \LS \RelUnion \NLS$ [from (\ref{eqn-ls}) and (\ref{eqn-nls}) above]
    
    \item\label{constraint-disjoint} $\LS \RelIntersect \NLS = \emptyset$ [\LS\ and \NLS\ are disjoint]
    
    \item\label{constraint-notlondon} $\RelRestrict_{\City \ne \mathit{'London'}}\LS = \emptyset$ [from (\ref{eqn-ls}) above]
    
    \item\label{constraint-london} $\RelRestrict_{\City = \mathit{'London'}}\NLS = \emptyset$ [from (\ref{eqn-nls}) above]
    
    \item\label{constraint-relation-type-union} (a) $\T{S} = \T{\LS} \RelUnion \T{\NLS}$; (b) $\TT{S} = \TT{\LS} \RelUnion \TT{\NLS}$ [from C$_{\ref{constraint-union}}$]
    
    \item\label{constraint-relation-types} (a) $\T{\LS} \subset \T{S}$; (b) $\TT{\LS} \subset \TT{S}$ [from (\ref{eqn-ls}) above]
    
    \item\label{constraint-tuple-types} (a) $\T{\NLS} \subset \T{S}$; (b) $\TT{\NLS} \subset \TT{S}$ [from (\ref{eqn-nls}) above]
    
    \item\label{constraint-implied-types-london} $\CityLondon \subset \T{\City}$
    
    \item\label{constraint-implied-types-nonlondon} $(\TCityMinusLondon) \subset \T{\City}$
    
    \item\label{constraint-implied-types-TSSL} $\TSSL \subset \TSSC$
    
    \item\label{constraint-implied-types-TSSNL} $\TSSNL \subset \TSSC$
    
\end{list}

Constraints C$_{\ref{constraint-union}}$--C$_{\ref{constraint-london}}$ enforce that $\LS$ and $\NLS$ are disjoint restrictions of $S\!$, while C$_{\ref{constraint-relation-type-union}}$--C$_{\ref{constraint-implied-types-TSSNL}}$ are additional type constraints that specify specialisation associations implied by the subtypes and supertypes occurring in $\SC{0}$.

In most typical database schemas, only constraints C$_{\ref{constraint-key}}$, C$_{\ref{constraint-lsfk}}$, and C$_{\ref{constraint-nlsfk}}$ will be explicitly specified. Some database schemas may also explicitly specify constraints C$_{\ref{constraint-union}}$ and C$_{\ref{constraint-disjoint}}$, but it is more likely that these and all the remaining constraints will be implicit in the definition of $\SC{0}$. It is important for the purposes of this analysis to clearly identify all such implicit constraints, so that nothing is missed when constructing corresponding SIGs.

Let us partition $\SC{0}$ into three sub-schemas: $\SC{1} = \{S\}$, $\SC{2} = \{\LS, \NLS\}$, and $\SC{3} = \{LS\}$. We wish to compare the relative information capacities of the schema pairs $(\SC{1}, \SC{2})$ and $(\SC{1}, \SC{3})$, in order to determine whether the schemas in each pair are information equivalent. This will in turn characterise the view update compatibility of the two schemas.

Note that in the following analysis, we omit the foreign key constraints C$_{\ref{constraint-lsfk}}$ and C$_{\ref{constraint-nlsfk}}$, as we have yet to determine the best way to represent inclusion dependencies of this nature in a SIG, or indeed whether such dependencies are significant.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Schema $\SC{1}$}
\label{sec-constraints-s-i}

\noindent A user granted access only to $\SC{1}$ sees only relvar $S\!$. Constraint C$_{\ref{constraint-key}}$(a) can be propagated directly into $\SC{1}$ because it is expressed only in terms of $S\!$. Constraints C$_{\ref{constraint-implied-types-london}}$--C$_{\ref{constraint-implied-types-TSSNL}}$ do not mention any relvars and can thus also be directly propagated into $\SC{1}$. Constraints C$_{\ref{constraint-union}}$--C$_{\ref{constraint-tuple-types}}$ can be propagated to $\SC{1}$ by rewriting them only in terms of $S\!$. We can do this by substituting the definitions of $\LS$ (equation~\ref{eqn-ls}) and $\NLS$ (equation~\ref{eqn-nls}) into the constraints. For brevity, we shall denote these substitutions as $\LSsub$ and $\NLSsub$, so as to distinguish them from the actual $\LS$ and $\NLS$ relvars in $\SC{0}$, $\SC{2}$, and $\SC{3}$. The superscript ``1'' in the following denotes that these constraints have been rewritten in $\SC{1}$:
\begin{list}{\textbf{C$\bm{_{\arabic{constraint}}^{1}}$:}}{\usecounter{constraint}\setcounter{constraint}{3}}

\item $S = \LSsub \RelUnion \NLSsub$ (or even just $S = S$)
    
    \item $\LSsub \RelIntersect \NLSsub = \emptyset$

\item $\RelRestrict_{\City \ne \mathit{'London'}}\LSsub = \emptyset$
    
    \item $\RelRestrict_{\City = \mathit{'London'}}\NLSsub = \emptyset$
    
    \item (a) $\T{S} = \T{\LSsub} \RelUnion \T{\NLSsub}$; (b) $\TT{S} = \TT{\LSsub} \RelUnion \TT{\NLSsub}$ (or even just $\T{S} = \T{S}$ and $\TT{S} = \TT{S}$)
    
    \item (a) $\T{\LSsub} \subset \T{S}$; (b) $\TT{\LSsub} \subset \TT{S}$ [from (\ref{eqn-ls}) above]
    
    \item (a) $\T{\NLSsub} \subset \T{S}$; (b) $\TT{\NLSsub} \subset \TT{S}$ [from (\ref{eqn-nls}) above]
    
\end{list}
We can also rewrite C$_{\ref{constraint-key}}$(b) and C$_{\ref{constraint-key}}$(c) to C$_{\ref{constraint-key}}^{1}$(b) and C$_{\ref{constraint-key}}^{1}$(c), so that the functional dependency applies to $\LSsub$ and $\NLSsub$ instead of $\LS$ and $\NLS$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Schema $\SC{2}$}
\label{sec-constraints-s-ii}

\noindent A user granted access only to $\SC{2}$ sees only relvars $\LS$ and $\NLS$. Constraints C$_{\ref{constraint-key}}$(b),  C$_{\ref{constraint-key}}$(c), and C$_{\ref{constraint-disjoint}}$--C$_{\ref{constraint-london}}$ can be directly propagated into $\SC{2}$ as they are defined only in terms of $\LS$ and $\NLS$. C$_{\ref{constraint-implied-types-london}}$--C$_{\ref{constraint-implied-types-TSSNL}}$ can be directly propagated into $\SC{2}$ as they do not mention any relvars. C$_{\ref{constraint-union}}$, C$_{\ref{constraint-relation-type-union}}$, C$_{\ref{constraint-relation-types}}$, and C$_{\ref{constraint-tuple-types}}$ can be rewritten only in terms of $\LS$ and $\NLS$:
\begin{list}{\textbf{C$\bm{_{\arabic{constraint}}^{2}}$:}}{\usecounter{constraint}\setcounter{constraint}{3}}

\item $\LS \RelUnion \NLS = \LS \RelUnion \NLS$\setcounter{constraint}{7}
    
    \item (a) $\T{\LS} \RelUnion \T{\NLS} = \T{\LS} \RelUnion \T{\NLS}$; (b) $\TT{\LS} \RelUnion \TT{\NLS} = \TT{\LS} \RelUnion \TT{\NLS}$
    
    \item (a) $\T{\LS} \subset (\T{\LS} \RelUnion \T{\NLS})$; (b) $\TT{\LS} \subset (\TT{\LS} \RelUnion \TT{\NLS})$
    
    \item (a) $\T{\NLS} \subset \T{S}$; (b) $\TT{\NLS} \subset \TT{S}$
    
\end{list}
We can also rewrite C$_{\ref{constraint-key}}$(a) to C$_{\ref{constraint-key}}^{2}$(a), so that the functional dependency applies to $\LS \RelUnion \NLS$ instead of $S\!$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Schema $\SC{3}$}
\label{sec-constraints-s-iii}

\noindent A user granted access only to $\SC{3}$ sees only relvar $\LS$. Constraints C$_{\ref{constraint-key}}$(b) and C$_{\ref{constraint-notlondon}}$ can be directly propagated into $\SC{3}$ as they are defined only  in terms of $\LS$. C$_{\ref{constraint-implied-types-london}}$--C$_{\ref{constraint-implied-types-TSSNL}}$ can also be directly propagated as previously discussed. The remaining constraints cannot be rewritten only in terms of $\LS$, and therefore cannot be propagated to $\SC{3}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

\subsection{Constructing the SIGs}
\label{sec-date-sigs}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Schema $\SC{1}$}
\label{sec-sigs-s-i}

\noindent The SIG for $\SC{1}$ will not just be that for relvar $S\!$ (see Figure~\ref{fig-sig-s-alone}), as that schema did not incorporate the constraints implicit in $\SC{0}$. The process for constructing the SIG will be similar to that for the earlier example, however. The constraints appearing in $\SC{1}$ are C$_{\ref{constraint-key}}^{1}$, C$_{\ref{constraint-union}}^{1}$--C$_{\ref{constraint-tuple-types}}^{1}$, and C$_{\ref{constraint-implied-types-london}}$--C$_{\ref{constraint-implied-types-TSSNL}}$.

The SIG will include all of the edges that appear in Figure~\ref{fig-sig-s-alone}. $\T{S} \SurTotal \TT{S}$ implies $\T{\LSsub} \SurTotal \TT{\LSsub}$ and $\T{\NLSsub} \SurTotal \TT{\NLSsub}$. Constraints C$_{\ref{constraint-implied-types-london}}$--C$_{\ref{constraint-implied-types-TSSNL}}$ imply the existence of several subtype nodes, plus corresponding selection and projection edges:
\begin{itemize}
   
    \item $\TT{\LSsub} \ProjectionEdge \T{\Sno}$ and $\TT{\NLSsub} \ProjectionEdge \T{\Sno}$

\item $\TT{\LSsub} \ProjectionEdge \TSSL$

\item $\TT{\NLSsub} \ProjectionEdge \TSSNL$

\item $\TSSL \ProjectionEdge \T{\Sname}$

\item $\TSSL \ProjectionEdge \T{\Status}$

\item $\TSSL \ProjectionEdge\,\CityLondon$

\item $\TSSNL \ProjectionEdge \T{\Sname}$

\item $\TSSNL \ProjectionEdge \T{\Status}$

\item $\TSSNL \ProjectionEdge \T{\City} -$ \newline $\CityLondon$

\end{itemize}
C$_{\ref{constraint-key}}^{1}$(b) and C$_{\ref{constraint-key}}^{1}$(c) imply two additional key edges:
\begin{itemize}
   
    \item $\T{\Sno} \KeyEdge \TSSL$

\item $\T{\Sno} \KeyEdge \TSSNL$
   
\end{itemize}

C$_{\ref{constraint-relation-type-union}}^{1}$(a) implies node $\TLSPlusNLSsub$ to represent $\T{\LSsub} \RelUnion \T{\NLSsub}$. However, this node would be effectively identical to $\T{S}\!$, so we can ignore C$_{\ref{constraint-relation-type-union}}^{1}$(a) and just continue to use $\T{S}\!$. A similar argument applies for C$_{\ref{constraint-relation-type-union}}^{1}$(b) and $\TT{S}$.

C$_{\ref{constraint-relation-types}}^{1}$--C$_{\ref{constraint-implied-types-TSSNL}}^{1}$ imply the following selection edges:
\begin{itemize}

\item $\T{S} \SelectionEdge \T{\LSsub}$ and $\TT{S} \SelectionEdge \TT{\LSsub}$
    
    \item $\T{S} \SelectionEdge \T{\NLSsub}$ and $\TT{S} \SelectionEdge \TT{\NLSsub}$
    
    \item $\T{\City} \SelectionEdge\,\CityLondon$
    
    \item $\T{\City} \SelectionEdge\,(\TCityMinusLondon)$
    
    \item $\TSSC \SelectionEdge$ $\TSSL$
    
    \item $\TSSC \SelectionEdge \T{\Sname} \times \T{\Status} \times \T{\City} -$ \newline $\CityLondon$

\end{itemize}
The complete SIG for $\SC{1}$ is shown in Figure~\ref{fig-sig-s}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\T{\LSsub}$};   &[7mm]  \node (TTLS) {$\TT{\LSsub}$};     &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                            &                                           &                                   &   \node (TSname) {$\T{\Sname}$};    \\
            \node (TLSPlusNLS) {$\T{S}$}; &       \node (TTLSPlusNLS) {$\TT{S}$};   &   \node (TSno) {$\T{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\T{City}$};   \\
                                            &                                           &                                   &                                       &                                           &   \node (TStatus) {$\T{\Status}$};  \\
            \node (TNLS) {$\T{\NLSsub}$}; &       \node (TTNLS) {$\TT{\NLSsub}$};   &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) },
            };
            
            % FDs
            (TSno) -- [funcdep left,bend left] (TSSL);
            (TSno) -- [funcdep right] (TSSNL);
            
            % projection edges
            { [edges=projection left]
                (TTLSPlusNLS) -- (TSno),
                (TTLS)  -- { (TSno), (TSSL) },
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TTNLS) -- { (TSno), (TSSNL) },
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (TStatus), (NLSCity), (TSname) },
            };
            
            % selection edges
            { [edges=selection left]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) },
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) },
                (TCity) -- (LSCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
                (TTLSPlusNLS) -- [bend right] (TSSC),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
            (TTLSPlusNLS) -- [projection right,bend right] (TSSC),
        };
    \end{tikzpicture}
    \caption{SIG for schema $\bm{\SC{1} = \{S\}}$.}
    \label{fig-sig-s}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Schema $\SC{2}$}
\label{sec-sigs-s-ii}

\noindent The constraints appearing in $\SC{2}$ are C$_{\ref{constraint-key}}^{2}$, C$_{\ref{constraint-union}}^{2}$, C$_{\ref{constraint-disjoint}}$--C$_{\ref{constraint-london}}$, C$_{\ref{constraint-relation-type-union}}^{2}$--C$_{\ref{constraint-tuple-types}}^{2}$, and C$_{\ref{constraint-implied-types-london}}$--C$_{\ref{constraint-implied-types-TSSNL}}$. This is the same set of constraints, albeit with different rewritings, as $\SC{1}$. This means that the SIG for $\SC{2}$ can be built in essentially the same manner as for $\SC{1}$, with some nodes differently named. In particular, C$_{\ref{constraint-relation-type-union}}$ implies that there will be nodes $\TLSPlusNLS$ and $\TTLSPlusNLS$ instead of $\T{S}$ and $\TT{S}$. There will also be $\T{\LS}$ instead of $\T{\LSsub}$, etc. The complete SIG for $\SC{2}$ is shown in Figure~\ref{fig-sig-ls-nls}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\T{\LS}$};                  &[7mm]  \node (TTLS) {$\TT{\LS}$};                    &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                                        &                                                       &                                   &   \node (TSname) {$\T{\Sname}$};    \\
            \node (TLSPlusNLS) {$\StackTLSPlusNLS$};  &       \node (TTLSPlusNLS) {$\StackTTLSPlusNLS$};    &   \node (TSno) {$\T{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\T{City}$};   \\
                                                        &                                                       &                                   &                                       &                                           &   \node (TStatus) {$\T{\Status}$};  \\
            \node (TNLS) {$\T{\NLS}$};                &       \node (TTNLS) {$\TT{\NLS}$};                  &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) },
            };
            
            % FDs
            (TSno) -- [funcdep left,bend left] (TSSL);
            (TSno) -- [funcdep right] (TSSNL);
            
            % projection edges
            { [edges=projection left]
                (TTLSPlusNLS) -- (TSno),
                (TTLS)  -- { (TSno), (TSSL) },
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TTNLS) -- { (TSno), (TSSNL) },
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (TStatus), (NLSCity), (TSname) },
            };
            
            % selection edges
            { [edges=selection left]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) },
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) },
                (TCity) -- (LSCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
                (TTLSPlusNLS) -- [bend right] (TSSC),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
            (TTLSPlusNLS) -- [projection right,bend right] (TSSC),
        };
    \end{tikzpicture}
    \caption{SIG for schema $\bm{\SC{2} = \{\LS, \NLS\}}$.}
    \label{fig-sig-ls-nls}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Schema $\SC{3}$}
\label{sec-sigs-s-iii}

\noindent The constraints appearing in $\SC{3}$ are C$_{\ref{constraint-key}}$(b), C$_{\ref{constraint-notlondon}}$, and C$_{\ref{constraint-implied-types-london}}$--C$_{\ref{constraint-implied-types-TSSNL}}$. The SIG will only include edges and nodes that relate either directly to $\LS$ or are independent of any relvar, such as the subtype node $\TSSNL$. The complete SIG for $\SC{3}$ is shown in Figure~\ref{fig-sig-ls}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\T{\LS}$};  &[7.5mm]  \node (TTLS) {$\TT{\LS}$};  &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                        &                                       &                                   &   \node (TSname) {$\T{\Sname}$};    \\
                                        &                                       &   \node (TSno) {$\T{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\T{City}$};   \\
                                        &                                       &                                   &                                       &                                           &   \node (TStatus) {$\T{\Status}$};  \\
                                        &                                       &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TLS) -- (TTLS),
            };
            
            % projection edges
            { [edges=projection left]
                (TTLS)  -- { (TSno), (TSSL) },
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (TStatus), (NLSCity), (TSname) },
            };
            
            % selection edges
            { [edges=selection left]
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                (TCity) -- (LSCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
        };
    \end{tikzpicture}
    \caption{SIG for schema $\bm{\SC{3} = \{\LS\}}$.}
    \label{fig-sig-ls}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Transforming the SIGs}
\label{sec-transforming}

\noindent Now that we have constructed SIGs for all three schemas, we can compare the relative information capacities of the schema pairs $(\SC{1}, \SC{2})$ and $(\SC{1}, \SC{3})$. We do this by attempting to transform the SIG for one of the schemas until it is isomorphic with the other.

Looking at the SIGs for $\SC{1}$ and $\SC{2}$, we can see immediately that both SIGs have the same node and edge structure; indeed the only difference is in the names of some of the nodes. However, we already know from the discussion in Sections~\ref{sec-constraints-s-i} and~\ref{sec-constraints-s-ii} that these differently-named nodes represent the same domains. We therefore have an immediate isomorphism between the two SIGs and can conclude that the information capacities of $\SC{1}$ and $\SC{2}$ are identical, i.e., $\Equivalent{\SC{1}}{\SC{2}}$.

The comparison between $\SC{1}$ and $\SC{3}$ is more interesting. There are some shared structures, but the two SIGs are clearly different. Since the SIG transformations discussed in Section~\ref{sec-sig-compare} only ever preserve or enhance information capacity, intuitively we should attempt to transform the ``smaller'' SIG for $\SC{3}$ into the ``larger'' SIG for $\SC{1}$. SIG transformations can generally be carried out in two broad phases (using the current transformation as an example):
\begin{description}

\item[Duplicate nodes] in $\SC{3}$ (producing $\SC{3}'$) that correspond to supertypes in $\SC{1}$, to produce nodes corresponding to the extra nodes that $\SC{1}$ has over $\SC{3}$. The goal is to produce a node structure similar to that of $\SC{1}$. The duplicated nodes will be connected to their original nodes by trivial selection edges. These transformations (if applicable) will not alter the information capacity of $\SC{3}'$ (i.e., $\equiv$).
    
    \item[Copy and move edges] across the bijective paths connecting the new duplicated nodes in $\SC{3}'$. The goal is to produce an edge structure similar to that of $\SC{1}$. The paths that the edges are copied or moved across are bijective, so the annotations of the affected edges will remain unchanged. These transformations (if applicable) will \emph{increase} the information capacity of $\SC{3}'$ (i.e., $\preceq$).
    
\end{description}

These two phases generally occur in order, but may be interspersed throughout with the following:
\begin{description}

\item[Delete duplicate nodes] in $\SC{3}'$ that are no longer needed, especially those that do not exist in $\SC{1}$. These transformations (if applicable) will not alter the information capacity of $\SC{3}'$ ($\equiv$).
    
    \item[Remove totality annotations] from trivial selection edges in $\SC{3}'$ to change them into non-trivial selection edges. This decouples subtypes and supertypes to ensure that they represent different domains. These transformations (if applicable) will \emph{increase} the information capacity of $\SC{3}'$ ($\preceq$).
    
\end{description}
For example, we might duplicate some nodes, remove some totality annotations, move or copy some edges, delete some no longer needed nodes, move or copy some more edges, then remove some more totality annotations.

Following this process, we first duplicate node $\T{\LS}$ to $\T{S}$ and node $\TT{\LS}$ to $\TT{S}$. To obtain a node structure similar to that of $\SC{1}$, we also need to further duplicate $\T{S}$ to $\T{\NLS}$ and $\TT{S}$ to $\TT{\NLS}$. The result of these duplications is shown in Figure~\ref{fig-transform-all-duplicates}. The transformed SIG now has essentially the same node structure as that for $\SC{1}$, albeit with different node semantics.

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node [input keep] (TLS) {$\T{\LS}$};     &[7mm]  \node [input keep] (TTLS) {$\TT{\LS}$};   &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                                        &                                                   &                                   &   \node (TSname) {$\T{\Sname}$};    \\
            \node [output] (TLSPlusNLS) {$\T{S}$};    &       \node [output] (TTLSPlusNLS) {$\TT{S}$};  &   \node (TSno) {$\T{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\T{City}$};   \\
                                                        &                                                   &                                   &                                       &                                           &   \node (TStatus) {$\T{\Status}$};  \\
            \node [output] (TNLS) {$\T{\NLS}$};       &       \node [output] (TTNLS) {$\TT{\NLS}$};     &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TLS) -- (TTLS),
            };
            
            % projection edges
            { [edges=projection left]
                (TTLS)  -- { (TSno), (TSSL) },
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (TStatus), (NLSCity), (TSname) },
            };
            
            % selection edges
            { [edges=selection left]
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                (TCity) -- (LSCity),
            };
            
            { [edges={bijection,output},edge label={\scriptsize$\RelRestrict$}]
                { (TLS), (TTLS) } -- { (TLSPlusNLS), (TTLSPlusNLS) },
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) },
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
        };
    \end{tikzpicture}
    \caption{SIG for modified schema $\bm{S_{3}'}$ after duplicating various nodes to produce analogues of the additional nodes in $\bm{S_{1}}$. Inputs to the transformation are coloured \textcolor{blue}{blue}, outputs are coloured \textcolor{red}{red}. [$\equiv$]}
    \label{fig-transform-all-duplicates}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%

The next phase is to copy and move edges. To produce the edge $\T{S} \SurTotal \TT{S}$, we first \emph{copy} $\T{\LS} \SurTotal \TT{\LS}$ across the path $\TT{S} \TrivialSelection \TT{\LS}$ (giving $\T{\LS} \SurTotal \TT{S}$), then \emph{move} it across $\T{S} \TrivialSelection \T{\LS}$. We can carry out a similar series of transformations to copy and move $\T{S} \SurTotal \TT{S}$ to $\T{\NLS} \SurTotal \TT{\NLS}$.

Next, we copy and move the projection edge $\TT{\LS} \ProjectionEdge \T{\Sno}$ to produce the edges $\TT{S} \ProjectionEdge \T{\Sno}$ and $\TT{\NLS} \ProjectionEdge \T{\Sno}$. Similarly, we copy and move $\TT{\LS} \ProjectionEdge \TSSL$ to produce $\TT{S} \LabelledEdge{\sigedge{03}}{\RelProject} \TSSC$ and $\TT{\NLS} \LabelledEdge{\sigedge{02}}{\RelProject} \TSSNL$. Note that the first edge in the latter pair of transformations loses the totality annotation, and the second edge the surjectivity annotation, as a result of composition with the pre-existing non-bijective selection edges in the middle of the SIG.

Finally, we carry out similar copy and move transformations to copy the key edge $\T{\Sno} \KeyEdge \TSSC$ to $\T{\Sno} \KeyEdge \TSSL$ and $\T{\Sno} \KeyEdge \TSSNL$. The annotations for these edges are unaffected. The result of all these edge transformations is shown in Figure~\ref{fig-transform-edge-moves}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\T{\LS}$};      &[7mm]  \node (TTLS) {$\TT{\LS}$};        &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                            &                                           &                                   &   \node (TSname) {$\T{\Sname}$};    \\
            \node (TLSPlusNLS) {$\T{S}$}; &       \node (TTLSPlusNLS) {$\TT{S}$};   &   \node (TSno) {$\T{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\T{City}$};   \\
                                            &                                           &                                   &                                       &                                           &   \node (TStatus) {$\T{\Status}$};  \\
            \node (TNLS) {$\T{\NLS}$};    &       \node (TTNLS) {$\TT{\NLS}$};      &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TLS)        -- [input keep] (TTLS),
                (TLSPlusNLS) -- [output] (TTLSPlusNLS),
                (TNLS)       -- [output] (TTNLS),
            };
            
            % FDs
            (TSno) -- [funcdep left,bend left,output] (TSSL);
            (TSno) -- [funcdep right,output] (TSSNL);
            
            % projection edges
            { [edges=projection left]
                (TTLS)  -- [input keep] { (TSno), (TSSL) },
                (TSSL)  -- { (TStatus), (LSCity) },
                (TTLSPlusNLS) -- [output] (TSno),
            };
            
            { [edges=projection right]
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (TStatus), (NLSCity), (TSname) },
                (TTNLS)  -- [output] (TSno),
                (TTNLS)  -- [arrows={:->},output] (TSSNL),
            };
            
            % selection edges
            { [edges=selection left]
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                (TCity) -- (LSCity),
            };
            
            { [edges=bijection,edge label={\scriptsize$\RelRestrict$}]
                { (TLS), (TTLS) } -- { (TLSPlusNLS), (TTLSPlusNLS) },
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) },
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
                (TTLSPlusNLS) -- [bend right] (TSSC),
            };
            (TSno) -- [funcdep right,input keep] (TSSC);
            (TSSC) -- [projection left] (TCity);
            (TTLSPlusNLS) -- [projection right,arrows={:-{crossbar}>},bend right,output] (TSSC),
        };
    \end{tikzpicture}
    \caption{SIG for modified schema $\bm{S_{3}'}$ after copying and moving edges. [$\preceq$]}
    \label{fig-transform-edge-moves}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%

If we compare Figure~\ref{fig-sig-s} with Figure~\ref{fig-transform-edge-moves}, we can see that the transformed SIG for $\SC{3}'$ now has the same node and edge structure as the SIG for $\SC{1}$. We still need to remove the totality annotations from the four remaining trivial selection edges, but even when this is done, we do not achieve the desired isomorphism, as the projection edges $\TT{S} \LabelledEdge{\sigedge{03}}{\RelProject} \TSSC$ and $\TT{\NLS} \LabelledEdge{\sigedge{02}}{\RelProject} \TSSNL$ have different annotations from the corresponding edges in the SIG for $\SC{1}$. Technically this means that the information capacities of $\SC{1}$ and $\SC{3}$ are incomparable, but the structural differences are so small that it is probably safe to say that the information capacity of $\SC{3}$ is less than that of $\SC{1}$ (i.e., \Dominates{\SC{1}}{\SC{3}}). Either way, it is clear that view updates based on this pair of schemas will be problematic.

Incidentally, if we created a fourth sub-schema $\SC{4} = \{\NLS\}$ and compared this with $\SC{1}$, the result would be similar to $\SC{3}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusion \& future work}
\label{sec-conclusion}

\noindent In this paper we have shown how to use the concept of relative information capacity and the schema intension graph (SIG) formalism to characterise information equivalence with respect to view updating. The results of our analysis of Date's restriction view example \cite{Date.C-2013a-View} are consistent with his findings. The information capacities of $\SC{1} = \{S\}$ and $\SC{2} = \{\LS,\NLS\}$ are equivalent, so it should be possible to effectively propagate view updates between $\SC{1}$ and $\SC{2}$, regardless of the configuration of views and base relvars. Conversely, the information capacity of $\SC{3} = \{\LS\}$ is at best less than that of $\SC{1}$, implying that there are updates that cannot be propagated between $\SC{1}$ and $\SC{3}$.

Our analysis shows the importance of clearly identifying all explicit and implicit constraints that can be propagated from the base schema $\SC{0}$ to its sub-schemas. Omission of constraints could lead to a different SIG structure that might not be comparable, or might lead to an erroneous conclusion about a schema's information capacity relative to another.

There are several avenues for future work. While the discussion in this paper focused on the relational context, relative information capacity and the SIG formalism are data model agnostic, and thus should be applicable in non-relational view contexts (e.g., XML).

It is unclear at present whether the omission of inclusion dependencies such as foreign keys is significant. If such dependencies are significant, the question then is how best to represent them in the SIG formalism. It is also unclear at present whether copying or moving an edge across a bijective path should necessarily increase information capacity, as no annotations are lost in such a transformation. It may be that the structural change to the SIG implies an increase in information capacity regardless. Further investigation is required.

Finally, we have only examined disjoint restriction views in this paper, which are arguably the simplest case of view updating. Further work needs to be done to characterise other types of view configuration, such as projection views, join views, and union views. This will enable us to further refine the application of SIGs in this context.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\balance

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibliographystyle{abbrv}
\bibliography{VLDB2016_Stanger}

\end{document}

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Computer Journal}} @string{crpit = {Conferences in Research and Practice in Information Techology}} @string{database = {ACM SIGMIS Database}} @string{dbj = {Database Journal}} @string{dbms = {DBMS Magazine}} @string{dbpd = {Database Programming {\&} Design}} @string{ddj = {Dr. Dobb's Journal}} @string{develop = {{d}evelop, The Apple Technical Journal}} @string{directions = {Apple Directions}} @string{dke = {Data {\&} Knowledge Engineering}} @string{dlib = {D-Lib Magazine}} @string{ejis = {European Journal of Information Systems}} @string{fm = {First Monday}} @string{hbr = {Harvard Business Review}} @string{ibmjrd = {IBM Journal of Research and Development}} @string{ibmsj = {IBM Systems Journal}} @string{idt = {Internet Development Trends}} @string{ieeeahc = {IEEE Annals of the History of Computing}} @string{ieeec = {IEEE Computer}} @string{ieeedeb = {IEEE Data Engineering Bulletin}} @string{ieeeic = {IEEE Internet Computing}} @string{ieeeis = {IEEE Intelligent Systems}} @string{ieeepc = {IEEE Pervasive Computing}} @string{ieees = {IEEE Software}} @string{ieeesp = {IEEE Spectrum}} @string{ieeetnn = {IEEE Transactions on Neural Networks}} @string{ieeetse = {IEEE Transactions on Software Engineering}} @string{ijast = {International Journal of Applied Software Technology}} @string{ijcatr = {International Journal of Computer Applications Technology and Research}} @string{ijgis = {International Journal of Geographical Information Science}} @string{ijgisold = {International Journal of Geographical Information Systems}} @string{ijhcs = {International Journal of Human-Computer Studies}} @string{ijmms = {International Journal of Man-Machine Studies}} @string{ijseke = {International Journal of Software Engineering and Knowledge Engineering}} @string{ijswis = {International Journal on Semantic Web and Information Systems}} @string{ijwet = {International Journal of Web Engineering and Technology}} @string{ipm = {Information Processing and Management}} @string{is = {Information Systems}} @string{isedj = {Information Systems Education Journal}} @string{isj = {Information Systems Journal}} @string{ist = {Information and Software Technology}} @string{jacm = {Journal of the ACM}} @string{jasist = {Journal of the American Society for Information Science and Technology}} @string{jdim = {Journal of Digital Information Management}} @string{jdmm = {Journal of Digital Media Management}} @string{jiis = {Journal of Intelligent Information Systems}} @string{jlp = {Journal of Logic Programming}} @string{jodi = {Journal of Digital Information}} @string{jot = {Journal of Object Technology}} @string{jrpit = {Journal of Research and Practice in Information Technology}} @string{jss = {The Journal of Systems and Software}} @string{lncs = {Lecture Notes in Computer Science}} @string{misq = {MIS Quarterly}} @string{nzjc = {New Zealand Journal of Computing}} @string{nzjis = {New Zealand Journal of Information Systems}} @string{nzlimj = {New Zealand Library {\&} Information Management Journal}} @string{oracle = {Oracle Magazine}} @string{oss = {OCLC Systems {\&} Services: International Digital Library Perspectives}} @string{pcbi = {PLoS Computational Biology}} @string{pnas = {Proceedings of the National Academy of Sciences of the United States of America}} @string{pvldb = {Proceedings of the VLDB Endowment}} @string{sciam = {Scientific American}} @string{sej = {Software Engineering Journal}} @string{sigmetrics = {ACM SIGMETRICS Performance Evaluation Review}} @string{sigmod = {ACM SIGMOD Record}} @string{sigplan = {ACM SIGPLAN Notices}} @string{sigsoft = {ACM SIGSOFT Software Engineering Notes}} @string{spe = {Software---Practice and Experience}} @string{swj = {Semantic Web}} @string{tkde = {IEEE Transactions on Knowledge and Data Engineering}} @string{vldb = {The VLDB Journal}} @article{Bancilhon.F-1981a-Update, Author = {Fran{\c c}ois Bancilhon and Nicolas Spyratos}, Doi = {10.1145/319628.319634}, Journal = acmtods, Month = dec, Number = {4}, Pages = 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Navathe}, Isbn = {978-0-8053-0244-1}, Publisher = {Benjamin/Cummings}, Title = {Conceptual Database Design: An Entity-Relationship Approach}, Year = {1992}} @inproceedings{Behrend.A-2008a-Transformation-based, Author = {Andreas Behrend and Rainer Manthey}, Crossref = {Hartmann.S-2008a-FoIKs}, Doi = {10.1007/978-3-540-77684-0_18}, Pages = {253--271}, Title = {A transformation-based approach to view updating in stratifiable deductive databases}} @inproceedings{Caroprese.L-2012a-View-update, Author = {Luciano Caroprese and Irina Trubitsyna and Miros{\l}aw Truszczy{\'n}ski and Ester Zumpano}, Crossref = {delCerro.L-2012a-JELIA}, Doi = {10.1007/978-3-642-33353-8_11}, Pages = {134--146}, Title = {The view-update problem for indefinite databases}} @article{Chen.H-2011a-Survey, Author = {Haitao Chen and Husheng Liao}, Doi = {10.7763/IJCTE.2011.V3.278}, Journal = {International Journal of Computer Theory and Engineering}, Month = feb, Number = {1}, Pages = {1793--8201}, Title = {A survey to view update problem}, Url = {http://www.ijcte.org/papers/278-D038.pdf}, Volume = {3}, Year = {2011}} @article{Codd.E-1970a-Relational, Author = {Edgar F. Codd}, Doi = {10.1145/362384.362685}, Journal = cacm, Month = jun, Number = {6}, Pages = {377--387}, Title = {A relational model of data for large shared data banks}, Volume = {13}, Year = {1970}} @incollection{Codd.E-1972a-Completeness, Author = {Edgar F. Codd}, Crossref = {Rustin.R-1972a-Data}, Pages = {65--98}, Title = {Relational completeness of data base sublanguages}} @inproceedings{Codd.E-1974a-Recent, Author = {Edgar F. Codd}, Crossref = {Rosenfeld.J-1974a-IFIP}, Pages = {1017--1021}, Title = {Recent investigations in relational data base systems}} @article{Console.L-1995a-Abduction, Author = {Luca Console and Maria Luisa Sapino and Daniele Theseider Dupr{\'e}}, Doi = {10.1007/BF00961655}, Journal = jiis, Keywords = {deductive databases, view updates, abduction}, Month = may, Number = {3}, Pages = {261--280}, Title = {The role of abduction in database view updating}, Volume = {4}, Year = {1995}} @article{Cosmadakis.S-1984a-Updates, Author = {Stavros S. Cosmadakis and Christos H. Papadimitriou}, Doi = {10.1145/1634.1887}, Journal = jacm, Month = sep, Number = {4}, Pages = {742--760}, Title = {Updates of relational views}, Volume = {31}, Year = {1984}} @book{Date.C-2013a-View, Author = {Chris J. Date}, Booktitle = {View Updating \{\&} Relational Theory: Solving the View Update Problem}, Date-Added = {2016-01-25 02:14:14 +0000}, Date-Modified = {2016-01-25 02:15:43 +0000}, Isbn = {9781449357849}, Publisher = {O'Reilly}, Series = {Theory in Practice}, Title = {View Updating {\&} Relational Theory: Solving the View Update Problem}, Year = {2013}} @book{Date.C-2012a-SQL-and-Relational, Author = {Chris J. Date}, Booktitle = {SQL and Relational Theory: How to Write Accurate SQL Code}, Date-Added = {2016-01-25 02:12:09 +0000}, Date-Modified = {2016-01-25 02:16:01 +0000}, Edition = {second}, Isbn = {9781449316402}, Publisher = {O'Reilly}, Series = {Theory in Practice}, Title = {SQL and Relational Theory: How to Write Accurate SQL Code}, Year = {2012}} @article{Dayal.U-1982a-Correct, Author = {Umeshwar Dayal and Philip A. Bernstein}, Doi = {10.1145/319732.319740}, Journal = acmtods, Month = sep, Number = {3}, Pages = {381--416}, Title = {On the correct translation of update operations on relational views}, Volume = {7}, Year = {1982}} @article{Gottlob.G-1988a-Properties, Author = {Georg Gottlob and Paolo Paolini and Roberto Zicari}, Doi = {10.1145/49346.50068}, Journal = acmtods, Month = dec, Number = {4}, Pages = {486--524}, Title = {Properties and update semantics of consistent views}, Volume = {13}, Year = {1988}} @article{Hegner.S-2004a-Order-based, Author = {Stephen J. Hegner}, Doi = {10.1023/A:1026158013113}, Journal = {Annals of Mathematics and Artificial Intelligence}, Month = jan, Number = {1--2}, Pages = {63--125}, Title = {An order-based theory of updates for closed database views}, Volume = {40}, Year = {2004}} @article{Hull.R-1986a-Relative, Author = {Richard Hull}, Journal = {SIAM Journal on Computing}, Month = aug, Number = {3}, Pages = {856--886}, Title = {Relative information capacity of simple relational database schemata}, Volume = {15}, Year = {1986}} @inproceedings{Keller.A-1985a-Algorithms, Author = {Arthur Michael Keller}, Crossref = {PODS-1985a-Proceedings}, Doi = {10.1145/325405.325423}, Pages = {154--163}, Title = {Algorithms for translating view updates to database updates for views involving selections, projections, and joins}} @article{Keller.A-1986a-Role, Author = {Arthur Michael Keller}, Doi = {10.1109/MC.1986.1663034}, Journal = ieeec, Month = jan, Number = {1}, Pages = {63--73}, Title = {The role of semantics in translating view updates}, Volume = {19}, Year = {1986}} @article{Keller.A-1987a-Comment, Author = {Arthur Michael Keller}, Doi = {10.1145/27629.214296}, Journal = acmtods, Month = sep, Number = {3}, Pages = {521--523}, Title = {Comments on {B}ancilhon and {S}pyratos' ``{U}pdate semantics and relational views''}, Volume = {12}, Year = {1987}} @inproceedings{Kotidis.Y-2006a-Updates, Annote = {Article 2.}, Author = {Yannis Kotidis and Divesh Srivastava and Yannis Velegrakis}, Crossref = {Liu.L-2006a-ICDE}, Doi = {10.1109/ICDE.2006.167}, Pages = {2}, Read = {1}, Title = {Updates through views: {A} new hope}} @article{Laurent.D-2001a-Monotonic, Author = {Dominique Laurent and Jens Lechtenb{\"o}rger and Nicolas Spyratos and Gottfried Vossen}, Doi = {10.1007/s007780100055}, Journal = vldb, Month = dec, Number = {4}, Pages = {295--315}, Title = {Monotonic complements for independent data warehouses}, Volume = {10}, Year = {2001}} @inproceedings{Lechtenborger.J-2003a-Impact, Author = {Jens Lechtenb{\"o}rger}, Crossref = {Neven.F-2003a-PODS}, Doi = {10.1145/773153.773159}, Pages = {49--55}, Title = {The impact of the constant complement approach towards view updating}} @article{Lechtenborger.J-2003a-Computation, Author = {Jens Lechtenb{\"o}rger and Gottfried Vossen}, Doi = {10.1145/777943.777946}, Journal = acmtods, Month = jun, Number = {2}, Pages = {175--208}, Title = {On the computation of relational view complements}, Volume = {28}, Year = {2003}} @inproceedings{Liefke.H-2000a-View, Author = {Hartmut Liefke and Susan B. Davidson}, Crossref = {Kambayashi.Y-2000a-DaWaK}, Doi = {10.1007/3-540-44466-1_12}, Pages = {114--125}, Title = {View maintenance for hierarchical semistructured data}, Url = {http://repository.upenn.edu/cis_papers/125/}} @article{Mantri.R-2010a-Data, Author = {Rashmi Mantri and Junkang Feng}, Journal = {Nirma University Journal of Engineering and Technology}, Keywords = {information bearing capability, information content, propositional content, semantic conflicts, semantic content, digitalization, semantic theories of information}, Month = jul # {--} # dec, Number = {2}, Pages = {26--30}, Title = {Data semantic conflicts: {A}n approach based on the notion of `information bearing capability'}, Url = {http://nujet.org.in/index.php/nujet/article/view/96}, Volume = {1}, Year = {2010}} @inproceedings{Masunaga.Y-1984a-Relational, Author = {Yoshifumi Masunaga}, Crossref = {Dayal.U-1984a-VLDB}, Pages = {309--320}, Title = {A relational database view update translation mechanism}, Url = {http://www.vldb.org/conf/1984/P309.PDF}} @article{Mayol.E-2003a-Consistency, Author = {Enric Mayol and Ernest Teniente}, Doi = {10.1016/S0169-023X(03)00061-2}, Journal = dke, Keywords = {deductive databases, integrity constraints, updates, view updating, integrity constraint maintenance}, Month = oct, Number = {1}, Pages = {61--103}, Title = {Consistency preserving updates in deductive databases}, Volume = {47}, Year = {2003}} @inproceedings{Miller.R-1993a-Information, Author = {Ren{\'e}e Jean Miller and Y. E. Ioannidis and Raghu Ramakrishnan}, Crossref = {Agrawal.R-1993a-VLDB}, Pages = {120--133}, Title = {The use of information capacity in schema integration and translation}, Url = {http://www.vldb.org/conf/1993/P120.PDF}} @phdthesis{Miller.R-1994a-PhD, Address = {Madison, Wisconsin, USA}, Author = {Miller, Ren{\'e}e Jean}, School = {University of Wisconsin-Madison}, Title = {Managing Structural Heterogeneity}, Type = {{PhD} thesis}, Url = {http://www.cs.toronto.edu/~miller/papers/dissertation.ps}, Year = {1994}} @techreport{Miller.R-1994a-SIG, Address = {Madison, Wisconsin, USA}, Author = {Ren{\'e}e Jean Miller and Y. E. Ioannidis and Raghu Ramakrishnan}, Institution = {Department of Computer Sciences, University of Wisconsin-Madison}, Month = jan, Number = {CS-TR-94-1185}, Title = {Schema intension graphs: {A} formal model for the study of schema equivalence}, Type = {Technical report}, Url = {http://www.cs.toronto.edu/~miller/papers/MIR93c.ps}, Year = {1994}} @article{Miller.R-1994a-Schema, Author = {Ren{\'e}e Jean Miller and Y. E. Ioannidis and Raghu Ramakrishnan}, Doi = {10.1016/0306-4379(94)90024-8}, Journal = is, Month = jan, Number = {1}, Pages = {3--31}, Title = {Schema equivalence in heterogeneous systems: {B}ridging theory and practice}, Url = {http://www.cs.toronto.edu/~miller/papers/MIR94b.ps}, Volume = {19}, Year = {1994}} @article{Pan.W-1996a-Object, Author = {Wen-Wei Pan and Wei-Pang Yang}, Doi = {10.1016/S0020-0255(96)00125-9}, Journal = {Information Sciences}, Month = nov, Number = {1--2}, Pages = {29--48}, Title = {An object model at conceptual level to support updatable views on object-oriented databases}, Volume = {95}, Year = {1996}} @inproceedings{Scholl.M-1991a-Updatable, Author = {Marc H. Scholl and Christian Laasch and Markus Tresch}, Crossref = {Delobel.C-1991a-DOOD}, Doi = {10.1007/3-540-55015-1_10}, Keywords = {object-oriented databases, object model, object algebra, views, view updates}, Pages = {189--207}, Title = {Updatable views in object-oriented databases}, Url = {https://kops.uni-konstanz.de/bitstream/handle/123456789/6438/Updatable_Views_in_Object_Oriented_Databases_1991.pdf}} @article{Shu.H-2000a-Using, Author = {Hua Shu}, Doi = {10.1023/A:1008726123320}, Journal = jiis, Month = sep, Number = {2}, Pages = {147--173}, Title = {Using constraint satisfaction for view update}, Volume = {15}, Year = {2000}} @techreport{Qian.X-1995a-Correct, Address = {Menlo Park, California, USA}, Author = {Xiaolei Qian}, Institution = {Computer Science Laboratory, SRI International}, Month = jul, Number = {SRI-CSL-95-08}, Title = {Correct schema transformations}, Type = {Technical report}, Url = {http://www.csl.sri.com/reports/postscript/sri-csl-95-08.ps.gz}, Year = {1995}} @article{Wang.L-2006a-Updating, Author = {Ling Wang and Elke A. Rundensteiner and Murali Mani}, Doi = {10.1016/j.datak.2005.07.003}, Journal = dke, Keywords = {XML view, XQuery, update translatability}, Month = sep, Number = {3}, Pages = {236--298}, Title = {Updating {XML} views published over relational databases: {T}owards the existence of a correct update mapping}, Volume = {58}, Year = {2006}} @article{Wang.X-2005a-SIG, Author = {Xi Wang and Junkang Feng and Hongwei Xu}, Journal = {Computing and Information Systems}, Month = oct, Number = {3}, Pages = {article 9}, Title = {A study of the schema intension graphs model ({SIG}) approach to the relative information capacity ({RIC}) of data schemas}, Url = {http://cis.uws.ac.uk/research/journal/V9/V9N3/xiwang.doc}, Volume = {9}, Year = {2005}} @article{Xu.K-2009a-Defining, Author = {Kaibo Xu and Junkang Feng and Malcolm Crowe}, Doi = {10.1007/s10115-008-0129-3}, Journal = {Knowledge and Information Systems}, Keywords = {information content, reasoning, data semantics, semantic information theory, inference rules}, Month = jan, Number = {1}, Pages = {29--59}, Title = {Defining the notion of `information content' and reasoning about it in a database}, Volume = {18}, Year = {2009}} %% crossrefs @proceedings{Agrawal.R-1993a-VLDB, Address = {Dublin, Ireland}, Booktitle = {Proceedings of the Nineteenth International Conference on Very Large Data Bases (VLDB)}, Editor = {Rakesh Agrawal and Se{\'a}n Baker and David A. Bell}, Isbn = {978-1-55860-152-X}, Month = aug # {~24--27}, Publisher = {Morgan Kaufmann}, Title = {Proceedings of the Nineteenth International Conference on Very Large Data Bases (VLDB)}, Year = {1993}} @proceedings{Delobel.C-1991a-DOOD, Address = {Munich, Germany}, Booktitle = {Proceedings of the 2nd International Conference on Deductive and Object-Oriented Databases (DOOD'91)}, Doi = {10.1007/3-540-55015-1}, Editor = {Claude Delobel and Michael Kifer and Yoshifumi Masunaga}, Isbn = {978-3-540-55015-0}, Month = dec # {~16--18}, Publisher = {Springer}, Series = lncs, Title = {Proceedings of the 2nd International Conference on Deductive and Object-Oriented Databases (DOOD'91)}, Volume = {566}, Year = {1991}} @proceedings{Dayal.U-1984a-VLDB, Address = {Singapore}, Booktitle = {Proceedings of the Tenth International Conference on Very Large Data Bases (VLDB 1984)}, Editor = {Umeshwar Dayal and Gunter Schlageter and Lim Huat Seng}, Isbn = {0-934613-16-8}, Month = aug # {~27--31}, Publisher = {Morgan Kaufmann}, Title = {Proceedings of the Tenth International Conference on Very Large Data Bases (VLDB 1984)}, Year = {1984}} @proceedings{delCerro.L-2012a-JELIA, Address = {Toulouse, France}, Booktitle = {Proceedings of the Thirteenth European Conference on Logics in Artificial Intelligence (JELIA 2012)}, Doi = {10.1007/978-3-642-33353-8}, Editor = {Luis Fari{\~n}as del Cerro and Andreas Herzig and J{\'e}r{\^o}me Mengin}, Isbn = {978-3-642-33352-1}, Month = sep # {~26--28}, Series = lncs, Title = {Proceedings of the Thirteenth European Conference on Logics in Artificial Intelligence (JELIA 2012)}, Volume = {7519}, Year = {2012}} @proceedings{Hartmann.S-2008a-FoIKs, Address = {Pisa, Italy}, Booktitle = {Proceedings of the 5th International Symposium on Foundations of Information and Knowledge Systems (FoIKS 2008)}, Doi = {10.1007/978-3-540-77684-0}, Editor = {Sven Hartmann and Gabriele Kern-Isberner}, Isbn = {978-3-540-77683-3}, Month = feb # {~11--15}, Publisher = {Springer}, Series = lncs, Title = {Proceedings of the 5th International Symposium on Foundations of Information and Knowledge Systems (FoIKS 2008)}, Volume = {4932}, Year = {2008}} @proceedings{Kambayashi.Y-2000a-DaWaK, Address = {London, UK}, Booktitle = {Proceedings of the Second International Conference on Data Warehousing and Knowledge Discovery (DaWaK 2000)}, Doi = {10.1007/3-540-44466-1}, Editor = {Yahiko Kambayashi and Mukesh K. Mohania and A. Min Tjoa}, Isbn = {3-540-67980-4}, Month = sep # {~4--6}, Publisher = {Springer}, Series = lncs, Title = {Proceedings of the Second International Conference on Data Warehousing and Knowledge Discovery (DaWaK 2000)}, Volume = {1874}, Year = {2000}} @proceedings{Liu.L-2006a-ICDE, Address = {Atlanta, Georgia, USA}, Booktitle = {Proceedings of the 22nd International Conference on Data Engineering (ICDE 2006)}, Editor = {Ling Liu and Andreas Reuter and Kyu-Young Whang and Jianjun Zhang}, Isbn = {0-7695-2570-9}, Month = {3--8~} # apr, Publisher = {IEEE Computer Society}, Title = {Proceedings of the 22nd International Conference on Data Engineering (ICDE 2006)}, Year = {2006}} @proceedings{Neven.F-2003a-PODS, Address = {San Diego, California, USA}, Booktitle = {Proceedings of the Twenty-Second ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS 2003)}, Editor = {Frank Neven and Catriel Beeri and Tova Milo}, Isbn = {1-58113-670-6}, Month = jun # {~9--12}, Organization = {ACM}, Title = {Proceedings of the Twenty-Second ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS 2003)}, Year = {2003}} @proceedings{PODS-1985a-Proceedings, Address = {Portland, Oregon, USA}, Booktitle = {Proceedings of the Fourth ACM SIGACT-SIGMOD Symposium on Principles of Database Systems (PODS '85)}, Isbn = {0-89791-153-9}, Month = mar # {~25--27}, Publisher = {ACM}, Title = {Proceedings of the Fourth ACM SIGACT-SIGMOD Symposium on Principles of Database Systems (PODS '85)}, Year = {1985}} @proceedings{Rosenfeld.J-1974a-IFIP, Address = {Stockholm, Sweden}, Booktitle = {Proceedings of the IFIP Congress '74 (Information Processing '74)}, Editor = {Jack L. Rosenfeld}, Isbn = {978-0-7204-2803-3}, Month = aug # {~5--10}, Publisher = {North-Holland}, Title = {Proceedings of the IFIP Congress '74 (Information Processing '74)}, Year = {1974}} @book{Rustin.R-1972a-Data, Address = {Englewood Cliffs, New Jersey, USA}, Booktitle = {Data Base Systems}, Editor = {Randall Rustin}, Publisher = {Prentice-Hall}, Series = {Courant Computer Science Symposia Series 6}, Title = {Data Base Systems}, Year = {1972}}

Ignore Space Show notes View

APCCM_2017/APCCM2017_Stanger.bib 0 → 100644

@inproceedings{Behrend.A-2008a-Transformation-based,
	Author = {Andreas Behrend and Rainer Manthey},
	Crossref = {Hartmann.S-2008a-FoIKs},
	Doi = {10.1007/978-3-540-77684-0_18},
	Pages = {253--271},
	Title = {A transformation-based approach to view updating in stratifiable deductive databases}}

@inproceedings{Caroprese.L-2012a-View-update,
	Author = {Luciano Caroprese and Irina Trubitsyna and Miros{\l}aw Truszczy{\'n}ski and Ester Zumpano},
	Crossref = {delCerro.L-2012a-JELIA},
	Doi = {10.1007/978-3-642-33353-8_11},
	Pages = {134--146},
	Title = {The view-update problem for indefinite databases}}

@incollection{Codd.E-1972a-Completeness,
	Author = {Edgar F. Codd},
	Crossref = {Rustin.R-1972a-Data},
	Pages = {65--98},
	Title = {Relational completeness of data base sublanguages}}

@inproceedings{Codd.E-1974a-Recent,
	Author = {Edgar F. Codd},
	Crossref = {Rosenfeld.J-1974a-IFIP},
	Pages = {1017--1021},
	Title = {Recent investigations in relational data base systems}}

@article{Console.L-1995a-Abduction,
	Author = {Luca Console and Maria Luisa Sapino and Daniele Theseider Dupr{\'e}},
	Doi = {10.1007/BF00961655},
	Journal = jiis,
	Keywords = {deductive databases, view updates, abduction},
	Month = may,
	Number = {3},
	Pages = {261--280},
	Title = {The role of abduction in database view updating},
	Volume = {4},
	Year = {1995}}

@inproceedings{Liefke.H-2000a-View,
	Author = {Hartmut Liefke and Susan B. Davidson},
	Crossref = {Kambayashi.Y-2000a-DaWaK},
	Doi = {10.1007/3-540-44466-1_12},
	Pages = {114--125},
	Title = {View maintenance for hierarchical semistructured data},
	Url = {http://repository.upenn.edu/cis_papers/125/}}

@article{Mantri.R-2010a-Data,
	Author = {Rashmi Mantri and Junkang Feng},
	Journal = {Nirma University Journal of Engineering and Technology},
	Keywords = {information bearing capability, information content, propositional content, semantic conflicts, semantic content, digitalization, semantic theories of information},
	Month = jul # {--} # dec,
	Number = {2},
	Pages = {26--30},
	Title = {Data semantic conflicts: {A}n approach based on the notion of `information bearing capability'},
	Url = {http://nujet.org.in/index.php/nujet/article/view/96},
	Volume = {1},
	Year = {2010}}

@article{Mayol.E-2003a-Consistency,
	Author = {Enric Mayol and Ernest Teniente},
	Doi = {10.1016/S0169-023X(03)00061-2},
	Journal = dke,
	Keywords = {deductive databases, integrity constraints, updates, view updating, integrity constraint maintenance},
	Month = oct,
	Number = {1},
	Pages = {61--103},
	Title = {Consistency preserving updates in deductive databases},
	Volume = {47},
	Year = {2003}}

@article{Pan.W-1996a-Object,
	Author = {Wen-Wei Pan and Wei-Pang Yang},
	Doi = {10.1016/S0020-0255(96)00125-9},
	Journal = {Information Sciences},
	Month = nov,
	Number = {1--2},
	Pages = {29--48},
	Title = {An object model at conceptual level to support updatable views on object-oriented databases},
	Volume = {95},
	Year = {1996}}

@inproceedings{Scholl.M-1991a-Updatable,
	Author = {Marc H. Scholl and Christian Laasch and Markus Tresch},
	Crossref = {Delobel.C-1991a-DOOD},
	Doi = {10.1007/3-540-55015-1_10},
	Keywords = {object-oriented databases, object model, object algebra, views, view updates},
	Pages = {189--207},
	Title = {Updatable views in object-oriented databases},
	Url = {https://kops.uni-konstanz.de/bitstream/handle/123456789/6438/Updatable_Views_in_Object_Oriented_Databases_1991.pdf}}

@article{Wang.L-2006a-Updating,
	Author = {Ling Wang and Elke A. Rundensteiner and Murali Mani},
	Doi = {10.1016/j.datak.2005.07.003},
	Journal = dke,
	Keywords = {XML view, XQuery, update translatability},
	Month = sep,
	Number = {3},
	Pages = {236--298},
	Title = {Updating {XML} views published over relational databases: {T}owards the existence of a correct update mapping},
	Volume = {58},
	Year = {2006}}

@article{Wang.X-2005a-SIG,
	Author = {Xi Wang and Junkang Feng and Hongwei Xu},
	Journal = {Computing and Information Systems},
	Month = oct,
	Number = {3},
	Pages = {article 9},
	Title = {A study of the schema intension graphs model ({SIG}) approach to the relative information capacity ({RIC}) of data schemas},
	Url = {http://cis.uws.ac.uk/research/journal/V9/V9N3/xiwang.doc},
	Volume = {9},
	Year = {2005}}

@article{Xu.K-2009a-Defining,
	Author = {Kaibo Xu and Junkang Feng and Malcolm Crowe},
	Doi = {10.1007/s10115-008-0129-3},
	Journal = {Knowledge and Information Systems},
	Keywords = {information content, reasoning, data semantics, semantic information theory, inference rules},
	Month = jan,
	Number = {1},
	Pages = {29--59},
	Title = {Defining the notion of `information content' and reasoning about it in a database},
	Volume = {18},
	Year = {2009}}

%% crossrefs

@proceedings{Delobel.C-1991a-DOOD,
	Address = {Munich, Germany},
	Booktitle = {Proceedings of the 2nd International Conference on Deductive and Object-Oriented Databases (DOOD'91)},
	Doi = {10.1007/3-540-55015-1},
	Editor = {Claude Delobel and Michael Kifer and Yoshifumi Masunaga},
	Isbn = {978-3-540-55015-0},
	Month = dec # {~16--18},
	Publisher = {Springer},
	Series = lncs,
	Title = {Proceedings of the 2nd International Conference on Deductive and Object-Oriented Databases (DOOD'91)},
	Volume = {566},
	Year = {1991}}

@proceedings{delCerro.L-2012a-JELIA,
	Address = {Toulouse, France},
	Booktitle = {Proceedings of the Thirteenth European Conference on Logics in Artificial Intelligence (JELIA 2012)},
	Doi = {10.1007/978-3-642-33353-8},
	Editor = {Luis Fari{\~n}as del Cerro and Andreas Herzig and J{\'e}r{\^o}me Mengin},
	Isbn = {978-3-642-33352-1},
	Month = sep # {~26--28},
	Series = lncs,
	Title = {Proceedings of the Thirteenth European Conference on Logics in Artificial Intelligence (JELIA 2012)},
	Volume = {7519},
	Year = {2012}}

@proceedings{Hartmann.S-2008a-FoIKs,
	Address = {Pisa, Italy},
	Booktitle = {Proceedings of the 5th International Symposium on Foundations of Information and Knowledge Systems (FoIKS 2008)},
	Doi = {10.1007/978-3-540-77684-0},
	Editor = {Sven Hartmann and Gabriele Kern-Isberner},
	Isbn = {978-3-540-77683-3},
	Month = feb # {~11--15},
	Publisher = {Springer},
	Series = lncs,
	Title = {Proceedings of the 5th International Symposium on Foundations of Information and Knowledge Systems (FoIKS 2008)},
	Volume = {4932},
	Year = {2008}}

@proceedings{Kambayashi.Y-2000a-DaWaK,
	Address = {London, UK},
	Booktitle = {Proceedings of the Second International Conference on Data Warehousing and Knowledge Discovery (DaWaK 2000)},
	Doi = {10.1007/3-540-44466-1},
	Editor = {Yahiko Kambayashi and Mukesh K. Mohania and A. Min Tjoa},
	Isbn = {3-540-67980-4},
	Month = sep # {~4--6},
	Publisher = {Springer},
	Series = lncs,
	Title = {Proceedings of the Second International Conference on Data Warehousing and Knowledge Discovery (DaWaK 2000)},
	Volume = {1874},
	Year = {2000}}

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% APCCM2017_Stanger.tex % Author: Nigel Stanger % % Built using Tex Live 2016 distribution. % All packages except perhaps PGF should be in the standard TeX Live packages. % REQUIRES PGF v3.0 or later for diagrams (custom arrow tips). % REQUIRES TeX Gyre fonts (newtxmath package). % NOTE: the last two figures use a small amount of colour. % % Revisions: 22 Aug 2016 Initial submission. % 07 Nov 2016 Camera-ready version. %% %% MAX 10 PAGES (all material) %% \documentclass{sig-alternate-05-2015} \usepackage{balance} \usepackage{newtxtext} \usepackage{newtxmath} \usepackage{bm} \usepackage{subfig} \usepackage{booktabs} \usepackage{tikz} \usetikzlibrary{positioning} \usetikzlibrary{graphs} \usetikzlibrary{decorations.pathreplacing} \usetikzlibrary{calc} \usetikzlibrary{arrows} % Relational algebra operators \newcommand{\RelRestrict}{\ensuremath{\sigma}} \newcommand{\RelProject}{\ensuremath{\pi}} \newcommand{\RelUnion}{\ensuremath{\cup}} \newcommand{\RelIntersect}{\ensuremath{\cap}} % "empty" arrow tip \pgfarrowsdeclare{:}{:}{}{} % custom bar arrow tip, offset from end of line (use "empty" tip at line ends if no >) \tikzset{crossbar/.tip={\|[scale width=1.75,sep=0.25em]}} % various edge styles for SIG diagrams in TikZ \tikzset{ function/.style={arrows={->}}, injection/.style={arrows={<-}}, total/.style={arrows={:{crossbar}-}}, surjection/.style={arrows={-{crossbar}:}}, bijection/.style={arrows={<{crossbar}-{crossbar}>}}, projection/.style={arrows={:{crossbar}-{crossbar}>}}, projection left/.style={arrows={:{crossbar}-{crossbar}>},edge label={\scriptsize\(\RelProject\)}}, projection right/.style={arrows={:{crossbar}-{crossbar}>},edge label'={\scriptsize\(\RelProject\)}}, selection left/.style={arrows={<-{crossbar}>},edge label={\scriptsize\(\RelRestrict\)}}, selection right/.style={arrows={<-{crossbar}>},edge label'={\scriptsize\(\RelRestrict\)}}, funcdep left/.style={arrows={:{crossbar}-{crossbar}>},edge node={node[sloped,midway,above] {\scriptsize\emph{key}}}}, funcdep right/.style={arrows={:{crossbar}-{crossbar}>},edge node={node[sloped,midway,below] {\scriptsize\emph{key}}}}, surtotal/.style={arrows={:{crossbar}-{crossbar}:}}, input keep/.style={blue,thick}, input delete/.style={blue!40,thick,dashed}, output/.style={red,thick}, output temp/.style={red,thick,dashed}, path 1/.style={green!60!black,thick}, path 2/.style={orange,thick}, } % projection and selection edge annotations for TikZ \newcommand{\ProjectionAnnotation}[3][]{% \path (#2) to node[above,#1] {\scriptsize\(\RelProject\)} (#3);% } \newcommand{\SelectionAnnotation}[3][]{% \path (#2) to node[above,#1] {\scriptsize\(\RelRestrict\)} (#3);% } \newcounter{constraint} % misc % \newcommand{\todo}[1]{\textbf{!!TODO!!} {[#1]}} % commonly used elements \newcommand{\LS}{\ensuremath{\mathit{LS}}} \newcommand{\NLS}{\ensuremath{\mathit{NLS}}} \newcommand{\LSsub}{\ensuremath{\mathit{L}}} \newcommand{\NLSsub}{\ensuremath{\mathit{N}}} \newcommand{\Sno}{\ensuremath{\mathit{Sno}}} \newcommand{\Sname}{\ensuremath{\mathit{Sname}}} \newcommand{\Status}{\ensuremath{\mathit{Status}}} \newcommand{\City}{\ensuremath{\mathit{City}}} \newcommand{\Type}[1]{\ensuremath{T_{#1}}} \newcommand{\TT}[1]{\ensuremath{T_{\{#1\}}}} \newcommand{\CityLondon}{\ensuremath{\{\City\colon\allowbreak\text{`\emph{London}'}\}}} \newcommand{\TCityMinusLondon}{\ensuremath{\Type{\City} \setminus \CityLondon}} \newcommand{\TSSC}{\ensuremath{\Type{\Sname} \times \Type{\Status} \times \Type{\City}}} \newcommand{\TSSL}{\ensuremath{\Type{\Sname} \times \Type{\Status} \times \CityLondon}} \newcommand{\TSSNL}{\ensuremath{\Type{\Sname} \times \Type{\Status} \times (\TCityMinusLondon)}} \newcommand{\TLSPlusNLS}{\ensuremath{\Type{\LS} + \Type{\NLS}}} \newcommand{\TTLSPlusNLS}{\ensuremath{\TT{\LS} + \TT{\NLS}}} \newcommand{\TLSPlusNLSsub}{\ensuremath{\Type{\LSsub} + \Type{\NLSsub}}} \newcommand{\TTLSPlusNLSsub}{\ensuremath{\TT{\LSsub} + \TT{\NLSsub}}} \newcommand{\StackTLSPlusNLS}{\ensuremath{\begin{array}{c}\Type{\LS}\,+ \\ \Type{\NLS}\end{array}}} \newcommand{\StackTTLSPlusNLS}{\ensuremath{\begin{array}{c}\TT{\LS}\,+ \\ \TT{\NLS}\end{array}}} \newcommand{\StackTLSPlusNLSsub}{\ensuremath{\begin{array}{c}\Type{\LSsub}\,+ \\ \Type{\NLSsub}\end{array}}} \newcommand{\StackTTLSPlusNLSsub}{\ensuremath{\begin{array}{c}\TT{\LSsub}\,+ \\ \TT{\NLSsub}\end{array}}} \newcommand{\StackTSSC}{\ensuremath{\begin{array}{c}\Type{\Sname}\,\times \\ \Type{\Status} \times \Type{\City}\end{array}}} \newcommand{\StackTSSL}{\ensuremath{\begin{array}{c}\Type{\Sname} \times \Type{\Status}\,\times \\ \CityLondon\end{array}}} \newcommand{\StackTSSNL}{\ensuremath{\begin{array}{c}\Type{\Sname} \times \Type{\Status}\,\times \\ (\TCityMinusLondon)\end{array}}} \newcommand{\StackTCityMinusLondon}{\ensuremath{\begin{array}{c}\Type{\City}\,\setminus \\ \CityLondon\end{array}}} \newcommand{\SC}[1]{\ensuremath{\mathcal{S}_{#1}}} % dominates \newcommand{\Dominates}[2]{\ensuremath{#2 \preceq #1}} \newcommand{\Equivalent}[2]{\ensuremath{#1 \equiv #2}} % SIG notation (in text) \newcommand{\Sedge}[1]{\ensuremath{\sigma_{\textrm{#1}}}} \newcommand{\SedgeP}[1]{\ensuremath{\sigma_{\textrm{#1}}^{'}}} \newcommand{\medmid}{\raise.125ex\hbox{\scalebox{1}[0.75]{$\mid$}}} % General SIG edges for use in formulas. % Adapted from: http://tex.stackexchange.com/questions/96330/adding-symbols-at-the-ends-of-a-horizontal-line \makeatletter \newlength{\@annotskipleft} \newlength{\@annotskipright} % #1 = left edge component % #2 = right edge component % #3 = left bar annotation % #4 = right bar annotation \newcommand\@sig@edge[4]{% \let\@middle\joinrel% \ifx#1\relbar% \@annotskipleft=.3em% % scrunch up the bare line so it's similar length to \long...arrow \ifx#2\relbar\def\@middle{\joinrel\joinrel\relbar\joinrel\joinrel}\fi% \else% % arrows need a little more clearance \@annotskipleft=.4em% \fi% \ifx#2\relbar\@annotskipright=.3em\else\@annotskipright=.4em\fi% \mathrel{\ooalign{$#1\@middle#2$\cr\hskip\@annotskipleft$#3$\hfil$#4$\hskip\@annotskipright\cr}}% } % 0 = nothing % 1 = bar % 2 = arrowhead % 3 = both \def\@sigedge#1#2{% \ifcase\numexpr#14+#2\relax% \@sig@edge{\relbar}{\relbar}{}{}\or % 00 = ----- \@sig@edge{\relbar}{\relbar}{}{\medmid}\or % 01 = ---+- \@sig@edge{\relbar}{\rightarrow}{}{}\or % 02 = ----> \@sig@edge{\relbar}{\rightarrow}{}{\medmid}\or % 03 = ---+> \@sig@edge{\relbar}{\relbar}{\medmid}{}\or % 10 = -+--- \@sig@edge{\relbar}{\relbar}{\medmid}{\medmid}\or % 11 = -+-+- \@sig@edge{\relbar}{\rightarrow}{\medmid}{}\or % 12 = -+--> \@sig@edge{\relbar}{\rightarrow}{\medmid}{\medmid}\or % 13 = -+-+> \@sig@edge{\leftarrow}{\relbar}{}{}\or % 20 = <---- \@sig@edge{\leftarrow}{\relbar}{}{\medmid}\or % 21 = <--+- \@sig@edge{\leftarrow}{\rightarrow}{}{}\or % 22 = <---> \@sig@edge{\leftarrow}{\rightarrow}{}{\medmid}\or % 23 = <--+> \@sig@edge{\leftarrow}{\relbar}{\medmid}{}\or % 30 = <+--- \@sig@edge{\leftarrow}{\relbar}{\medmid}{\medmid}\or % 31 = <+-+- \@sig@edge{\leftarrow}{\rightarrow}{\medmid}{}\or % 32 = <+--> \@sig@edge{\leftarrow}{\rightarrow}{\medmid}{\medmid} % 33 = <+-+> \fi% } \newcommand{\sigedge}[1]{\ensuremath{\@sigedge#1}} \makeatother \newcommand{\Edge}{\sigedge{00}} \newcommand{\Total}{\sigedge{10}} \newcommand{\Surjective}{\sigedge{01}} \newcommand{\SurTotal}{\sigedge{11}} \newcommand{\Functional}{\sigedge{02}} \newcommand{\Injective}{\sigedge{20}} \newcommand{\Bijective}{\sigedge{33}} % Hack to get the overset symbols to appear at the right height. % \smash removes the spacing around the operator, hence \mathop. \newcommand{\LabelledEdge}[2]{\mathop{\overset{\raisebox{0.3em}{\scriptsize\ensuremath{#2}}}{\smash[t]{#1}}}} \newcommand{\ProjectionEdge}{\LabelledEdge{\sigedge{13}}{\RelProject}} \newcommand{\SelectionEdge}{\LabelledEdge{\sigedge{23}}{\RelRestrict}} \newcommand{\TrivialProjection}{\ensuremath{\LabelledEdge{\Bijective}{\RelProject}}} \newcommand{\TrivialSelection}{\ensuremath{\LabelledEdge{\Bijective}{\RelRestrict}}} \newcommand{\KeyEdge}{\ensuremath{\LabelledEdge{\sigedge{13}}{\mathit{key}}}} % Constraints. \newcommand{\Constraint}[2][]{C\ensuremath{_{#2}\ifx&#1&\else^{#1}\fi}} \newenvironment{ConstraintList}[1][]{% \begin{list}{% \bfseries% \ifx&#1&% \Constraint{\ensuremath{\bm{\arabic{constraint}}}}% \else% \Constraint[\ensuremath{\bm{#1}}]{\ensuremath{\bm{\arabic{constraint}}}}% \fi% }% {\usecounter{constraint}}% }{\end{list}} \hyphenation{co-pied sche-ma} \pagestyle{empty} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \CopyrightYear{2017} \setcopyright{acmlicensed} \conferenceinfo{ACSW '17,}{January 31-February 03, 2017, Geelong, Australia} \isbn{978-1-4503-4768-6/17/01}\acmPrice{\$15.00} \doi{http://dx.doi.org/10.1145/3014812.3014847} \title{Characterising relational view updates \\ using relative information capacity} \numberofauthors{1} \author{ \alignauthor Nigel Stanger \\ \affaddr{Department of Information Science, University of Otago}\\ \affaddr{Dunedin, New Zealand}\\ \email{nigel.stanger@otago.ac.nz} } \begin{CCSXML} <ccs2012> <concept> <concept_id>10002951.10002952.10002953.10002955</concept_id> <concept_desc>Information systems~Relational database model</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10002951.10002952.10003190.10003205</concept_id> <concept_desc>Information systems~Database views</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10003752.10010070.10010111.10010112</concept_id> <concept_desc>Theory of computation~Data modeling</concept_desc> <concept_significance>300</concept_significance> </concept> <concept> <concept_id>10003752.10010070.10010111.10011733</concept_id> <concept_desc>Theory of computation~Data integration</concept_desc> <concept_significance>300</concept_significance> </concept> </ccs2012> \end{CCSXML} \ccsdesc[500]{Information systems~Relational database model} \ccsdesc[500]{Information systems~Database views} \ccsdesc[300]{Theory of computation~Data modeling} \ccsdesc[300]{Theory of computation~Data integration} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} In 2013, Chris Date published a book about the view update problem in the context of the Relational Model. He presented several detailed examples of different varieties of view updates and characterised their behaviour, in the process deriving a set of principles for updating views based on the notion of \emph{information equivalence} of view schemas. In this paper we discuss work in progress that examines how Date's operational definition of information equivalence can be formally characterised using Hull's concept of \emph{relative information capacity}. As a proof of concept, we use an extension of Miller's \emph{schema intension graph} formalism to model the information equivalence of Date's restriction view example. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \keywords{view update problem, information equivalence, relative information capacity, schema intension graph, schema transformation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \noindent The view update problem is a long-standing problem of great practical significance in the database field. Date \cite{Date.C-2013a-View} likens it to significant unsolved problems in other fields such as the Riemann Hypothesis in mathematics or ``\(P = \mathit{NP}\)?'' in computational complexity theory. In his 2013 book \cite{Date.C-2013a-View} Date developed a set of principles for updating relational views, based on the notion of \emph{information equivalence}. However, that notion was defined more operationally than formally. View updating is a special case of schema translation: we have one schema comprising a set of views, another schema comprising a set of base relations, and we need to translate updates from one to the other.\footnote{These schemas may be actually separate, or more likely may be sub-schemas of a larger database schema, with effective separation achieved via the access privilege mechanism.} It therefore makes sense that formalisms developed to characterise schema translations are also applicable to view updates. We propose to use Hull's notion of \emph{relative information capacity} \cite{Hull.R-1986a-Relative} as a means to formally characterise the information equivalence of the schemas that make up a given view configuration. In particular, we develop an extension of Miller's \emph{schema intension graph} (SIG) formalism, which was originally designed to facilitate the comparison of relative information capacity across schemas. We first extend the SIG formalism with relevant relational concepts such as relation types, tuple types, and key constraints. We then show a proof of concept of how this extended formalism can be used to model the information equivalence of a view configuration, by applying it to Date's example of disjoint restriction views \cite{Date.C-2013a-View}. Our results are consistent with Date's operational characterisation of information equivalence for this example. In Section~\ref{sec-view-update-problem}, we briefly review the view update problem and relevant research, and in Section~\ref{sec-relations-types} we introduce some core concepts that are used throughout the remainder of the paper. In Section~\ref{sec-info-capacity} we discuss relative information capacity and the SIG formalism, then explain in Section~\ref{sec-relational-sigs} how to extend the SIG formalism to more precisely model a relational schema. In Section~\ref{sec-characterising} we analyse Date's restriction view example \cite{Date.C-2013a-View} in detail, and characterise the information equivalence of its various sub-schemas. We discuss conclusions and future work in Section~\ref{sec-conclusion}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The view update problem} \label{sec-view-update-problem} \noindent The view update problem was first discussed in the relational context by Codd in 1974 \cite{Codd.E-1974a-Recent}. When an update operation is applied to a view, it needs to be translated into corresponding updates on the base relations from which the view is derived \cite{Keller.A-1985a-Algorithms}. Sometimes there are multiple translations that can produce the desired view update, but produce different---possibly incompatible---changes in the base relations. Worse, there may be translations that have side effects (e.g., for some views based on joins), which produce a different result in the view than if the updates were applied directly to an equivalent base relation \cite{Keller.A-1985a-Algorithms}. The problem then is how to choose the most appropriate translation. As yet there has been no solution that completely and automatically resolves this ambiguity. There has been considerable research into this problem over the last four decades, and numerous different approaches have been suggested in the context of relational, non-relational (e.g., \cite{Liefke.H-2000a-View,Pan.W-1996a-Object,Scholl.M-1991a-Updatable,Wang.L-2006a-Updating}), and deductive databases (e.g., \cite{Behrend.A-2008a-Transformation-based,Caroprese.L-2012a-View-update,Console.L-1995a-Abduction,Mayol.E-2003a-Consistency}). We shall now briefly review some key developments in the relational context. We leave consideration of non-relational and deductive contexts for future work. Banchilhon and Spyratos proposed the concept of \emph{constant complement} \cite{Bancilhon.F-1981a-Update} in 1981. The \emph{complement} of a view describes all the information from the original database that does not appear in the view, so composing a view with its complement provides enough information to reconstruct the original database. Keeping the complement invariant under view update facilitates translation construction. This is a useful approach that has been developed further by other authors \cite{Hegner.S-2004a-Order-based,Lechtenborger.J-2003a-Impact}, but it has been shown that there are reasonable translations that cannot be carried out under constant complement \cite{Keller.A-1987a-Comment}. Computing the complement of a view is also difficult, and several authors have worked on more efficient algorithms \cite{Cosmadakis.S-1984a-Updates,Laurent.D-2001a-Monotonic,Lechtenborger.J-2003a-Computation}. Finally, this approach is purely structural in nature and does not consider the semantics of the schema, which is necessary to properly resolve translation ambiguity \cite{Keller.A-1986a-Role}. Gottlob et al.\ generalised the idea of constant complement into the class of \emph{consistent views} \cite{Gottlob.G-1988a-Properties}. They define consistent views as having the property that ``if the effect of a view update on a view state is determined, then the effect of the corresponding database update is unambiguously determined'' \cite{Gottlob.G-1988a-Properties}. However, there are still reasonable views that are not consistent under this definition, and this approach again does not consider schema semantics. Keller noted that semantic information is essential in order to choose an appropriate translation \cite{Keller.A-1986a-Role}, and suggested that the da\-ta\-base administrator (DBA) could provide this at view definition time \cite{Keller.A-1985a-Algorithms}. Masunaga argued, however, that this would be insufficient to completely resolve ambiguity, and suggested collecting further semantic information from end users at update time \cite{Masunaga.Y-1984a-Relational}. Shu \cite{Shu.H-2000a-Using} discussed an interesting approach to this kind of information gathering, by treating the problem of generating suitable view translations as a constraint satisfaction problem. This enabled constraints to be added incrementally at view update time. However, while collecting such information from the DBA seems reasonable, collecting it from the end user at update time could be seen as too burdensome, and could impact the usability of query interfaces. It is also unclear which information needs to be captured, and how best to capture it. Kotidis et al.\ \cite{Kotidis.Y-2006a-Updates} discussed a different approach to avoiding side effects in view translations. They proposed creating a materialised copy of the view that could be updated directly, rather than mapping updates from the (virtual) view onto its underlying base tables. Clearly this enables the view to be updated without translation side effects, but does imply that the physical manifestation (extension) of the view may no longer match its defining query (intension). Finally, Date has published several books on the theoretical foundations of the Relational Model over the last decade, including one specifically targeting the view update problem \cite{Date.C-2013a-View} (indeed, the book is ambitiously subtitled ``Solving the View Update Problem''). He developed a comprehensive set of rules for updating views in a relational context, supported by an array of worked examples. Date advocated explicitly stating all relevant constraints at view creation time, similar to the information gathering approaches discussed above \cite{Keller.A-1985a-Algorithms,Masunaga.Y-1984a-Relational,Shu.H-2000a-Using}. He argued that if a complete set of constraints was available, then ambiguity would be diminished or eliminated, and it would then be possible to automatically generate a set of appropriate \emph{compensatory rules} to automatically manage the view update process. (Compensatory rules are similar in principle to database triggers, except that they are intended to be declarative so that their definitions can made explicitly visible to end users.) Date's work is interesting and compelling, and raises several questions worthy of further investigation. However, there are still situations where the correct view translation may be ambiguous, e.g., views based on certain types of join, as previously noted \cite{Keller.A-1985a-Algorithms}. Somewhat disappointingly, given his stated goal of ``solving'' the view update problem, Date took a pragmatic approach in such cases. While his justification was well argued and logical, even he acknowledged that it involved ``a certain degree of pragma'' \cite{Date.C-2013a-View}. Date characterises the ability to define an appropriate and effective view translation using the notion of \emph{information equivalence}. Two schemas \(\SC{1}\) and \(\SC{2}\) are said to be information equivalent ``if and only if the constraints that apply to [\(\SC{1}\)] and [\(\SC{2}\)] are such that every proposition that can be represented by [\(\SC{1}\)] can be represented by [\(\SC{2}\)] and vice versa'' \cite{Date.C-2013a-View}. Date's information equivalence is a formal concept, but its definition and characterisation in \cite{Date.C-2013a-View} are more operational in nature. This is what we explore in this paper. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Relations and types} \label{sec-relations-types} \noindent We generally follow Date's \cite{Date.C-2012a-SQL-and-Relational} notation and terminology regarding the Relational Model. We do not use Date's Tutorial D notation, however, preferring instead a more classical algebraic style. In this paper, \(A\) represents an attribute (comprising a \emph{name}:\emph{type} pair), \(R\) represents a relation variable (\emph{relvar}), and \(v\) represents a value of an attribute, tuple, relation, etc. Attribute values may be disambiguated where necessary by prefixing the value with the attribute name, i.e., \(\mathit{name}\colon\mathit{value}\). Date defines a \emph{type} \(\Type{i}\) as ``essentially a \emph{named, finite set of values}'' \cite{Date.C-2012a-SQL-and-Relational}, representing every possible value of some specific kind, e.g., every possible ASCII string of length 5, every possible city name, or every possible relation value with a particular heading. That is: \begin{equation}\label{eqn-type} \Type{i} = \{v_{1}, v_{2}, \dotsc, v_{m}\}, 0 < m < k \end{equation} where \(k\) is some finite upper bound. Thus, the type of attribute \(A_{i}\) (denoted \(\Type{A_{i}}\)) is by definition the set of all possible values of \(A_{i}\). Consider a relvar \(R\) with heading \(\{A_{1}, A_{2}, \dotsc, A_{n}\}\). Following from equation~\ref{eqn-type}, the \emph{relation type} of \(R\), which we denote here as \(\Type{R}\), is the set of all possible relation values with \(R\)'s heading. Similarly, the \emph{tuple type} of \(R\)'s tuples, which we denote here as \TT{R}, is the set of all possible tuple values with \(R\)'s heading, which equates to the Cartesian product of \(R\)'s attribute types, i.e.: \begin{equation}\label{eqn-tuple-type} \TT{R} = \Type{A_{1}} \times \Type{A_{2}} \times\dotsb\times \Type{A_{n}}\text{.} \end{equation} (This construction can be applied to any tuple type, including projections of other tuple types.) By definition any given value of a relvar \(R\) must comprise a set (possibly empty) of tuples \(\{t_{1}, t_{2}, \dotsc, t_{m}\} \subseteq \TT{R}\). \(\Type{R}\) therefore comprises every possible \(k\)-combination of the elements of \TT{R}, for \(0 \le k \le \left\|\TT{R}\right\|\). For example, Date's usual suppliers-and-parts schema \cite{Date.C-2013a-View} includes a relvar \(S\) with heading \(\{\Sno, \Sname, \Status, \City\}\), which contains details of suppliers. From equation~\ref{eqn-tuple-type}, the tuples of \(S\) have type \(\TT{S} = \Type{\Sno} \times \TSSC\). This notion of type as a set of possible values is synonymous with Codd's original definition of domains in the Relational Model \cite{Codd.E-1970a-Relational,Date.C-2012a-SQL-and-Relational}, and also with the notion of domains as ``sets of values'' in schema intension graphs \cite{Miller.R-1994a-Schema,Miller.R-1994a-SIG}. The implications of the latter point will be explored further in Section~\ref{sec-relational-sigs}, after we have introduced relative information capacity and schema intension graphs in Section~\ref{sec-info-capacity}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Nulls} \noindent We also follow Date's prohibition of nulls. Philosophical objections aside, however, this makes little practical difference to our approach, as we are dealing exclusively with types rather than attributes anyway. While an attribute could be null, it makes little sense to say that a \emph{type} could be ``null''. It could be claimed that a type ``includes'' nulls, but it is unclear what this would even mean in practice, and runs the risk of treating nulls like ``normal'' values rather than the special markers that they are. In any case, this would effectively be no different from the situation without nulls, so we omit further consideration of nulls in the interest of clarity. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \newpage \section{Relative information capacity} \label{sec-info-capacity} \noindent A schema conveys information about the phenomenon it models. Hull \cite{Hull.R-1986a-Relative} defined the concept of \emph{relative information capacity} to describe the information content of different schemas. The relative information capacity of a schema determines the set of all valid instances of that schema \cite{Hull.R-1986a-Relative,Miller.R-1994a-PhD,Miller.R-1993a-Information}. Relative information capacity, as its name implies, is not a quantitative measure; rather it is an indicator of the relative information content of different schemas. Relative information capacity provides a means to identify the differences between schema instances, and can therefore be used as an aid to schema integration and translation \cite{Miller.R-1994a-PhD}. As noted earlier, view updating is a special case of this, so it follows that information capacity can also be used to characterise the information content of views compared to the base relvar(s) from which they are derived. Miller et al.\ \cite{Miller.R-1993a-Information} defined relative information capacity in terms of two types of mapping. An \emph{information (capacity) preserving mapping} from schema \(\SC{1}\) to schema \(\SC{2}\) maps every element of \(\SC{1}\) to an element of \(\SC{2}\), but only some elements of \(\SC{2}\) to elements of \(\SC{1}\). \(\SC{2}\) has an information capacity greater than \(\SC{1}\), and is said to \emph{dominate} \(\SC{1}\), denoted \Dominates{\SC{2}}{\SC{1}}. An \emph{equivalence preserving mapping} from \(\SC{1}\) to \(\SC{3}\) maps every element of \(\SC{1}\) to an element of \(\SC{3}\) and vice versa (i.e., both the mapping and its inverse are information preserving). The information capacity of \(\SC{1}\) is identical to \(\SC{3}\), and the schemas are said to be \emph{equivalent}, denoted \Equivalent{\SC{1}}{\SC{3}}. (Note that Miller et al.\ \cite{Miller.R-1994a-Schema} use stricter definitions of equivalence and dominance than those used here; strictly speaking, the discussion in this section should refer to \emph{absolute} dominance and equivalence, whereas the following discussion on SIG transformations should refer to \emph{internal} dominance and equivalence \cite{Miller.R-1994a-Schema}. The distinction, however, is not crucial to the discussion in this paper.) Relative information capacity can only be used comparatively. To this end, Miller et al.\ \cite{Miller.R-1994a-SIG} defined the \emph{schema intension graph} formalism as a tool for comparing the information capacity of schemas. A schema intension graph (SIG) describes the associations between domains in a schema, and provides a way to ``reason about constraints on \emph{collections of entities} in an instance of the entity set, rather than about the internal structure of a single entity'' \cite{Miller.R-1994a-SIG}. In particular, SIGs provide a useful abstract formalism within which to perform schema transformations, as transforming the SIG for a schema is effectively the same as transforming the schema itself. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Constructing SIGs} \label{sec-sig-build} \noindent A SIG is a graph \(G = (N, E)\) \cite{Miller.R-1994a-Schema}. The nodes \(n_{i} \in N\) represent typed domains (i.e., sets of values). Nodes may be \emph{simple} (atomic) or \emph{constructed} (composite). Simple nodes are denoted by the name of the domain that they represent. Constructed nodes represent a collection of domains and are built using the \(\times\) (Cartesian product) and \(+\) (union) operators. The edges \(e_{i} \in E\) represent binary relations (in the mathematical sense) between domains. Edges are denoted by lines with an optional label, and may be composed to form a \emph{path} between two nodes (the notation \(e_{2} \circ e_{1}\) means ``edge \(e_{1}\) followed by edge \(e_{2}\)''). Some edges may be designated as \emph{selection} edges, labelled \(\RelRestrict\), or \emph{projection} edges, labelled \(\RelProject\). These correspond to the relational algebra operators of the same names \cite{Codd.E-1972a-Completeness}. Edges may also be annotated with up to four properties denoting simple constraints on the binary relation represented by the edge: totality, surjectivity, functionality and injectivity. These properties are defined as follows \cite{Miller.R-1994a-Schema}: \begin{quotation} \noindent Let \(A\) and \(B\) be two sets. A binary relation \(r : A \sigedge{00} B\) is \emph{total} (denoted \(e : A \Total B\)) if it is defined for all elements of \(A\); \emph{surjective} (\(e : A \Surjective B\)) if it is defined for all elements of \(B\); \emph{functional} (\(e : A \Functional B\)) if an element of \(A\) determines at most one element of \(B\); and \emph{injective} (\(e : A \Injective B\)) if an element of \(B\) determines at most one element of \(A\). \end{quotation} A \emph{bijection} is a binary relation that has all four properties \cite{Miller.R-1994a-SIG}. The use of the term ``binary relation'' here, while mathematically correct, is potentially confusing in any discussion that also involves the Relational Model. From now on, we shall therefore refer to the binary relations represented by SIG edges as ``associations''. A selection edge \(e : A \LabelledEdge{\sigedge{00}}{\RelRestrict} B\) indicates that the elements of \(B\) comprise a subset or specialisation of the elements of \(A\). If \(B = A\), we have what we term a \emph{trivial} selection. Every element of B must appear in A, and vice versa, and every element of \(A\) is associated with exactly one element of \(B\), and vice versa. Trivial selection edges are therefore bijective, i.e., \(A \TrivialSelection B\). If \(B \subset A\), we have what we term a \emph{non-trivial} selection. Every element of \(B\) must appear in \(A\), but not vice versa. The mapping remains one-to-one, as before. From the perspective of the superset \(A\), non-trivial selection edges are therefore surjective, functional, and injective, i.e., \(A \SelectionEdge B\). A projection edge \(e : A \LabelledEdge{\sigedge{00}}{\RelProject} B\) indicates that the elements of \(B\) are a projection of \(A\), where \(A\) is a constructed node of the form \(C \times D \times \dotsb\) (i.e., a tuple type). A bijective projection edge \(A \TrivialProjection B\) indicates that \(B\) is unique with respect to \(A\). For non-unique projections, every element of \(A\) must match an element in \(B\), and vice versa. Every element of \(A\) is also associated with at most one element of \(B\), but not vice versa. Non-unique projection edges are therefore total, surjective, and functional, i.e., \(A \ProjectionEdge B\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Comparing SIGs} \label{sec-sig-compare} \noindent It is possible to determine the relationship between the relative information capacities of two schemas by comparing SIGs for the two schemas. Consider two schemas \(\SC{1}\) and \(\SC{2}\), with corresponding SIGs \(G_{\SC{1}}\) and \(G_{\SC{2}}\). Intuitively, if \(G_{\SC{1}}\) and \(G_{\SC{2}}\) are isomorphic, then \(\SC{1}\) and \(\SC{2}\) are equivalent \cite{Miller.R-1994a-SIG}. \(G_{\SC{1}}\) and \(G_{\SC{2}}\) are considered isomorphic if they have identical structure and semantics, i.e., the same nodes, edges, node types, constraints, and annotations. Miller et al.\ \cite{Miller.R-1994a-SIG} identify three possible results from comparing two SIGs \(G_{\SC{1}}\) and \(G_{\SC{2}}\): \begin{enumerate} \item \(G_{\SC{1}}\) and \(G_{\SC{2}}\) are either immediately isomorphic, or one can be manipulated by a series of equivalence preserving transformations (described below) until it is isomorphic to the other. The two schemas are equivalent, i.e., \(\SC{1} \equiv \SC{2}\). \item \(G_{\SC{1}}\) and \(G_{\SC{2}}\) are not immediately isomorphic, but one can be manipulated by a series of information capacity augmenting transformations (described below) until it is isomorphic to the other. The transformed schema is dominated by the other, e.g., if we transform \(G_{\SC{1}}\), then \(\Dominates{\SC{2}}{\SC{1}}\). \item \(G_{\SC{1}}\) and \(G_{\SC{2}}\) are not immediately isomorphic and cannot be transformed to be isomorphic. The information capacities of \(\SC{1}\) and \(\SC{2}\) cannot be compared. \end{enumerate} Miller et al.\ \cite{Miller.R-1994a-Schema} describe the following equivalence preserving and information capacity augmenting transformations on SIGs: \begin{description} \item[Annotation transformations] allow the removal of annotations or projection and selection constraints from edges (e.g., \(n_{1} \sigedge{23} n_{2} \Rightarrow n_{1} \sigedge{03} n_{2}\), or \(n_{3} \TrivialSelection n_{4} \Rightarrow n_{3} \Bijective n_{4}\)). If an annotation transformation on \(G_{\SC{1}}\) makes it isomorphic to \(G_{\SC{2}}\), then the information capacity of \(\SC{2}\) dominates that of \(\SC{1}\) (i.e., \Dominates{\SC{2}}{\SC{1}}). \item[Composition transformations] allow edges to be moved and copied around the graph. Consider an edge \(e : n_{1} \sigedge{00} n_{2}\), as shown in Figure~\ref{fig-sig-composition}\subref{fig-sig-composition-original}. If there is a path \(P = e_{m} \circ \dotsb \circ e_{k}\) of functional edges from some other node \(n_{k}\) to \(n_{2}\), \(e\) may be copied to a new edge \(e' : n_{1} \sigedge{00} n_{k}\), as shown in Figure~\ref{fig-sig-composition}\subref{fig-sig-composition-copy}. The original \(e\) can then be deleted if desired, effectively moving \(e\) to \(e'\) as shown in Figure~\ref{fig-sig-composition}\subref{fig-sig-composition-move}. The annotation of \(e'\) is determined by composing \(P\) with \(e\), i.e., \(e \circ P\). If an edge with a particular annotation (such as surjectivity) is composed with an edge without that annotation, the resulting edge also does not have the annotation. For example, the composition of the edges \(\Functional\) and \(\Surjective\) would be an edge with no annotations (\(\Edge\)), whereas the composition of \(\Functional\) and \(\sigedge{03}\) would be \(\Functional\). If a composition transformation on \(G_{\SC{1}}\) makes it isomorphic to \(G_{\SC{2}}\), then \Dominates{\SC{2}}{\SC{1}}. %%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] \centering \subfloat[Original state\label{fig-sig-composition-original}]{ \begin{tikzpicture} \node (n1) {\(n_{1}\)}; \node (n2) [below=1cm of n1] {\(n_{2}\)}; \node (n3) [left=1cm of n2] {\(n_{3}\)}; \node (ee) [left=0.75cm of n3] {\ldots}; \node (nk) [left=0.75cm of ee] {\(n_{k}\)}; \graph{ (n1) -- [edge label={\(e\)}] (n2); (nk) -> [edge label={\(e_{k}\)},swap] (ee) -> [edge label={\(e_{m+1}\)},swap] (n3) -> [edge label={\(e_{m}\)},swap] (n2); }; \node (Pleft) [above=2mm of nk.north east] {}; \node (Pright) [above=2mm of n2.north west] {}; \node (P) at ($(Pleft)!0.5!(Pright)$) {\(P\)}; \draw[decorate,decoration=brace] (nk.north east) -- (n2.north west); \end{tikzpicture} } \subfloat[Copy edge \(e\) to \(e'\) across path \(P\)\label{fig-sig-composition-copy}]{ \begin{tikzpicture} \node (n1) {\(n_{1}\)}; \node (n2) [below=1cm of n1] {\(n_{2}\)}; \node (n3) [left=1cm of n2] {\(n_{3}\)}; \node (ee) [left=0.75cm of n3] {\ldots}; \node (nk) [left=0.75cm of ee] {\(n_{k}\)}; \graph{ (n1) -- [edge label={\(e\)}] (n2); (n1) -- [edge label={\(e'\)},swap] (nk); (nk) -> [edge label={\(e_{k}\)},swap] (ee) -> [edge label={\(e_{m+1}\)},swap] (n3) -> [edge label={\(e_{m}\)},swap] (n2); }; \end{tikzpicture} } \subfloat[Move edge \(e\) to \(e'\) across path \(P\)\label{fig-sig-composition-move}]{ \begin{tikzpicture} \node (n1) {\(n_{1}\)}; \node (n2) [below=1cm of n1] {\(n_{2}\)}; \node (n3) [left=1cm of n2] {\(n_{3}\)}; \node (ee) [left=0.75cm of n3] {\ldots}; \node (nk) [left=0.75cm of ee] {\(n_{k}\)}; \graph{ (n1) -- [edge label={\(e'\)},swap] (nk); (nk) -> [edge label={\(e_{k}\)},swap] (ee) -> [edge label={\(e_{m+1}\)},swap] (n3) -> [edge label={\(e_{m}\)},swap] (n2); }; \end{tikzpicture} } \caption{Composition transformations.} \label{fig-sig-composition} \end{figure} %%%%%%%%%%%%%%%%%%%% \item[Selection transformations] allow the creation and deletion of new nodes and edges as follows: \begin{enumerate} \item An existing node \(n\) may be duplicated to create a new node \(n'\!\). A trivial selection edge \(e:n \TrivialSelection n'\) is added to enforce the constraint that \(n'\) is a duplicate of \(n\). \item A node \(n_{1}\) may be deleted if it is associated with exactly one other node \(n_{2}\) by a single trivial selection edge \(e:n_{1} \TrivialSelection n_{2}\) (implying \(n_{1}\) is a duplicate of \(n_{2}\)). This also deletes the trivial selection edge. \item A trivial selection edge \(e:n \TrivialSelection n\) may be created that connects a node \(n\) to itself. \item A trivial selection edge \(e:n_{1} \TrivialSelection n_{2}\) may be deleted, removing the association between \(n_{1}\) and \(n_{2}\). \end{enumerate} If we can make \(G_{\SC{1}}\) isomorphic to \(G_{\SC{2}}\) using any of the first three selection transformations, then \Equivalent{\SC{1}}{\SC{2}}. If we can make \(G_{\SC{1}}\) isomorphic to \(G_{\SC{2}}\) using the fourth selection transformation, then \Dominates{\SC{2}}{\SC{1}}. The latter may seem somewhat counter-intuitive \cite{Qian.X-1995a-Correct}, but is nonetheless correct \cite{Miller.R-1994a-Schema}. A trivial selection edge \(e:n_{1} \TrivialSelection n_{2}\) denotes that \(n_{1}\) and \(n_{2}\) correspond to the same domain, so removing the edge also removes this constraint. The information capacity of \(\SC{2}\) must therefore be greater than \(\SC{1}\). \end{description} These SIG transformations and their effects are summarised in Table~\ref{Tab.SIG.Transformations}. %%%%%%%%%%%%%%%%%%%% \begin{table}[htb] \centering \small \begin{tabular}{lcc} \toprule & & \textbf{Effect on infor-} \\ \textbf{Transformation} & \textbf{Type} & \textbf{mation capacity} \\ \midrule Delete annotation or \(\RelProject\)/\(\RelRestrict\) constraint & annotation & augmented \\ Move or copy edge & composition & augmented \\ Create or delete duplicate node & selection & no change \\ Create trivial selection edge & selection & no change \\ Delete trivial selection edge & selection & augmented \\ \bottomrule \end{tabular} \caption{Summary of SIG transformations.} \label{Tab.SIG.Transformations} \end{table} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Extending SIGs to model relational concepts} \label{sec-relational-sigs} \noindent The direct correspondence between relational types and SIG domains noted in Section~\ref{sec-relations-types} enables us to use SIGs to represent the structure and constraints of relational schemas. Equating tuple types to the Cartesian product of their attributes (equation \ref{eqn-tuple-type}) means that these can also be mapped directly to a constructed SIG node. Let us again consider the relvar \(S\{\Sno, \Sname, \Status, \City\}\) from Date's suppliers-and-parts schema \cite{Date.C-2013a-View}. In addition to type constraints on the attributes, tuples, and the relvar as a whole, \(S\) is subject to a (primary) key constraint on \Sno. Taken together with the collection of types \(\Type{\Sno}\), \(\Type{\Sname}\), \(\Type{\Status}\), \(\Type{\City}\), \(\TT{S}\), and \(\Type{S}\), and assuming that \(S\) is the only relvar in the schema, we can construct the SIG shown in Figure~\ref{fig-sig-s-alone} as explained below. %%%%%%%%%%%%%%%%%%%% \begin{figure}[htb] \centering \begin{tikzpicture}[level distance=12mm] \node (TS) {\(\Type{S}\)}; \node (TTS) [right=1cm of TS] {\(\TT{S}\)} [sibling distance=3cm] child[arrows={<{crossbar}-{crossbar}>}] {node (TSno) {\(\Type{\Sno}\)}} child[arrows={:{crossbar}-{crossbar}>}] {node (TSSC) {\(\TSSC\)} [sibling distance=15mm] child[arrows={:{crossbar}-{crossbar}>}] {node (TSname) {\(\Type{\Sname}\)}} child[arrows={:{crossbar}-{crossbar}>}] {node (TStatus) {\(\Type{\Status}\)}} child[arrows={:{crossbar}-{crossbar}>}] {node (TCity) {\(\Type{\City}\)}}}; % TS to T{S} \draw[arrows={:{crossbar}-{crossbar}:}] (TS) to (TTS); % FD \draw[arrows={:{crossbar}-{crossbar}>}] (TSno) to node[below=-2pt] {\scriptsize\emph{key}} (TSSC); % projection edges \ProjectionAnnotation[above left=-2pt and -4pt]{TTS}{TSno} \ProjectionAnnotation[above right=-2pt and -4pt]{TTS}{TSSC} \ProjectionAnnotation[above left=-2pt and -4pt]{TSSC}{TSname} \ProjectionAnnotation[right=-2pt]{TSSC}{TStatus} \ProjectionAnnotation[above right=-2pt and -4pt]{TSSC}{TCity} \end{tikzpicture} \caption{SIG for relvar \(S\{\Sno, \Sname, \Status, \City\}\).} \label{fig-sig-s-alone} \end{figure} %%%%%%%%%%%%%%%%%%%% As discussed in Section~\ref{sec-relations-types}, any given value of \(S\) must comprise a subset of \(\TT{S}\), so the association between \(\Type{S}\) and \(\TT{S}\) is total. Similarly, every tuple in \(\TT{S}\) must appear in at least one value of \(S\), so the association is also surjective. The association is neither functional nor injective, however, as a value of \(S\) in general may contain multiple tuples, and each tuple in \(\TT{S}\) can appear in multiple values of \(S\). This gives us \(\Type{S}\SurTotal\TT{S}\). The key constraint on \Sno\ implies that the functional dependency \(\{\Sno\} \rightarrow \{\Sname, \Status, \City\}\) holds in \(S\). From equation \ref{eqn-tuple-type}, we can infer the type of \(\{\Sname, \Status, \City\}\) as \(\TSSC\). The association is functional and not injective by definition. Every supplier number (when used in a value of \(S\)) must be associated with an element of \(\TSSC\), and vice versa. This implies that the association is both total and surjective. Indeed, this holds for every edge that represents a key constraint \cite{Miller.R-1994a-SIG}. We introduce the edge label \(\mathit{key}\) to denote an edge representing a key constraint, which gives us \(\Type{\Sno} \KeyEdge \TSSC\) in Figure~\ref{fig-sig-s-alone}. Both \(\Type{\Sno}\) and \(\TSSC\) are projections of \(\TT{S}\), so we can add a non-unique projection edge \(\TT{S} \ProjectionEdge \Type{\Sname} \times \Type{\Status} \times T_{\City}\) and a unique projection edge \(\TT{S} \TrivialProjection \Type{\Sno}\). Similarly, \(\Type{\Sname}\), \(\Type{\Status}\), and \(\Type{\City}\) are all projections of \(\TSSC\), so we can add further non-unique projection edges to these nodes, as shown in Figure~\ref{fig-sig-s-alone}. Looking at Figure~\ref{fig-sig-s-alone}, it could be argued that the \(T_{\dots}\) type notation is redundant and unnecessarily complicates the diagram. Why not just use, e.g., \(S\) instead of \(\Type{S}\), \(\Sno\) instead of \(\Type{\Sno}\), and \(\Sno \times \Sname \times \Status \times \City\) instead of \(\TT{S}\)? It is important to remember, however, that SIGs like that shown in Figure~\ref{fig-sig-s-alone} refer to multiple elements of quite different kinds, including relations, tuples, and various combinations of attributes. Discarding the type notation might be reasonable in informal situations, but could lead to confusion between the different kinds of elements represented. The explicit type notation protects the semantics of the various elements of the schema and makes it clear what is being represented. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Characterising view updates} \label{sec-characterising} \noindent In this section we show how the extended SIG formalism can be used to model the information equivalence of relational view updates, using Date's restriction view example as a proof of concept \cite{Date.C-2013a-View}. Views and the base relvars they are derived from normally exist within the same ``base'' database schema (although this is not strictly required), which we shall refer to as \(\SC{0}\). However, an end user may be granted access only to the views, and thus not be aware of the existence of the base relvar(s), or vice versa. This effectively partitions the schema into logical sub-schemas \(\SC{1}, \SC{2}, \dotsc, \SC{n}\). There are explicit and implicit constraints in \(\SC{0}\) that should also apply to the sub-schemas \cite{Date.C-2013a-View}. Ignoring these constraints in a sub-schema \(\SC{k}\) will affect its information capacity, and could negatively impact the information equivalence of view updates involving \(\SC{k}\). In the next section we will show how such constraints can be propagated into sub-schemas by rewriting them in terms of the sub-schema alone, which enables us to explicitly incorporate into the sub-schema semantic information implicit in \(\SC{0}\) that would otherwise be lost. We will then discuss the construction of SIGs for Date's restriction view example (Section~\ref{sec-date-sigs}), and show how to transform and compare these (Section~\ref{sec-transforming}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Constraint derivation and propagation} \label{sec-constraints} \noindent Date's motivating example in Chapter 4 of \cite{Date.C-2013a-View} explores what is arguably the simplest case of view updating: disjoint restriction views. The example includes a base relvar \(S\) and two views \(\LS\) and \(\NLS\) derived\footnote{Although as Date notes \cite{Date.C-2013a-View}, we could just as easily define \(\LS\) and \(\NLS\) as base relvars and then derive \(S = \LS \RelUnion \NLS\), or even define all three as base relvars. The constraints remain the same regardless.} from \(S\): \begin{align} \LS &= \RelRestrict_{\City = \mathit{'London'}}S \label{eqn-ls} \\ \NLS &= \RelRestrict_{\City \ne \mathit{'London'}}S \label{eqn-nls} \end{align} Informally, \(\LS\) and \(\NLS\) contain details of London and non-London suppliers, respectively. They represent disjoint restrictions of \(S\) with the same heading, and have the following tuple types: \begin{align} \TT{\LS} &= \Type{\Sno} \times \Type{\Sname} \times \Type{\Status} \times \CityLondon \nonumber \\ \TT{\NLS} &= \Type{\Sno} \times \Type{\Sname} \times \Type{\Status} \times (\TCityMinusLondon) \nonumber \end{align} The base schema for this example is \(\SC{0} = \{S, \LS, \NLS\}\). This is subject to type constraints as specified by the various types noted previously, and the following additional constraints: \begin{ConstraintList} \item\label{constraint-key} \Sno\ is a key for each of \(S\), \(\LS\), and \(\NLS\), therefore the functional dependency \(\{\Sno\} \rightarrow \{\Sname, \Status, \City\}\) also holds in each of (a) \(S\); (b) \(\LS\); and (c) \(\NLS\). % not sure how relevant the FK constraints are? % how to represent? FK is a subset of the original key, so selection edge between S.Sno and each of LS.Sno and NLS.Sno % or perhaps a selection edge from Sno to itself? (labelled FK) % Notation: \Type{\Sno}, \Type{\LS.\Sno}, \Type{\NLS.\Sno}? \item\label{constraint-lsfk} \(\RelProject_{\{\Sno\}}\LS \subseteq \RelProject_{\{\Sno\}}S\) [foreign key from \(\LS\) to \(S\)] \item\label{constraint-nlsfk} \(\RelProject_{\{\Sno\}}\NLS \subseteq \RelProject_{\{\Sno\}}S\) [foreign key from \(\NLS\) to \(S\)] \item\label{constraint-union} \(S = \LS \RelUnion \NLS\) [from (\ref{eqn-ls}) and (\ref{eqn-nls}) above] \item\label{constraint-notlondon} \(\RelRestrict_{\City \ne \mathit{'London'}}\LS = \emptyset\) [from (\ref{eqn-ls}) above] \item\label{constraint-london} \(\RelRestrict_{\City = \mathit{'London'}}\NLS = \emptyset\) [from (\ref{eqn-nls}) above] \item\label{constraint-disjoint} \(\LS \RelIntersect \NLS = \emptyset\) [from \Constraint{\ref{constraint-notlondon}} and \Constraint{\ref{constraint-london}}] \item\label{constraint-relation-type-union} (a) \(\Type{S} = \Type{\LS} \RelUnion \Type{\NLS}\); (b) \(\TT{S} = \TT{\LS} \RelUnion \TT{\NLS}\) [from \Constraint{\ref{constraint-union}}] \item\label{constraint-relation-types} (a) \(\Type{\LS} \subset \Type{S}\); (b) \(\TT{\LS} \subset \TT{S}\) [from (\ref{eqn-ls}) above] \item\label{constraint-tuple-types} (a) \(\Type{\NLS} \subset \Type{S}\); (b) \(\TT{\NLS} \subset \TT{S}\) [from (\ref{eqn-nls}) above] \item\label{constraint-implied-types-london} \(\CityLondon \subset \Type{\City}\) \item\label{constraint-implied-types-nonlondon} \((\TCityMinusLondon) \subset \Type{\City}\) \item\label{constraint-implied-types-TSSL} \(\TSSL \subset \TSSC\) \item\label{constraint-implied-types-TSSNL} \(\TSSNL \subset \TSSC\) \end{ConstraintList} Constraints \Constraint{\ref{constraint-union}}--\Constraint{\ref{constraint-disjoint}} enforce that \(\LS\) and \(\NLS\) are disjoint restrictions of \(S\), while \Constraint{\ref{constraint-relation-type-union}}--\Constraint{\ref{constraint-implied-types-TSSNL}} are additional type constraints that enforce specialisation associations implied by the subtypes and supertypes occurring in \(\SC{0}\). In most typical database schemas, only constraints \Constraint{\ref{constraint-key}}, \Constraint{\ref{constraint-lsfk}}, and \Constraint{\ref{constraint-nlsfk}} will be explicitly declared. Some database schemas may also explicitly declare constraints \Constraint{\ref{constraint-union}} and \Constraint{\ref{constraint-disjoint}}, but it is more likely that these and all the remaining constraints will be implicit in the definition of \(\SC{0}\). It is important for the purposes of this analysis to clearly identify all such implicit constraints, so that nothing is missed when constructing corresponding SIGs. Let us partition \(\SC{0}\) into three sub-schemas: \(\SC{1} = \{S\}\), \(\SC{2} = \{\LS, \NLS\}\), and \(\SC{3} = \{LS\}\). We wish to compare the relative information capacities of \((\SC{1}, \SC{2})\) and \((\SC{1}, \SC{3})\), in order to characterise the view update compatibility of the schemas in each pair. Note that in the following analysis, we omit the foreign key constraints \Constraint{\ref{constraint-lsfk}} and \Constraint{\ref{constraint-nlsfk}}, as we have yet to determine the best way to represent inclusion dependencies of this nature in a SIG\footnote{Miller et al.\ \cite{Miller.R-1994a-SIG} briefly mention inclusion dependencies, but do not discuss how to represent them in a SIG.}, or indeed whether such dependencies are significant. %%%%%%%%%%%%%%%%%%%% \begin{table} \centering \begin{tabular}{cl} \toprule \textbf{Constraint} & \textbf{Implied edge(s)} \\ \midrule {\Constraint[\SC{1}]{\ref{constraint-key}}}(b) & \(\Type{\Sno} \KeyEdge \TSSL\) \\ \phantom{\Constraint[\SC{1}]{\ref{constraint-key}}}(c) & \(\Type{\Sno} \KeyEdge \TSSNL\) \\ \midrule {\Constraint[\SC{1}]{\ref{constraint-relation-types}}} & \(\Type{\LSsub} \SurTotal \TT{\LSsub}\) \quad\quad \(\Type{S} \SelectionEdge \Type{\LSsub}\) \quad\quad \(\TT{S} \SelectionEdge \TT{\LSsub}\) \\ \midrule {\Constraint[\SC{1}]{\ref{constraint-tuple-types}}} & \(\Type{\NLSsub} \SurTotal \TT{\NLSsub}\) \quad\quad \(\Type{S} \SelectionEdge \Type{\NLSsub}\) \quad\quad \(\TT{S} \SelectionEdge \TT{\NLSsub}\) \\ \midrule {\Constraint{\ref{constraint-implied-types-london}}} & \(\Type{\City} \SelectionEdge \CityLondon\) \\ \midrule {\Constraint{\ref{constraint-implied-types-nonlondon}}} & \(\Type{\City} \SelectionEdge (\TCityMinusLondon)\) \\ \midrule {\Constraint{\ref{constraint-implied-types-TSSL}}} & \(\TSSC \SelectionEdge \TSSL\) \\ & \(\TT{\LSsub} \ProjectionEdge \TSSL\) \\ & \(\TSSL \ProjectionEdge \Type{\Sname}\) \\ & \(\TSSL \ProjectionEdge \Type{\Status}\) \\ & \(\TSSL \ProjectionEdge\,\CityLondon\) \\ \midrule {\Constraint{\ref{constraint-implied-types-TSSNL}}} & \(\TSSC \SelectionEdge \TSSNL\) \\ & \(\TT{\NLSsub} \ProjectionEdge \TSSNL\) \\ & \(\TSSNL \ProjectionEdge \Type{\Sname}\) \\ & \(\TSSNL \ProjectionEdge \Type{\Status}\) \\ & \(\TSSNL \ProjectionEdge \TCityMinusLondon\) \\ \bottomrule \end{tabular} \caption{Constraints and the SIG edges they imply for schema \(\SC{1}\).} \label{Tab.SIG.Implied.Edges} \end{table} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Schema \(\SC{1}\)} \label{sec-constraints-s-i} \noindent A user granted access only to \(\SC{1}\) can see only relvar \(S\). Constraint \Constraint{\ref{constraint-key}}(a) can be propagated directly from \(\SC{0}\) into \(\SC{1}\) as it is already expressed in terms of \(S\) only. Constraints \Constraint{\ref{constraint-implied-types-london}}--\Constraint{\ref{constraint-implied-types-TSSNL}} do not include any named relvars in \(\SC{0}\), and thus can also be propagated directly from \(\SC{0}\) into \(\SC{1}\). Constraints \Constraint{\ref{constraint-key}}(b), \Constraint{\ref{constraint-key}}(c), and \Constraint{\ref{constraint-union}}--\Constraint{\ref{constraint-tuple-types}} cannot be propagated directly from \(\SC{0}\) into \(\SC{1}\), but can all be rewritten in terms of \(S\) only, by substituting the definitions of \(\LS\) (equation~\ref{eqn-ls}) and \(\NLS\) (equation~\ref{eqn-nls}). To simplify expressions in the following discussion, we shall use \(\LSsub\) to denote \(\RelRestrict_{\City = \mathit{'London'}}S\) (\ref{eqn-ls}) and \(\NLSsub\) to denote \(\RelRestrict_{\City \ne \mathit{'London'}}S\) (\ref{eqn-nls}). (This also clearly distinguishes the substitutions from the \(\LS\) and \(\NLS\) relvars in \(\SC{0}\), \(\SC{2}\), and \(\SC{3}\).) A constraint that has been rewritten so that it can be propagated from \(\SC{0}\) into \(\SC{1}\) is denoted by, e.g., \Constraint[\SC{1}]{\ref{constraint-union}}. Consequently we can derive the following set of rewritten constraints for \(\SC{1}\): \begin{ConstraintList}[\SC{1}] \item \Sno\ is a key for each of \(\LSsub\) and \(\NLSsub\), therefore the functional dependency \(\{\Sno\} \rightarrow \{\Sname, \Status, \City\}\) also holds in each of (b) \(\LSsub\); and (c) \(\NLSsub\). [The fact that we are dealing with two disjoint subsets of \(\City\) does not change this.] \setcounter{constraint}{3} \item \(S = \LSsub \RelUnion \NLSsub\) [or even just \(S = S\)] \item \(\RelRestrict_{\City \ne \mathit{'London'}}\LSsub = \emptyset\) \item \(\RelRestrict_{\City = \mathit{'London'}}\NLSsub = \emptyset\) \item \(\LSsub \RelIntersect \NLSsub = \emptyset\) \item (a) \(\Type{S} = \Type{\LSsub} \RelUnion \Type{\NLSsub}\); (b) \(\TT{S} = \TT{\LSsub} \RelUnion \TT{\NLSsub}\) \newline {[or even just \(\Type{S} = \Type{S}\) and \(\TT{S} = \TT{S}\)]} \item (a) \(\Type{\LSsub} \subset \Type{S}\); (b) \(\TT{\LSsub} \subset \TT{S}\) [from (\ref{eqn-ls}) above] \item (a) \(\Type{\NLSsub} \subset \Type{S}\); (b) \(\TT{\NLSsub} \subset \TT{S}\) [from (\ref{eqn-nls}) above] \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Schema \(\SC{2}\)} \label{sec-constraints-s-ii} \noindent A user granted access only to \(\SC{2}\) can see only relvars \(\LS\) and \(\NLS\). Constraints \Constraint{\ref{constraint-key}}(b), \Constraint{\ref{constraint-key}}(c), and \Constraint{\ref{constraint-notlondon}}--\Constraint{\ref{constraint-disjoint}} can be propagated directly from \(\SC{0}\) into \(\SC{2}\) as they are defined only in terms of \(\LS\) and \(\NLS\). \Constraint{\ref{constraint-implied-types-london}}--\Constraint{\ref{constraint-implied-types-TSSNL}} can be propagated directly into \(\SC{2}\) as they are not expressed in terms of any named relvars in \(\SC{0}\). \Constraint{\ref{constraint-key}}(a), \Constraint{\ref{constraint-union}}, \Constraint{\ref{constraint-relation-type-union}}, \Constraint{\ref{constraint-relation-types}}, and \Constraint{\ref{constraint-tuple-types}} can be rewritten in terms of \(\LS\) and \(\NLS\) only, giving us: \begin{ConstraintList}[\SC{2}] \item \Sno\ is a key for \(\LS \RelUnion \NLS\), therefore the functional dependency \(\{\Sno\} \rightarrow \{\Sname, \Status, \City\}\) also holds in (a) \(\LS \RelUnion \NLS\). \setcounter{constraint}{3} \item \(\LS \RelUnion \NLS = \LS \RelUnion \NLS\) \setcounter{constraint}{7} \item (a) \(\Type{\LS} \RelUnion \Type{\NLS} = \Type{\LS} \RelUnion \Type{\NLS}\); \newline (b) \(\TT{\LS} \RelUnion \TT{\NLS} = \TT{\LS} \RelUnion \TT{\NLS}\) \item (a) \(\Type{\LS} \subset \Type{\LS} \RelUnion \Type{\NLS}\); (b) \(\TT{\LS} \subset \TT{\LS} \RelUnion \TT{\NLS}\) \item (a) \(\Type{\NLS} \subset \Type{\LS} \RelUnion \Type{\NLS}\); (b) \(\TT{\NLS} \subset \TT{\LS} \RelUnion \TT{\NLS}\) \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Schema \(\SC{3}\)} \label{sec-constraints-s-iii} \noindent A user granted access only to \(\SC{3}\) can see only relvar \(\LS\). Constraints \Constraint{\ref{constraint-key}}(b) and \Constraint{\ref{constraint-notlondon}} can be propagated directly into \(\SC{3}\), as they are defined in terms of \(\LS\) only. \Constraint{\ref{constraint-implied-types-london}}--\Constraint{\ref{constraint-implied-types-TSSNL}} can also be propagated directly into \(\SC{3}\) as previously discussed. The remaining constraints cannot be rewritten in terms of \(\LS\) only, and so cannot be propagated into \(\SC{3}\). As we were able to propagate all constraints into \(\SC{1}\), this indicates a qualitative difference between \(\SC{1}\) and \(\SC{3}\). It therefore seems likely that the relative information capacities of \(\SC{1}\) and \(\SC{3}\) will also differ. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Constructing the SIGs} \label{sec-date-sigs} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Schema \(\SC{1}\)} \label{sec-sigs-s-i} \noindent The SIG for \(\SC{1}\) will not just be that for relvar \(S\) as shown in Figure~\ref{fig-sig-s-alone}, as that schema did not incorporate all the constraints implicit in \(\SC{0}\). The process for constructing the SIG will be similar to that for the earlier example, however. The constraints appearing in \(\SC{1}\) are \Constraint{\ref{constraint-key}}(a), \Constraint[\SC{1}]{\ref{constraint-key}}(b), \Constraint[\SC{1}]{\ref{constraint-key}}(c), \Constraint[\SC{1}]{\ref{constraint-union}}--\Constraint[\SC{1}]{\ref{constraint-tuple-types}}, and \Constraint{\ref{constraint-implied-types-london}}--\Constraint{\ref{constraint-implied-types-TSSNL}}. The SIG will include all of the edges that appear in Figure~\ref{fig-sig-s-alone}. The constraints within the schema also imply the existence of several subtype nodes (e.g., the two subtypes of \(\Type{City}\) implied by \Constraint{\ref{constraint-implied-types-london}} and \Constraint{\ref{constraint-implied-types-nonlondon}}), along with corresponding selection and projection edges, which are listed in Table~\ref{Tab.SIG.Implied.Edges} above. \Constraint[\SC{1}]{\ref{constraint-union}}, \Constraint[\SC{1}]{\ref{constraint-relation-type-union}}(a), and \Constraint[\SC{1}]{\ref{constraint-relation-type-union}}(b) are effectively tautologies that add no new information, and can thus be safely ignored. The completed SIG for \(\SC{1}\) is shown in Figure~\ref{fig-sig-s}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node (TLS) {\(\Type{\LSsub}\)}; &[7mm] \node (TTLS) {\(\TT{\LSsub}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\Type{\Sname}\)}; \\ \node (TLSPlusNLS) {\(\Type{S}\)}; & \node (TTLSPlusNLS) {\(\TT{S}\)}; & \node (TSno) {\(\Type{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\Type{City}\)}; \\ & & & & & \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TNLS) {\(\Type{\NLSsub}\)}; & \node (TTNLS) {\(\TT{\NLSsub}\)}; & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) }, }; % FDs (TSno) -- [funcdep left,bend left] (TSSL); (TSno) -- [funcdep right] (TSSNL); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTLSPlusNLS), (TTLS) } -- (TSno) }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTNLS) -- (TSno) }; { [edges=projection left] (TTLS) -- (TSSL), (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TTNLS) -- (TSSNL), (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (TStatus), (NLSCity), (TSname) }, }; % selection edges { [edges=selection left] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) }, (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, (TCity) -- (LSCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right] (TSSC), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,bend right] (TSSC), }; \end{tikzpicture} \caption{SIG for schema \(\SC{1} = \{S\}\).} \label{fig-sig-s} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node (TLS) {\(\Type{\LS}\)}; &[7mm] \node (TTLS) {\(\TT{\LS}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\Type{\Sname}\)}; \\ \node (TLSPlusNLS) {\(\StackTLSPlusNLS\)}; & \node (TTLSPlusNLS) {\(\StackTTLSPlusNLS\)}; & \node (TSno) {\(\Type{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\Type{City}\)}; \\ & & & & & \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TNLS) {\(\Type{\NLS}\)}; & \node (TTNLS) {\(\TT{\NLS}\)}; & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) }, }; % FDs (TSno) -- [funcdep left,bend left] (TSSL); (TSno) -- [funcdep right] (TSSNL); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTLSPlusNLS), (TTLS) } -- (TSno) }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTNLS) -- (TSno) }; { [edges=projection left] (TTLS) -- (TSSL), (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TTNLS) -- (TSSNL), (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (TStatus), (NLSCity), (TSname) }, }; % selection edges { [edges=selection left] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) }, (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, (TCity) -- (LSCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right] (TSSC), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,bend right] (TSSC), }; \end{tikzpicture} \caption{SIG for schema \(\SC{2} = \{\LS, \NLS\}\).} \label{fig-sig-ls-nls} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node (TLS) {\(\Type{\LS}\)}; &[7.5mm] \node (TTLS) {\(\TT{\LS}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\Type{\Sname}\)}; \\ & & \node (TSno) {\(\Type{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\Type{City}\)}; \\ & & & & & \node (TStatus) {\(\Type{\Status}\)}; \\ & & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TLS) -- (TTLS), }; % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTLS) -- (TSno) }; { [edges=projection left] (TTLS) -- (TSSL), (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (TStatus), (NLSCity), (TSname) }, }; % selection edges { [edges=selection left] (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] (TCity) -- (LSCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); }; \end{tikzpicture} \caption{SIG for schema \(\SC{3} = \{\LS\}\).} \label{fig-sig-ls} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Schema \(\SC{2}\)} \label{sec-sigs-s-ii} \noindent The constraints appearing in \(\SC{2}\) are \Constraint[\SC{2}]{\ref{constraint-key}}(a), \Constraint{\ref{constraint-key}}(b), \Constraint{\ref{constraint-key}}(c), \Constraint[\SC{2}]{\ref{constraint-union}}, \Constraint{\ref{constraint-notlondon}}--\Constraint{\ref{constraint-disjoint}}, \Constraint[\SC{2}]{\ref{constraint-relation-type-union}}--\Constraint[\SC{2}]{\ref{constraint-tuple-types}}, and \Constraint{\ref{constraint-implied-types-london}}--\Constraint{\ref{constraint-implied-types-TSSNL}}. This is essentially the same set of constraints as \(\SC{1}\), but with different rewritings. This means that the SIG for \(\SC{2}\) can be built in the same manner as for \(\SC{1}\), with some nodes named differently. In particular, \Constraint[\SC{2}]{\ref{constraint-relation-type-union}} implies that there will be nodes \(\TLSPlusNLS\) and \(\TTLSPlusNLS\) instead of \(\Type{S}\) and \(\TT{S}\), respectively. There will also be \(\Type{\LS}\) instead of \(\Type{\LSsub}\), etc. The complete SIG for \(\SC{2}\) is shown in Figure~\ref{fig-sig-ls-nls}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Schema \(\SC{3}\)} \label{sec-sigs-s-iii} \noindent The constraints appearing in \(\SC{3}\) are \Constraint{\ref{constraint-key}}(b), \Constraint{\ref{constraint-notlondon}}, and \Constraint{\ref{constraint-implied-types-london}}--\Constraint{\ref{constraint-implied-types-TSSNL}}. The SIG will only include edges and nodes that relate either directly to \(\LS\) or are independent of any relvar, such as the subtype node \(\TSSNL\). The complete SIG for \(\SC{3}\) is shown in Figure~\ref{fig-sig-ls} on the previous page. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Transforming the SIGs} \label{sec-transforming} \noindent Now that we have constructed SIGs for all three schemas, we can compare the relative information capacities of the schema pairs \((\SC{1}, \SC{2})\) and \((\SC{1}, \SC{3})\). We do this by attempting to transform the SIG for one of the schemas until it is isomorphic with the other. Looking at the SIGs for \(\SC{1}\) and \(\SC{2}\), we can see almost immediately that both SIGs have the same node and edge structure; indeed the only difference is in the names of some of the nodes. However, we already know from the discussion in Sections~\ref{sec-constraints-s-i} and~\ref{sec-constraints-s-ii} that these differently named nodes are just rewritings that represent the same domains. We therefore have an immediate isomorphism between the two SIGs and can conclude that the information capacities of \(\SC{1}\) and \(\SC{2}\) are equivalent, i.e., \(\Equivalent{\SC{1}}{\SC{2}}\). The comparison between \(\SC{1}\) and \(\SC{3}\) is more interesting. There are some shared structures, but the two SIGs are clearly different. Since the SIG transformations discussed in Section~\ref{sec-sig-compare} only ever preserve or enhance information capacity, intuitively we should attempt to transform the ``smaller'' SIG for \(\SC{3}\) into the ``larger'' SIG for \(\SC{1}\). SIG transformations can generally be carried out in two broad phases (using the \(\SC{3} \rightarrow \SC{1}\) transformation as an example): \begin{description} \item[Duplicate nodes] in \(\SC{3}\) (producing \(\SC{3}'\)) that correspond to supertypes in \(\SC{1}\), in order to create nodes corresponding to the extra nodes that \(\SC{1}\) has over \(\SC{3}\). The goal is to produce a node structure similar to \(\SC{1}\). The duplicated nodes are connected to their original nodes by trivial selection edges. These transformations (if applicable) do not alter the information capacity of \(\SC{3}'\) (i.e., \(\Equivalent{\SC{3}}{\SC{3}'}\)). \item[Copy and move edges] around the SIG to produce an edge structure similar to \(\SC{1}\). The annotations of each moved or copied edge are determined by the path it is composed with, as discussed in Section~\ref{sec-sig-compare}. These transformations (if applicable) \emph{increase} the information capacity of \(\SC{3}'\) (i.e., \(\Dominates{\SC{3}'}{\SC{3}}\)). \end{description} These two phases generally occur in order, but may be interspersed throughout with the following: \begin{description} \item[Delete duplicate nodes] in \(\SC{3}'\) that are no longer needed, especially those that do not exist in \(\SC{1}\). These transformations (if applicable) do not alter the information capacity of \(\SC{3}'\) (\(\equiv\)). \item[Remove totality annotations] from trivial selection edges in \(\SC{3}'\) to change them into non-trivial selection edges. This decouples duplicate nodes into subtypes and supertypes, ensuring that they represent different domains. These transformations (if applicable) \emph{increase} the information capacity of \(\SC{3}'\) (\(\preceq\)). \end{description} For example, we might duplicate some nodes, remove some totality annotations, move or copy some edges, delete some no longer needed nodes, move or copy some more edges, then remove some more totality annotations. Applying this process to \(\SC{3}\), we first duplicate node \(\Type{\LS}\) to \(\Type{S}\) and node \(\TT{\LS}\) to \(\TT{S}\). To obtain a node structure similar to that of \(\SC{1}\), we also need to further duplicate \(\Type{S}\) to \(\Type{\NLS}\) and \(\TT{S}\) to \(\TT{\NLS}\). The result of these duplications is shown in Figure~\ref{fig-transform-all-duplicates}. The transformed SIG now has the same node structure as that for \(\SC{1}\). %%%%%%%%%%%%%%%%%%%% \begin{figure}[t] \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node [input keep] (TLS) {\(\Type{\LS}\)}; &[7mm] \node [input keep] (TTLS) {\(\TT{\LS}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\Type{\Sname}\)}; \\ \node [output] (TLSPlusNLS) {\(\Type{S}\)}; & \node [output] (TTLSPlusNLS) {\(\TT{S}\)}; & \node (TSno) {\(\Type{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\Type{City}\)}; \\ & & & & & \node (TStatus) {\(\Type{\Status}\)}; \\ \node [output] (TNLS) {\(\Type{\NLS}\)}; & \node [output] (TTNLS) {\(\TT{\NLS}\)}; & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TLS) -- (TTLS), }; % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTLS) -- (TSno) }; { [edges=projection left] (TTLS) -- (TSSL), (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (TStatus), (NLSCity), (TSname) }, }; % selection edges { [edges=selection left] (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] (TCity) -- (LSCity), }; { [edges={bijection,output},edge label'={\scriptsize\(\RelRestrict\)}] { (TLS), (TTLS) } -- { (TLSPlusNLS), (TTLSPlusNLS) }, { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); % transformation indicators { [edges={->,gray,dashed,bend left=45}] (TLS) -- [edge node={node[sloped,above]{\small\emph{copy}}}] (TLSPlusNLS), (TLSPlusNLS) -- [edge node={node[sloped,above,rotate=180]{\small\emph{copy}}}] (TNLS) }; }; \end{tikzpicture} \caption{SIG for transformed schema \(\SC{3}'\) after duplicating \Type{\LS} and \TT{\LS} to produce analogues of the additional nodes in \(\SC{1}\). Inputs to the transformations are coloured \textcolor{blue}{blue}, outputs are coloured \textcolor{red}{red}. [\(\equiv\)]} \label{fig-transform-all-duplicates} \end{figure} %%%%%%%%%%%%%%%%%%%% The next phase is to copy and move edges. To produce the edge \(\Type{S} \SurTotal \TT{S}\), we can first \emph{copy} \(\Type{\LS} \SurTotal \TT{\LS}\) across the path \(\TT{S} \TrivialSelection \TT{\LS}\) (giving \(\Type{\LS} \SurTotal \TT{S}\)), then \emph{move} it across \(\Type{S} \TrivialSelection \Type{\LS}\). We can carry out a similar series of transformations to copy and move \(\Type{S} \SurTotal \TT{S}\) to \(\Type{\NLS} \SurTotal \TT{\NLS}\) (see Figure~\ref{fig-transform-edge-moves}). We next copy and move the unique projection edge \(\TT{\LS} \TrivialProjection \Type{\Sno}\) to produce the edges \(\TT{S} \TrivialProjection \Type{\Sno}\) and \(\TT{\NLS} \TrivialProjection \Type{\Sno}\). Similarly, we copy and move \(\TT{\LS} \ProjectionEdge \TSSL\) to produce \(\TT{S} \LabelledEdge{\sigedge{12}}{\RelProject} \TSSC\) and \(\TT{\NLS} \LabelledEdge{\sigedge{02}}{\RelProject} \TSSNL\). Note that the first edge in the latter pair of transformations loses its totality annotation, and the second edge its surjectivity annotation, as a result of composition with the pre-existing non-bijective selection edges between the constructed tuple types running vertically down the centre of the SIG. Finally, we carry out similar copy and move transformations to copy the key edge \(\Type{\Sno} \KeyEdge \TSSC\) to \(\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSL\), and \(\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSNL\). Both new edges lose their totality annotations for the same reason as the projection edges above. The final result of all these edge transformations is shown in Figure~\ref{fig-transform-edge-moves}. %%%%%%%%%%%%%%%%%%%% \begin{figure}[t] \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node (TLS) {\(\Type{\LS}\)}; &[7mm] \node (TTLS) {\(\TT{\LS}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\Type{\Sname}\)}; \\ \node (TLSPlusNLS) {\(\Type{S}\)}; & \node (TTLSPlusNLS) {\(\TT{S}\)}; & \node (TSno) {\(\Type{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\Type{City}\)}; \\ & & & & & \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TNLS) {\(\Type{\NLS}\)}; & \node (TTNLS) {\(\TT{\NLS}\)}; & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TLS) -- [input keep,edge node={node (c1a) {}}] (TTLS), (TLSPlusNLS) -- [output,edge node={node (c1b) {}}] (TTLSPlusNLS), (TNLS) -- [output,edge node={node (c1c) {}}] (TTNLS), }; % FDs (TSno) -- [funcdep left,arrows={-{crossbar}>},bend left,output] (TSSL); (TSno) -- [funcdep right,arrows={-{crossbar}>},output] (TSSNL); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTLS) -- [input keep] (TSno), (TTLSPlusNLS) -- [output] (TSno), }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTNLS) -- [output] (TSno), }; { [edges=projection left] (TTLS) -- [input keep,edge node={node (c2a) {}}] (TSSL), (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (TStatus), (NLSCity), (TSname) }, (TTNLS) -- [arrows={:->},output,edge node={node (c2c) {}}] (TSSNL), }; % selection edges { [edges=selection left] (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] (TCity) -- (LSCity), }; { [edges=bijection,edge label'={\scriptsize\(\RelRestrict\)}] { (TLS), (TTLS) } -- { (TLSPlusNLS), (TTLSPlusNLS) }, { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right] (TSSC), }; (TSno) -- [funcdep right,input keep] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,arrows={:{crossbar}->},bend right,output,edge node={node (c2b) {}}] (TSSC), % transformation indicators { [edges={->,gray,dashed}] (c1a) -- (c1b), (c1a) -- [bend right=12,edge node={node[above,sloped,pos=0.175]{\small\emph{copy \& move}}}] (c1c), (c2a) -- [bend right=12] (c2b), (c2a) -- [bend right=12,edge node={node[above,sloped,pos=0.175]{\small\emph{copy \& move}}}] (c2c) }; }; \end{tikzpicture} \caption{SIG for transformed schema \(\SC{3}'\) after copying and moving various edges. [\(\preceq\)]} \label{fig-transform-edge-moves} \end{figure*} %%%%%%%%%%%%%%%%%%%% If we compare Figure~\ref{fig-sig-s} with Figure~\ref{fig-transform-edge-moves}, we can see that the transformed SIG for \(\SC{3}'\) now has the same topology as the SIG for \(\SC{1}\). We still need to remove the totality annotations from the newly created trivial selection edges between the relation and tuple types on the left, but even when this is done, we do not achieve the desired isomorphism, as the edges \(\TT{S} \LabelledEdge{\sigedge{12}}{\RelProject} \TSSC\), \(\TT{\NLS} \LabelledEdge{\sigedge{02}}{\RelProject} \TSSNL\), \(\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSL\), and \(\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSNL\) all have different annotations from the corresponding edges in the SIG for \(\SC{1}\). This clearly shows that the information capacities of \(\SC{1}\) and \(\SC{3}\) are incomparable, and that view updates based on this pair of schemas will be problematic. If we were to create a fourth sub-schema \(\SC{4} = \{\NLS\}\) and compare this with \(\SC{1}\), the result would be similar to that for \(\SC{3}\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusion \& future work} \label{sec-conclusion} \noindent In this paper we have shown how to use the concept of relative information capacity and an extension of the schema intension graph (SIG) formalism to more formally model Date's notion of information equivalence with respect to relational view updates. Our analysis of Date's restriction view example \cite{Date.C-2013a-View} using these techniques is consistent with his findings, so the approach shows promise. The information capacities of \(\SC{1} = \{S\}\) and \(\SC{2} = \{\LS,\NLS\}\) are equivalent, so it should be possible to effectively propagate view updates between \(\SC{1}\) and \(\SC{2}\), regardless of the configuration of views and base relvars. Conversely, the information capacity of \(\SC{3} = \{\LS\}\) is not comparable with that of \(\SC{1}\), implying that there are updates that cannot be propagated between \(\SC{1}\) and \(\SC{3}\). Our analysis reinforces the importance of explicitly identifying all constraints that can be propagated from the base schema \(\SC{0}\) to its sub-schemas. Omission of constraints could produce a different SIG structure that might lead to an erroneous conclusion about a schema's information capacity relative to another. There are several avenues for future work. First, the proof of concept discussed in this paper only deals with disjoint restriction views, which are arguably the simplest case of view updating. Further work needs to be done to characterise other types of view configuration, such as projection views, union views, and in particular, join views. This will enable us to test and refine the application of SIGs in this context. The discussion in this paper focused only on the relational context, but relative information capacity and the SIG formalism are data model agnostic. It would therefore be interesting to investigate the application of these techniques to view updating contexts beyond the Relational Model (e.g., deductive databases). It is unclear at present whether the omission of inclusion dependencies such as foreign keys is significant. If they are significant, the question then is how best to represent them in the SIG formalism. Adding such constraints will clearly change the structure of a SIG, but we suspect that this will not change the relationship between the relative information capacities of different schemas. It is interesting to note that with the equivalent schemas \(\SC{1}\) and \(\SC{2}\), we were able to propagate all of the constraints from the base schema \(\SC{0}\) (Section~\ref{sec-constraints-s-ii}), whereas with the incomparable schemas \(\SC{1}\) and \(\SC{3}\), only a subset of the constraints could be propagated (Section~\ref{sec-constraints-s-iii}). A reasonable conjecture is that the degree to which constraints can be propagated from \(\SC{0}\) to its sub-schemas could be a direct indicator of information equivalence among the sub-schemas, but this needs to be further investigated. Finally, we have recently discovered further work that explores and expands upon the concepts of information capacity and meaning within data and database schemas (e.g., \cite{Mantri.R-2010a-Data,Wang.X-2005a-SIG,Xu.K-2009a-Defining}). Some of these ideas may be applicable to our work, and need to be investigated in more detail. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \balance %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibliographystyle{abbrv} \bibliography{APCCM2017_Stanger} \end{document}

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% APCCM2017_Stanger.tex
% Author: Nigel Stanger
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% Revisions: 22 Aug 2016 Initial submission.
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% Relational algebra operators
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\newcounter{constraint}

% misc
% \newcommand{\todo}[1]{\textbf{!!TODO!!} {[#1]}}

% commonly used elements
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\newcommand{\Type}[1]{\ensuremath{T_{#1}}}
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\newcommand{\StackTLSPlusNLS}{\ensuremath{\begin{array}{c}\Type{\LS}\,+ \\ \Type{\NLS}\end{array}}}
\newcommand{\StackTTLSPlusNLS}{\ensuremath{\begin{array}{c}\TT{\LS}\,+ \\ \TT{\NLS}\end{array}}}
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\newcommand{\StackTCityMinusLondon}{\ensuremath{\begin{array}{c}\Type{\City}\,\setminus \\ \CityLondon\end{array}}}

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% dominates
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\begin{document}

\CopyrightYear{2017} 
\setcopyright{acmlicensed}
\conferenceinfo{ACSW '17,}{January 31-February 03, 2017, Geelong, Australia}
\isbn{978-1-4503-4768-6/17/01}\acmPrice{\$15.00}
\doi{http://dx.doi.org/10.1145/3014812.3014847}

\title{Characterising relational view updates \\ using relative information capacity}

\numberofauthors{1}
\author{
\alignauthor Nigel Stanger \\
    \affaddr{Department of Information Science, University of Otago}\\
    \affaddr{Dunedin, New Zealand}\\
    \email{nigel.stanger@otago.ac.nz}
}

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\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}
In 2013, Chris Date published a book about the view update problem in the context of the Relational Model. He presented several detailed examples of different varieties of view updates and characterised their behaviour, in the process deriving a set of principles for updating views based on the notion of \emph{information equivalence} of view schemas. In this paper we discuss work in progress that examines how Date's operational definition of information equivalence can be formally characterised using Hull's concept of \emph{relative information capacity}. As a proof of concept, we use an extension of Miller's \emph{schema intension graph} formalism to model the information equivalence of Date's restriction view example.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\keywords{view update problem, information equivalence, relative information capacity, schema intension graph, schema transformation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

\noindent The view update problem is a long-standing problem of great practical significance in the database field. Date \cite{Date.C-2013a-View} likens it to significant unsolved problems in other fields such as the Riemann Hypothesis in mathematics or ``$P = \mathit{NP}$?'' in computational complexity theory. In his 2013 book \cite{Date.C-2013a-View} Date developed a set of principles for updating relational views, based on the notion of \emph{information equivalence}. However, that notion was defined more operationally than formally.

View updating is a special case of schema translation: we have one schema comprising a set of views, another schema comprising a set of base relations, and we need to translate updates from one to the other.\footnote{These schemas may be actually separate, or more likely may be sub-schemas of a larger database schema, with effective separation achieved via the access privilege mechanism.} It therefore makes sense that formalisms developed to characterise schema translations are also applicable to view updates. We propose to use Hull's notion of \emph{relative information capacity} \cite{Hull.R-1986a-Relative} as a means to formally characterise the information equivalence of the schemas that make up a given view configuration.

In particular, we develop an extension of Miller's \emph{schema intension graph} (SIG) formalism, which was originally designed to facilitate the comparison of relative information capacity across schemas. We first extend the SIG formalism with relevant relational concepts such as relation types, tuple types, and key constraints. We then show a proof of concept of how this extended formalism can be used to model the information equivalence of a view configuration, by applying it to Date's example of disjoint restriction views \cite{Date.C-2013a-View}. Our results are consistent with Date's operational characterisation of information equivalence for this example.

In Section~\ref{sec-view-update-problem}, we briefly review the view update problem and relevant research, and in Section~\ref{sec-relations-types} we introduce some core concepts that are used throughout the remainder of the paper. In Section~\ref{sec-info-capacity} we discuss relative information capacity and the SIG formalism, then explain in Section~\ref{sec-relational-sigs} how to extend the SIG formalism to more precisely model a relational schema. In Section~\ref{sec-characterising} we analyse Date's restriction view example \cite{Date.C-2013a-View} in detail, and characterise the information equivalence of its various sub-schemas. We discuss conclusions and future work in Section~\ref{sec-conclusion}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The view update problem}
\label{sec-view-update-problem}

\noindent The view update problem was first discussed in the relational context by Codd in 1974 \cite{Codd.E-1974a-Recent}. When an update operation is applied to a view, it needs to be translated into corresponding updates on the base relations from which the view is derived \cite{Keller.A-1985a-Algorithms}. Sometimes there are multiple translations that can produce the desired view update, but produce different---possibly incompatible---changes in the base relations. Worse, there may be translations that have side effects (e.g., for some views based on joins), which produce a different result in the view than if the updates were applied directly to an equivalent base relation \cite{Keller.A-1985a-Algorithms}. The problem then is how to choose the most appropriate translation. As yet there has been no solution that completely and automatically resolves this ambiguity.

There has been considerable research into this problem over the last four decades, and numerous different approaches have been suggested in the context of relational, non-relational (e.g., \cite{Liefke.H-2000a-View,Pan.W-1996a-Object,Scholl.M-1991a-Updatable,Wang.L-2006a-Updating}), and deductive databases (e.g., \cite{Behrend.A-2008a-Transformation-based,Caroprese.L-2012a-View-update,Console.L-1995a-Abduction,Mayol.E-2003a-Consistency}). We shall now briefly review some key developments in the relational context. We leave consideration of non-relational and deductive contexts for future work.

Banchilhon and Spyratos proposed the concept of \emph{constant complement} \cite{Bancilhon.F-1981a-Update} in 1981. The \emph{complement} of a view describes all the information from the original database that does not appear in the view, so composing a view with its complement provides enough information to reconstruct the original database. Keeping the complement invariant under view update facilitates translation construction. This is a useful approach that has been developed further by other authors \cite{Hegner.S-2004a-Order-based,Lechtenborger.J-2003a-Impact}, but it has been shown that there are reasonable translations that cannot be carried out under constant complement \cite{Keller.A-1987a-Comment}. Computing the complement of a view is also difficult, and several authors have worked on more efficient algorithms \cite{Cosmadakis.S-1984a-Updates,Laurent.D-2001a-Monotonic,Lechtenborger.J-2003a-Computation}. Finally, this approach is purely structural in nature and does not consider the semantics of the schema, which is necessary to properly resolve translation ambiguity \cite{Keller.A-1986a-Role}.

Keller noted that semantic information is essential in order to choose an appropriate translation \cite{Keller.A-1986a-Role}, and suggested that the da\-ta\-base administrator (DBA) could provide this at view definition time \cite{Keller.A-1985a-Algorithms}. Masunaga argued, however, that this would be insufficient to completely resolve ambiguity, and suggested collecting further semantic information from end users at update time \cite{Masunaga.Y-1984a-Relational}. Shu \cite{Shu.H-2000a-Using} discussed an interesting approach to this kind of information gathering, by treating the problem of generating suitable view translations as a constraint satisfaction problem. This enabled constraints to be added incrementally at view update time. However, while collecting such information from the DBA seems reasonable, collecting it from the end user at update time could be seen as too burdensome, and could impact the usability of query interfaces. It is also unclear which information needs to be captured, and how best to capture it.

Kotidis et al.\ \cite{Kotidis.Y-2006a-Updates} discussed a different approach to avoiding side effects in view translations. They proposed creating a materialised copy of the view that could be updated directly, rather than mapping updates from the (virtual) view onto its underlying base tables. Clearly this enables the view to be updated without translation side effects, but does imply that the physical manifestation (extension) of the view may no longer match its defining query (intension).

Finally, Date has published several books on the theoretical foundations of the Relational Model over the last decade, including one specifically targeting the view update problem \cite{Date.C-2013a-View} (indeed, the book is ambitiously subtitled ``Solving the View Update Problem''). He developed a comprehensive set of rules for updating views in a relational context, supported by an array of worked examples. Date advocated explicitly stating all relevant constraints at view creation time, similar to the information gathering approaches discussed above \cite{Keller.A-1985a-Algorithms,Masunaga.Y-1984a-Relational,Shu.H-2000a-Using}. He argued that if a complete set of constraints was available, then ambiguity would be diminished or eliminated, and it would then be possible to automatically generate a set of appropriate \emph{compensatory rules} to automatically manage the view update process. (Compensatory rules are similar in principle to database triggers, except that they are intended to be declarative so that their definitions can made explicitly visible to end users.)

Date's work is interesting and compelling, and raises several questions worthy of further investigation. However, there are still situations where the correct view translation may be ambiguous, e.g., views based on certain types of join, as previously noted \cite{Keller.A-1985a-Algorithms}. Somewhat disappointingly, given his stated goal of ``solving'' the view update problem, Date took a pragmatic approach in such cases. While his justification was well argued and logical, even he acknowledged that it involved ``a certain degree of pragma'' \cite{Date.C-2013a-View}.

Date characterises the ability to define an appropriate and effective view translation using the notion of \emph{information equivalence}. Two schemas $\SC{1}$ and $\SC{2}$ are said to be information equivalent ``if and only if the constraints that apply to [$\SC{1}$] and [$\SC{2}$] are such that every proposition that can be represented by [$\SC{1}$] can be represented by [$\SC{2}$] and vice versa'' \cite{Date.C-2013a-View}. Date's information equivalence is a formal concept, but its definition and characterisation in \cite{Date.C-2013a-View} are more operational in nature. This is what we explore in this paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Relations and types}
\label{sec-relations-types}

\noindent We generally follow Date's \cite{Date.C-2012a-SQL-and-Relational} notation and terminology regarding the Relational Model. We do not use Date's Tutorial D notation, however, preferring instead a more classical algebraic style.

In this paper, $A$ represents an attribute (comprising a \emph{name}:\emph{type} pair), $R$ represents a relation variable (\emph{relvar}), and $v$ represents a value of an attribute, tuple, relation, etc. Attribute values may be disambiguated where necessary by prefixing the value with the attribute name, i.e., $\mathit{name}\colon\mathit{value}$.

Date defines a \emph{type} $\Type{i}$ as ``essentially a \emph{named, finite set of values}'' \cite{Date.C-2012a-SQL-and-Relational}, representing every possible value of some specific kind, e.g., every possible ASCII string of length 5, every possible city name, or every possible relation value with a particular heading. That is:
\begin{equation}\label{eqn-type}
    \Type{i} = \{v_{1}, v_{2}, \dotsc, v_{m}\}, 0 < m < k
\end{equation}
where $k$ is some finite upper bound. Thus, the type of attribute $A_{i}$ (denoted $\Type{A_{i}}$) is by definition the set of all possible values of $A_{i}$.

Consider a relvar $R$ with heading $\{A_{1}, A_{2}, \dotsc, A_{n}\}$. Following from equation~\ref{eqn-type}, the \emph{relation type} of $R$, which we denote here as $\Type{R}$, is the set of all possible relation values with $R$'s heading. Similarly, the \emph{tuple type} of $R$'s tuples, which we denote here as \TT{R}, is the set of all possible tuple values with $R$'s heading, which equates to the Cartesian product of $R$'s attribute types, i.e.:
\begin{equation}\label{eqn-tuple-type}
\TT{R} = \Type{A_{1}} \times \Type{A_{2}} \times\dotsb\times \Type{A_{n}}\text{.}
\end{equation}
(This construction can be applied to any tuple type, including projections of other tuple types.)

By definition any given value of a relvar $R$ must comprise a set (possibly empty) of tuples $\{t_{1}, t_{2}, \dotsc, t_{m}\} \subseteq \TT{R}$. $\Type{R}$ therefore comprises every possible $k$-combination of the elements of \TT{R}, for $0 \le k \le \left|\TT{R}\right|$.

For example, Date's usual suppliers-and-parts schema \cite{Date.C-2013a-View} includes a relvar $S$ with heading $\{\Sno, \Sname, \Status, \City\}$, which contains details of suppliers. From equation~\ref{eqn-tuple-type}, the tuples of $S$ have type $\TT{S} = \Type{\Sno} \times \TSSC$.

This notion of type as a set of possible values is synonymous with Codd's original definition of domains in the Relational Model \cite{Codd.E-1970a-Relational,Date.C-2012a-SQL-and-Relational}, and also with the notion of domains as ``sets of values'' in schema intension graphs \cite{Miller.R-1994a-Schema,Miller.R-1994a-SIG}. The implications of the latter point will be explored further in Section~\ref{sec-relational-sigs}, after we have introduced relative information capacity and schema intension graphs in Section~\ref{sec-info-capacity}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Nulls}

\noindent We also follow Date's prohibition of nulls. Philosophical objections aside, however, this makes little practical difference to our approach, as we are dealing exclusively with types rather than attributes anyway. While an attribute could  be null, it makes little sense to say that a \emph{type} could be ``null''. It could be claimed that a type ``includes'' nulls, but it is unclear what this would even mean in practice, and runs the risk of treating nulls like ``normal'' values rather than the special markers that they are. In any case, this would effectively be no different from the situation without nulls, so we omit further consideration of nulls in the interest of clarity.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \newpage

\section{Relative information capacity}
\label{sec-info-capacity}

\noindent A schema conveys information about the phenomenon it models. Hull \cite{Hull.R-1986a-Relative} defined the concept of \emph{relative information capacity} to describe the information content of different schemas. The relative information capacity of a schema determines the set of all valid instances of that schema \cite{Hull.R-1986a-Relative,Miller.R-1994a-PhD,Miller.R-1993a-Information}. Relative information capacity, as its name implies, is not a quantitative measure; rather it is an indicator of the relative information content of different schemas.

Relative information capacity provides a means to identify the differences between schema instances, and can therefore be used as an aid to schema integration and translation \cite{Miller.R-1994a-PhD}. As noted earlier, view updating is a special case of this, so it follows that information capacity can also be used to characterise the information content of views compared to the base relvar(s) from which they are derived.

Miller et al.\ \cite{Miller.R-1993a-Information} defined relative information capacity in terms of two types of mapping. An \emph{information (capacity) preserving mapping} from schema $\SC{1}$ to schema $\SC{2}$ maps every element of $\SC{1}$ to an element of $\SC{2}$, but only some elements of $\SC{2}$ to elements of $\SC{1}$. $\SC{2}$ has an information capacity greater than $\SC{1}$, and is said to \emph{dominate} $\SC{1}$, denoted \Dominates{\SC{2}}{\SC{1}}. An \emph{equivalence preserving mapping} from $\SC{1}$ to $\SC{3}$ maps every element of $\SC{1}$ to an element of $\SC{3}$ and vice versa (i.e., both the mapping and its inverse are information preserving). The information capacity of $\SC{1}$ is identical to $\SC{3}$, and the schemas are said to be \emph{equivalent}, denoted \Equivalent{\SC{1}}{\SC{3}}.

(Note that Miller et al.\ \cite{Miller.R-1994a-Schema} use stricter definitions of equivalence and dominance than those used here; strictly speaking, the discussion in this section should refer to \emph{absolute} dominance and equivalence, whereas the following discussion on SIG transformations should refer to \emph{internal} dominance and equivalence \cite{Miller.R-1994a-Schema}. The distinction, however, is not crucial to the discussion in this paper.)

Relative information capacity can only be used comparatively. To this end, Miller et al.\ \cite{Miller.R-1994a-SIG} defined the \emph{schema intension graph} formalism as a tool for comparing the information capacity of schemas. A schema intension graph (SIG) describes the associations between domains in a schema, and provides a way to ``reason about constraints on \emph{collections of entities} in an instance of the entity set, rather than about the internal structure of a single entity'' \cite{Miller.R-1994a-SIG}. In particular, SIGs provide a useful abstract formalism within which to perform schema transformations, as transforming the SIG for a schema is effectively the same as transforming the schema itself.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Constructing SIGs}
\label{sec-sig-build}

The edges $e_{i} \in E$ represent binary relations (in the mathematical sense) between domains. Edges are denoted by lines with an optional label, and may be composed to form a \emph{path} between two nodes (the notation $e_{2} \circ e_{1}$ means ``edge $e_{1}$ followed by edge $e_{2}$''). Some edges may be designated as \emph{selection} edges, labelled $\RelRestrict$, or \emph{projection} edges, labelled $\RelProject$. These correspond to the relational algebra operators of the same names \cite{Codd.E-1972a-Completeness}.

Edges may also be annotated with up to four properties denoting simple constraints on the binary relation represented by the edge: totality, surjectivity, functionality and injectivity. These properties are defined as follows \cite{Miller.R-1994a-Schema}:
\begin{quotation}
    \noindent Let $A$ and $B$ be two sets. A binary relation $r : A \sigedge{00} B$ is \emph{total} (denoted $e : A \Total B$) if it is defined for all elements of $A$; \emph{surjective} ($e : A \Surjective B$) if it is defined for all elements of $B$; \emph{functional} ($e : A \Functional B$) if an element of $A$ determines at most one element of $B$; and \emph{injective} ($e : A \Injective B$) if an element of $B$ determines at most one element of $A$.
\end{quotation}
A \emph{bijection} is a binary relation that has all four properties \cite{Miller.R-1994a-SIG}.

The use of the term ``binary relation'' here, while mathematically correct, is potentially confusing in any discussion that also involves the Relational Model. From now on, we shall therefore refer to the binary relations represented by SIG edges as ``associations''.

A selection edge $e : A \LabelledEdge{\sigedge{00}}{\RelRestrict} B$ indicates that the elements of $B$ comprise a subset or specialisation of the elements of $A$. If $B = A$, we have what we term a \emph{trivial} selection. Every element of B must appear in A, and vice versa, and every element of $A$ is associated with exactly one element of $B$, and vice versa. Trivial selection edges are therefore bijective, i.e., $A \TrivialSelection B$. If $B \subset A$, we have what we term a \emph{non-trivial} selection. Every element of $B$ must appear in $A$, but not vice versa. The mapping remains one-to-one, as before. From the perspective of the superset $A$, non-trivial selection edges are therefore surjective, functional, and injective, i.e., $A \SelectionEdge B$.

A projection edge $e : A \LabelledEdge{\sigedge{00}}{\RelProject} B$ indicates that the elements of $B$ are a projection of $A$, where $A$ is a constructed node of the form $C \times D \times \dotsb$ (i.e., a tuple type). A bijective projection edge $A \TrivialProjection B$ indicates that $B$ is unique with respect to $A$. For non-unique projections, every element of $A$ must match an element in $B$, and vice versa. Every element of $A$ is also associated with at most one element of $B$, but not vice versa. Non-unique projection edges are therefore total, surjective, and functional, i.e., $A \ProjectionEdge B$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Comparing SIGs}
\label{sec-sig-compare}

\noindent It is possible to determine the relationship between the relative information capacities of two schemas by comparing SIGs for the two schemas. Consider two schemas $\SC{1}$ and $\SC{2}$, with corresponding SIGs $G_{\SC{1}}$ and $G_{\SC{2}}$. Intuitively, if $G_{\SC{1}}$ and $G_{\SC{2}}$ are isomorphic, then $\SC{1}$ and $\SC{2}$ are equivalent \cite{Miller.R-1994a-SIG}. $G_{\SC{1}}$ and $G_{\SC{2}}$ are considered isomorphic if they have identical structure and semantics, i.e., the same nodes, edges, node types, constraints, and annotations.

Miller et al.\ \cite{Miller.R-1994a-SIG} identify three possible results from comparing two SIGs $G_{\SC{1}}$ and $G_{\SC{2}}$:
\begin{enumerate}
    
    \item $G_{\SC{1}}$ and $G_{\SC{2}}$ are either immediately isomorphic, or one can be manipulated by a series of equivalence preserving transformations (described below) until it is isomorphic to the other. The two schemas are equivalent, i.e., $\SC{1} \equiv \SC{2}$.
    
    \item $G_{\SC{1}}$ and $G_{\SC{2}}$ are not immediately isomorphic, but one can be manipulated by a series of information capacity augmenting transformations (described below) until it is isomorphic to the other. The transformed schema is dominated by the other, e.g., if we transform $G_{\SC{1}}$, then $\Dominates{\SC{2}}{\SC{1}}$.
    
    \item $G_{\SC{1}}$ and $G_{\SC{2}}$ are not immediately isomorphic and cannot be transformed to be isomorphic. The information capacities of $\SC{1}$ and $\SC{2}$ cannot be compared.
    
\end{enumerate}

Miller et al.\ \cite{Miller.R-1994a-Schema} describe the following equivalence preserving and information capacity augmenting transformations on SIGs:
\begin{description}
    
    \item[Annotation transformations] allow the removal of annotations or projection and selection constraints from edges (e.g., $n_{1} \sigedge{23} n_{2} \Rightarrow n_{1} \sigedge{03} n_{2}$, or $n_{3} \TrivialSelection n_{4} \Rightarrow n_{3} \Bijective n_{4}$). If an annotation transformation on $G_{\SC{1}}$ makes it isomorphic to $G_{\SC{2}}$, then the information capacity of $\SC{2}$ dominates that of $\SC{1}$ (i.e., \Dominates{\SC{2}}{\SC{1}}).
    
    \item[Composition transformations] allow edges to be moved and copied around the graph. Consider an edge $e : n_{1} \sigedge{00} n_{2}$, as shown in Figure~\ref{fig-sig-composition}\subref{fig-sig-composition-original}. If there is a path $P = e_{m} \circ \dotsb \circ e_{k}$ of functional edges from some other node $n_{k}$ to $n_{2}$, $e$ may be copied to a new edge $e' : n_{1} \sigedge{00} n_{k}$, as shown in Figure~\ref{fig-sig-composition}\subref{fig-sig-composition-copy}. The original $e$ can then be deleted if desired, effectively moving $e$ to $e'$ as shown in Figure~\ref{fig-sig-composition}\subref{fig-sig-composition-move}.
    
    The annotation of $e'$ is determined by composing $P$ with $e$, i.e., $e \circ P$. If an edge with a particular annotation (such as surjectivity) is composed with an edge without that annotation, the resulting edge also does not have the annotation. For example, the composition of the edges $\Functional$ and $\Surjective$ would be an edge with no annotations ($\Edge$), whereas the composition of $\Functional$ and $\sigedge{03}$ would be $\Functional$.
    
    If a composition transformation on $G_{\SC{1}}$ makes it isomorphic to $G_{\SC{2}}$, then \Dominates{\SC{2}}{\SC{1}}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}[htb]
    \centering
    \subfloat[Original state\label{fig-sig-composition-original}]{
        \begin{tikzpicture}
            \node (n1)                      {$n_{1}$};
            \node (n2) [below=1cm of n1]    {$n_{2}$};
            \node (n3) [left=1cm of n2]     {$n_{3}$};
            \node (ee) [left=0.75cm of n3]  {\ldots};
            \node (nk) [left=0.75cm of ee]  {$n_{k}$};
        
            \graph{
                (n1) -- [edge label={$e$}] (n2);
                (nk) -> [edge label={$e_{k}$},swap] (ee) -> [edge label={$e_{m+1}$},swap] (n3) -> [edge label={$e_{m}$},swap] (n2);
            };
            
            \node (Pleft) [above=2mm of nk.north east] {};
            \node (Pright) [above=2mm of n2.north west] {};
            \node (P) at ($(Pleft)!0.5!(Pright)$) {$P$};
            \draw[decorate,decoration=brace] (nk.north east) -- (n2.north west);
        \end{tikzpicture}
    }
    
    \subfloat[Copy edge $e$ to $e'$ across path $P$\label{fig-sig-composition-copy}]{
        \begin{tikzpicture}
            \node (n1)                      {$n_{1}$};
            \node (n2) [below=1cm of n1]    {$n_{2}$};
            \node (n3) [left=1cm of n2]     {$n_{3}$};
            \node (ee) [left=0.75cm of n3]  {\ldots};
            \node (nk) [left=0.75cm of ee]  {$n_{k}$};
        
            \graph{
                (n1) -- [edge label={$e$}]        (n2);
                (n1) -- [edge label={$e'$},swap]  (nk);
                (nk) -> [edge label={$e_{k}$},swap] (ee) -> [edge label={$e_{m+1}$},swap] (n3) -> [edge label={$e_{m}$},swap] (n2);
            };
        \end{tikzpicture}
    }
    
    \subfloat[Move edge $e$ to $e'$ across path $P$\label{fig-sig-composition-move}]{
        \begin{tikzpicture}
            \node (n1)                      {$n_{1}$};
            \node (n2) [below=1cm of n1]    {$n_{2}$};
            \node (n3) [left=1cm of n2]     {$n_{3}$};
            \node (ee) [left=0.75cm of n3]  {\ldots};
            \node (nk) [left=0.75cm of ee]  {$n_{k}$};
        
            \graph{
                (n1) -- [edge label={$e'$},swap]  (nk);
                (nk) -> [edge label={$e_{k}$},swap] (ee) -> [edge label={$e_{m+1}$},swap] (n3) -> [edge label={$e_{m}$},swap] (n2);
            };
        \end{tikzpicture}
    }
    \caption{Composition transformations.}
    \label{fig-sig-composition}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

\item[Selection transformations] allow the creation and deletion of new nodes and edges as follows:
    \begin{enumerate}
        
        \item An existing node $n$ may be duplicated to create a new node $n'\!$. A trivial selection edge $e:n \TrivialSelection n'$ is added to enforce the constraint that $n'$ is a duplicate of $n$.
        
        \item A node $n_{1}$ may be deleted if it is associated with exactly one other node $n_{2}$ by a single trivial selection edge $e:n_{1} \TrivialSelection n_{2}$ (implying $n_{1}$ is a duplicate of $n_{2}$). This also deletes the trivial selection edge.
        
        \item A trivial selection edge $e:n \TrivialSelection n$ may be created that connects a node $n$ to itself.
        
        \item A trivial selection edge $e:n_{1} \TrivialSelection n_{2}$ may be deleted, removing the association between $n_{1}$ and $n_{2}$.
        
    \end{enumerate}
    
    If we can make $G_{\SC{1}}$ isomorphic to $G_{\SC{2}}$ using any of the first three selection transformations, then \Equivalent{\SC{1}}{\SC{2}}. If we can make $G_{\SC{1}}$ isomorphic to $G_{\SC{2}}$ using the fourth selection transformation, then \Dominates{\SC{2}}{\SC{1}}. The latter may seem somewhat counter-intuitive \cite{Qian.X-1995a-Correct}, but is nonetheless correct \cite{Miller.R-1994a-Schema}. A trivial selection edge $e:n_{1} \TrivialSelection n_{2}$ denotes that $n_{1}$ and $n_{2}$ correspond to the same domain, so removing the edge also removes this constraint. The information capacity of $\SC{2}$ must therefore be greater than $\SC{1}$.
    
\end{description}

These SIG transformations and their effects are summarised in Table~\ref{Tab.SIG.Transformations}.

%%%%%%%%%%%%%%%%%%%%

\begin{table}[htb]
    \centering
    \small
    \begin{tabular}{lcc}
        \toprule
                                                        &                   &   \textbf{Effect on infor-}   \\
        \textbf{Transformation}                         &   \textbf{Type}   &   \textbf{mation capacity}    \\
        \midrule
        Delete annotation or $\RelProject$/$\RelRestrict$ constraint    &   annotation      &   augmented   \\
        Move or copy edge                               &   composition     &   augmented   \\
        Create or delete duplicate node                 &   selection       &   no change   \\
        Create trivial selection edge                   &   selection       &   no change   \\
        Delete trivial selection edge                   &   selection       &   augmented   \\
        \bottomrule
    \end{tabular}
    \caption{Summary of SIG transformations.}
    \label{Tab.SIG.Transformations}
\end{table}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Extending SIGs to model relational concepts}
\label{sec-relational-sigs}

\noindent The direct correspondence between relational types and SIG domains noted in Section~\ref{sec-relations-types} enables us to use SIGs to represent the structure and constraints of relational schemas. Equating tuple types to the Cartesian product of their attributes (equation \ref{eqn-tuple-type}) means that these can also be mapped directly to a constructed SIG node.

Let us again consider the relvar $S\{\Sno, \Sname, \Status, \City\}$ from Date's suppliers-and-parts schema \cite{Date.C-2013a-View}. In addition to type constraints on the attributes, tuples, and the relvar as a whole, $S$ is subject to a (primary) key constraint on \Sno. Taken together with the collection of types $\Type{\Sno}$, $\Type{\Sname}$, $\Type{\Status}$, $\Type{\City}$, $\TT{S}$, and $\Type{S}$, and assuming that $S$ is the only relvar in the schema, we can construct the SIG shown in Figure~\ref{fig-sig-s-alone} as explained below.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}[htb]
    \centering
    \begin{tikzpicture}[level distance=12mm]
        \node (TS) {$\Type{S}$};
        \node (TTS) [right=1cm of TS] {$\TT{S}$}
            [sibling distance=3cm]
            child[arrows={<{crossbar}-{crossbar}>}] {node (TSno) {$\Type{\Sno}$}}
            child[arrows={:{crossbar}-{crossbar}>}] {node (TSSC) {$\TSSC$}
                [sibling distance=15mm]
                child[arrows={:{crossbar}-{crossbar}>}] {node (TSname) {$\Type{\Sname}$}}
                child[arrows={:{crossbar}-{crossbar}>}] {node (TStatus) {$\Type{\Status}$}}
                child[arrows={:{crossbar}-{crossbar}>}] {node (TCity) {$\Type{\City}$}}};
        
        % TS to T{S}
        \draw[arrows={:{crossbar}-{crossbar}:}] (TS) to (TTS);
        
        % FD
        \draw[arrows={:{crossbar}-{crossbar}>}] (TSno) to node[below=-2pt] {\scriptsize\emph{key}} (TSSC);
        
        % projection edges
        \ProjectionAnnotation[above left=-2pt and -4pt]{TTS}{TSno}
        \ProjectionAnnotation[above right=-2pt and -4pt]{TTS}{TSSC}

\ProjectionAnnotation[above left=-2pt and -4pt]{TSSC}{TSname}
        \ProjectionAnnotation[right=-2pt]{TSSC}{TStatus}
        \ProjectionAnnotation[above right=-2pt and -4pt]{TSSC}{TCity}
    \end{tikzpicture}
    \caption{SIG for relvar $S\{\Sno, \Sname, \Status, \City\}$.}
    \label{fig-sig-s-alone}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

As discussed in Section~\ref{sec-relations-types}, any given value of $S$ must comprise a subset of $\TT{S}$, so the association between $\Type{S}$ and $\TT{S}$ is total. Similarly, every tuple in $\TT{S}$ must appear in at least one value of $S$, so the association is also surjective. The association is neither functional nor injective, however, as a value of $S$ in general may contain multiple tuples, and each tuple in $\TT{S}$ can appear in multiple values of $S$. This gives us $\Type{S}\SurTotal\TT{S}$.
    
The key constraint on \Sno\ implies that the functional dependency $\{\Sno\} \rightarrow \{\Sname, \Status, \City\}$ holds in $S$. From equation \ref{eqn-tuple-type}, we can infer the type of $\{\Sname, \Status, \City\}$ as $\TSSC$. The association is functional and not injective by definition. Every supplier number (when used in a value of $S$) must be associated with an element of $\TSSC$, and vice versa. This implies that the association is both total and surjective. Indeed, this holds for every edge that represents a key constraint \cite{Miller.R-1994a-SIG}. We introduce the edge label $\mathit{key}$ to denote an edge representing a key constraint, which gives us $\Type{\Sno} \KeyEdge \TSSC$ in Figure~\ref{fig-sig-s-alone}. 
    
Both $\Type{\Sno}$ and $\TSSC$ are projections of $\TT{S}$, so we can add a non-unique projection edge $\TT{S} \ProjectionEdge \Type{\Sname} \times \Type{\Status} \times T_{\City}$ and a unique projection edge $\TT{S} \TrivialProjection \Type{\Sno}$. Similarly, $\Type{\Sname}$, $\Type{\Status}$, and $\Type{\City}$ are all projections of $\TSSC$, so we can add further non-unique projection edges to these nodes, as shown in Figure~\ref{fig-sig-s-alone}.

Looking at Figure~\ref{fig-sig-s-alone}, it could be argued that the $T_{\dots}$ type notation is redundant and unnecessarily complicates the diagram. Why not just use, e.g., $S$ instead of $\Type{S}$, $\Sno$ instead of $\Type{\Sno}$, and $\Sno \times \Sname \times \Status \times \City$ instead of $\TT{S}$? It is important to remember, however, that SIGs like that shown in Figure~\ref{fig-sig-s-alone} refer to multiple elements of quite different kinds, including relations, tuples, and various combinations of attributes. Discarding the type notation might be reasonable in informal situations, but could lead to confusion between the different kinds of elements represented. The explicit type notation protects the semantics of the various elements of the schema and makes it clear what is being represented.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Characterising view updates}
\label{sec-characterising}

\noindent In this section we show how the extended SIG formalism can be used to model the information equivalence of relational view updates, using Date's restriction view example as a proof of concept \cite{Date.C-2013a-View}.

There are explicit and implicit constraints in $\SC{0}$ that should also apply to the sub-schemas \cite{Date.C-2013a-View}. Ignoring these constraints in a sub-schema $\SC{k}$ will affect its information capacity, and could negatively impact the information equivalence of view updates involving $\SC{k}$. In the next section we will show how such constraints can be propagated into sub-schemas by rewriting them in terms of the sub-schema alone, which enables us to explicitly incorporate into the sub-schema semantic information implicit in $\SC{0}$ that would otherwise be lost. We will then discuss the construction of SIGs for Date's restriction view example (Section~\ref{sec-date-sigs}), and show how to transform and compare these (Section~\ref{sec-transforming}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Constraint derivation and propagation}
\label{sec-constraints}

\noindent Date's motivating example in Chapter 4 of \cite{Date.C-2013a-View} explores what is arguably the simplest case of view updating: disjoint restriction views. The example includes a base relvar $S$ and two views $\LS$ and $\NLS$ derived\footnote{Although as Date notes \cite{Date.C-2013a-View}, we could just as easily define $\LS$ and $\NLS$ as base relvars and then derive $S = \LS \RelUnion \NLS$, or even define all three as base relvars. The constraints remain the same regardless.} from $S$:
\begin{align}
    \LS  &= \RelRestrict_{\City = \mathit{'London'}}S \label{eqn-ls}   \\
    \NLS &= \RelRestrict_{\City \ne \mathit{'London'}}S \label{eqn-nls}
\end{align}

Informally, $\LS$ and $\NLS$ contain details of London and non-London suppliers, respectively. They represent disjoint restrictions of $S$ with the same heading, and have the following tuple types:
\begin{align}
    \TT{\LS}  &= \Type{\Sno} \times \Type{\Sname} \times \Type{\Status} \times \CityLondon \nonumber    \\
    \TT{\NLS} &= \Type{\Sno} \times \Type{\Sname} \times \Type{\Status} \times (\TCityMinusLondon) \nonumber
\end{align}

\item\label{constraint-key} \Sno\ is a key for each of $S$, $\LS$, and $\NLS$, therefore the functional dependency $\{\Sno\} \rightarrow \{\Sname, \Status, \City\}$ also holds in each of (a) $S$; (b) $\LS$; and (c) $\NLS$.
    
    % not sure how relevant the FK constraints are?
    % how to represent? FK is a subset of the original key, so selection edge between S.Sno and each of LS.Sno and NLS.Sno
    % or perhaps a selection edge from Sno to itself? (labelled FK)
    % Notation: \Type{\Sno}, \Type{\LS.\Sno}, \Type{\NLS.\Sno}?
    \item\label{constraint-lsfk} $\RelProject_{\{\Sno\}}\LS \subseteq \RelProject_{\{\Sno\}}S$ [foreign key from $\LS$ to $S$]
    
    \item\label{constraint-nlsfk} $\RelProject_{\{\Sno\}}\NLS \subseteq \RelProject_{\{\Sno\}}S$ [foreign key from $\NLS$ to $S$]
    
    \item\label{constraint-union} $S = \LS \RelUnion \NLS$ [from (\ref{eqn-ls}) and (\ref{eqn-nls}) above]
    
    \item\label{constraint-notlondon} $\RelRestrict_{\City \ne \mathit{'London'}}\LS = \emptyset$ [from (\ref{eqn-ls}) above]
    
    \item\label{constraint-london} $\RelRestrict_{\City = \mathit{'London'}}\NLS = \emptyset$ [from (\ref{eqn-nls}) above]
    
    \item\label{constraint-disjoint} $\LS \RelIntersect \NLS = \emptyset$ [from \Constraint{\ref{constraint-notlondon}} and \Constraint{\ref{constraint-london}}]
    
    \item\label{constraint-relation-type-union} (a) $\Type{S} = \Type{\LS} \RelUnion \Type{\NLS}$; (b) $\TT{S} = \TT{\LS} \RelUnion \TT{\NLS}$ [from \Constraint{\ref{constraint-union}}]
    
    \item\label{constraint-relation-types} (a) $\Type{\LS} \subset \Type{S}$; (b) $\TT{\LS} \subset \TT{S}$ [from (\ref{eqn-ls}) above]
    
    \item\label{constraint-tuple-types} (a) $\Type{\NLS} \subset \Type{S}$; (b) $\TT{\NLS} \subset \TT{S}$ [from (\ref{eqn-nls}) above]
    
    \item\label{constraint-implied-types-london} $\CityLondon \subset \Type{\City}$
    
    \item\label{constraint-implied-types-nonlondon} $(\TCityMinusLondon) \subset \Type{\City}$
    
    \item\label{constraint-implied-types-TSSL} $\TSSL \subset \TSSC$
    
    \item\label{constraint-implied-types-TSSNL} $\TSSNL \subset \TSSC$
    
\end{ConstraintList}

Constraints \Constraint{\ref{constraint-union}}--\Constraint{\ref{constraint-disjoint}} enforce that $\LS$ and $\NLS$ are disjoint restrictions of $S$, while \Constraint{\ref{constraint-relation-type-union}}--\Constraint{\ref{constraint-implied-types-TSSNL}} are additional type constraints that enforce specialisation associations implied by the subtypes and supertypes occurring in $\SC{0}$.

In most typical database schemas, only constraints \Constraint{\ref{constraint-key}}, \Constraint{\ref{constraint-lsfk}}, and \Constraint{\ref{constraint-nlsfk}} will be explicitly declared. Some database schemas may also explicitly declare constraints \Constraint{\ref{constraint-union}} and \Constraint{\ref{constraint-disjoint}}, but it is more likely that these and all the remaining constraints will be implicit in the definition of $\SC{0}$. It is important for the purposes of this analysis to clearly identify all such implicit constraints, so that nothing is missed when constructing corresponding SIGs.

Let us partition $\SC{0}$ into three sub-schemas: $\SC{1} = \{S\}$, $\SC{2} = \{\LS, \NLS\}$, and $\SC{3} = \{LS\}$. We wish to compare the relative information capacities of $(\SC{1}, \SC{2})$ and $(\SC{1}, \SC{3})$, in order to characterise the view update compatibility of the schemas in each pair.

Note that in the following analysis, we omit the foreign key constraints \Constraint{\ref{constraint-lsfk}} and \Constraint{\ref{constraint-nlsfk}}, as we have yet to determine the best way to represent inclusion dependencies of this nature in a SIG\footnote{Miller et al.\ \cite{Miller.R-1994a-SIG} briefly mention inclusion dependencies, but do not discuss how to represent them in a SIG.}, or indeed whether such dependencies are significant.

%%%%%%%%%%%%%%%%%%%%

\begin{table*}
    \centering
    \begin{tabular}{cl}
        \toprule
        \textbf{Constraint}                                     &   \textbf{Implied edge(s)}    \\
        \midrule
        {\Constraint[\SC{1}]{\ref{constraint-key}}}(b)          &   $\Type{\Sno} \KeyEdge \TSSL$   \\
        \phantom{\Constraint[\SC{1}]{\ref{constraint-key}}}(c)  &   $\Type{\Sno} \KeyEdge \TSSNL$   \\
        \midrule
        {\Constraint[\SC{1}]{\ref{constraint-relation-types}}}  &   $\Type{\LSsub} \SurTotal \TT{\LSsub}$ \quad\quad $\Type{S} \SelectionEdge \Type{\LSsub}$ \quad\quad $\TT{S} \SelectionEdge \TT{\LSsub}$   \\
        \midrule
        {\Constraint[\SC{1}]{\ref{constraint-tuple-types}}}     &   $\Type{\NLSsub} \SurTotal \TT{\NLSsub}$ \quad\quad $\Type{S} \SelectionEdge \Type{\NLSsub}$ \quad\quad $\TT{S} \SelectionEdge \TT{\NLSsub}$   \\
        \midrule
        {\Constraint{\ref{constraint-implied-types-london}}}    &   $\Type{\City} \SelectionEdge \CityLondon$   \\
        \midrule
        {\Constraint{\ref{constraint-implied-types-nonlondon}}} &   $\Type{\City} \SelectionEdge (\TCityMinusLondon)$   \\
        \midrule
        {\Constraint{\ref{constraint-implied-types-TSSL}}}      &   $\TSSC \SelectionEdge \TSSL$    \\
            &   $\TT{\LSsub} \ProjectionEdge \TSSL$   \\
            &   $\TSSL \ProjectionEdge \Type{\Sname}$   \\
            &   $\TSSL \ProjectionEdge \Type{\Status}$   \\
            &   $\TSSL \ProjectionEdge\,\CityLondon$  \\
        \midrule
        {\Constraint{\ref{constraint-implied-types-TSSNL}}}     &   $\TSSC \SelectionEdge \TSSNL$ \\
            &   $\TT{\NLSsub} \ProjectionEdge \TSSNL$  \\
            &   $\TSSNL \ProjectionEdge \Type{\Sname}$  \\
            &   $\TSSNL \ProjectionEdge \Type{\Status}$  \\
            &   $\TSSNL \ProjectionEdge \TCityMinusLondon$  \\
        \bottomrule
    \end{tabular}
    \caption{Constraints and the SIG edges they imply for schema $\SC{1}$.}
    \label{Tab.SIG.Implied.Edges}
\end{table*}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Schema $\SC{1}$}
\label{sec-constraints-s-i}

\noindent A user granted access only to $\SC{1}$ can see only relvar $S$. Constraint \Constraint{\ref{constraint-key}}(a) can be propagated directly from $\SC{0}$ into $\SC{1}$ as it is already expressed in terms of $S$ only. Constraints \Constraint{\ref{constraint-implied-types-london}}--\Constraint{\ref{constraint-implied-types-TSSNL}} do not include any named relvars in $\SC{0}$, and thus can also be propagated directly from $\SC{0}$ into $\SC{1}$. Constraints \Constraint{\ref{constraint-key}}(b), \Constraint{\ref{constraint-key}}(c), and \Constraint{\ref{constraint-union}}--\Constraint{\ref{constraint-tuple-types}} cannot be propagated directly from $\SC{0}$ into $\SC{1}$, but can all be rewritten in terms of $S$ only, by substituting the definitions of $\LS$ (equation~\ref{eqn-ls}) and $\NLS$ (equation~\ref{eqn-nls}). To simplify expressions in the following discussion, we shall use $\LSsub$ to denote $\RelRestrict_{\City = \mathit{'London'}}S$ (\ref{eqn-ls}) and $\NLSsub$ to denote $\RelRestrict_{\City \ne \mathit{'London'}}S$ (\ref{eqn-nls}). (This also clearly distinguishes the substitutions from the $\LS$ and $\NLS$ relvars in $\SC{0}$, $\SC{2}$, and $\SC{3}$.) A constraint that has been rewritten so that it can be propagated from $\SC{0}$ into $\SC{1}$ is denoted by, e.g., \Constraint[\SC{1}]{\ref{constraint-union}}. Consequently we can derive the following set of rewritten constraints for $\SC{1}$:
\begin{ConstraintList}[\SC{1}]
    
    \item \Sno\ is a key for each of $\LSsub$ and $\NLSsub$, therefore the functional dependency $\{\Sno\} \rightarrow \{\Sname, \Status, \City\}$ also holds in each of (b) $\LSsub$; and (c) $\NLSsub$. [The fact that we are dealing with two disjoint subsets of $\City$ does not change this.]
    
    \setcounter{constraint}{3}
    \item $S = \LSsub \RelUnion \NLSsub$ [or even just $S = S$]
    
    \item $\RelRestrict_{\City \ne \mathit{'London'}}\LSsub = \emptyset$
    
    \item $\RelRestrict_{\City = \mathit{'London'}}\NLSsub = \emptyset$
    
    \item $\LSsub \RelIntersect \NLSsub = \emptyset$

\item (a) $\Type{S} = \Type{\LSsub} \RelUnion \Type{\NLSsub}$; (b) $\TT{S} = \TT{\LSsub} \RelUnion \TT{\NLSsub}$ \newline
    {[or even just $\Type{S} = \Type{S}$ and $\TT{S} = \TT{S}$]}
    
    \item (a) $\Type{\LSsub} \subset \Type{S}$; (b) $\TT{\LSsub} \subset \TT{S}$ [from (\ref{eqn-ls}) above]
    
    \item (a) $\Type{\NLSsub} \subset \Type{S}$; (b) $\TT{\NLSsub} \subset \TT{S}$ [from (\ref{eqn-nls}) above]
    
\end{ConstraintList}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Schema $\SC{2}$}
\label{sec-constraints-s-ii}

\noindent A user granted access only to $\SC{2}$ can see only relvars $\LS$ and $\NLS$. Constraints \Constraint{\ref{constraint-key}}(b),  \Constraint{\ref{constraint-key}}(c), and \Constraint{\ref{constraint-notlondon}}--\Constraint{\ref{constraint-disjoint}} can be propagated directly from $\SC{0}$ into $\SC{2}$ as they are defined only in terms of $\LS$ and $\NLS$. \Constraint{\ref{constraint-implied-types-london}}--\Constraint{\ref{constraint-implied-types-TSSNL}} can be propagated directly into $\SC{2}$ as they are not expressed in terms of any named relvars in $\SC{0}$. \Constraint{\ref{constraint-key}}(a), \Constraint{\ref{constraint-union}}, \Constraint{\ref{constraint-relation-type-union}}, \Constraint{\ref{constraint-relation-types}}, and \Constraint{\ref{constraint-tuple-types}} can be rewritten in terms of $\LS$ and $\NLS$ only, giving us:
\begin{ConstraintList}[\SC{2}]

\item \Sno\ is a key for $\LS \RelUnion \NLS$, therefore the functional dependency $\{\Sno\} \rightarrow \{\Sname, \Status, \City\}$ also holds in (a) $\LS \RelUnion \NLS$.
    
    \setcounter{constraint}{3}
    \item $\LS \RelUnion \NLS = \LS \RelUnion \NLS$
    
    \setcounter{constraint}{7}
    \item (a) $\Type{\LS} \RelUnion \Type{\NLS} = \Type{\LS} \RelUnion \Type{\NLS}$; \newline
          (b) $\TT{\LS} \RelUnion \TT{\NLS} = \TT{\LS} \RelUnion \TT{\NLS}$
    
    \item (a) $\Type{\LS} \subset \Type{\LS} \RelUnion \Type{\NLS}$; (b) $\TT{\LS} \subset \TT{\LS} \RelUnion \TT{\NLS}$
    
    \item (a) $\Type{\NLS} \subset \Type{\LS} \RelUnion \Type{\NLS}$; (b) $\TT{\NLS} \subset \TT{\LS} \RelUnion \TT{\NLS}$
    
\end{ConstraintList}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Schema $\SC{3}$}
\label{sec-constraints-s-iii}

\noindent A user granted access only to $\SC{3}$ can see only relvar $\LS$. Constraints \Constraint{\ref{constraint-key}}(b) and \Constraint{\ref{constraint-notlondon}} can be propagated directly into $\SC{3}$, as they are defined in terms of $\LS$ only. \Constraint{\ref{constraint-implied-types-london}}--\Constraint{\ref{constraint-implied-types-TSSNL}} can also be propagated directly into $\SC{3}$ as previously discussed. The remaining constraints cannot be rewritten in terms of $\LS$ only, and so cannot be propagated into $\SC{3}$. As we were able to propagate all constraints into $\SC{1}$, this indicates a qualitative difference between $\SC{1}$ and $\SC{3}$. It therefore seems likely that the relative information capacities of $\SC{1}$ and $\SC{3}$ will also differ.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Constructing the SIGs}
\label{sec-date-sigs}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Schema $\SC{1}$}
\label{sec-sigs-s-i}

\noindent The SIG for $\SC{1}$ will not just be that for relvar $S$ as shown in Figure~\ref{fig-sig-s-alone}, as that schema did not incorporate all the constraints implicit in $\SC{0}$. The process for constructing the SIG will be similar to that for the earlier example, however. The constraints appearing in $\SC{1}$ are \Constraint{\ref{constraint-key}}(a), \Constraint[\SC{1}]{\ref{constraint-key}}(b), \Constraint[\SC{1}]{\ref{constraint-key}}(c), \Constraint[\SC{1}]{\ref{constraint-union}}--\Constraint[\SC{1}]{\ref{constraint-tuple-types}}, and \Constraint{\ref{constraint-implied-types-london}}--\Constraint{\ref{constraint-implied-types-TSSNL}}.

The SIG will include all of the edges that appear in Figure~\ref{fig-sig-s-alone}. The constraints within the schema also imply the existence of several subtype nodes (e.g., the two subtypes of $\Type{City}$ implied by \Constraint{\ref{constraint-implied-types-london}} and \Constraint{\ref{constraint-implied-types-nonlondon}}), along with corresponding selection and projection edges, which are listed in Table~\ref{Tab.SIG.Implied.Edges} above. \Constraint[\SC{1}]{\ref{constraint-union}}, \Constraint[\SC{1}]{\ref{constraint-relation-type-union}}(a), and \Constraint[\SC{1}]{\ref{constraint-relation-type-union}}(b) are effectively tautologies that add no new information, and can thus be safely ignored. The completed SIG for $\SC{1}$ is shown in Figure~\ref{fig-sig-s}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\Type{\LSsub}$};   &[7mm]  \node (TTLS) {$\TT{\LSsub}$};     &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                            &                                           &                                   &   \node (TSname) {$\Type{\Sname}$};    \\
            \node (TLSPlusNLS) {$\Type{S}$}; &       \node (TTLSPlusNLS) {$\TT{S}$};   &   \node (TSno) {$\Type{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\Type{City}$};   \\
                                            &                                           &                                   &                                       &                                           &   \node (TStatus) {$\Type{\Status}$};  \\
            \node (TNLS) {$\Type{\NLSsub}$}; &       \node (TTNLS) {$\TT{\NLSsub}$};   &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) },
            };
            
            % FDs
            (TSno) -- [funcdep left,bend left] (TSSL);
            (TSno) -- [funcdep right] (TSSNL);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                { (TTLSPlusNLS), (TTLS) } -- (TSno)
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTNLS) -- (TSno)
            };
            
            { [edges=projection left]
                (TTLS)  -- (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TTNLS) -- (TSSNL),
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (TStatus), (NLSCity), (TSname) },
            };
            
            % selection edges
            { [edges=selection left]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) },
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) },
                (TCity) -- (LSCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
                (TTLSPlusNLS) -- [bend right] (TSSC),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
            (TTLSPlusNLS) -- [projection right,bend right] (TSSC),
        };
    \end{tikzpicture}
    \caption{SIG for schema $\SC{1} = \{S\}$.}
    \label{fig-sig-s}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\Type{\LS}$};                  &[7mm]  \node (TTLS) {$\TT{\LS}$};                    &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                                        &                                                       &                                   &   \node (TSname) {$\Type{\Sname}$};    \\
            \node (TLSPlusNLS) {$\StackTLSPlusNLS$};  &       \node (TTLSPlusNLS) {$\StackTTLSPlusNLS$};    &   \node (TSno) {$\Type{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\Type{City}$};   \\
                                                        &                                                       &                                   &                                       &                                           &   \node (TStatus) {$\Type{\Status}$};  \\
            \node (TNLS) {$\Type{\NLS}$};                &       \node (TTNLS) {$\TT{\NLS}$};                  &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) },
            };
            
            % FDs
            (TSno) -- [funcdep left,bend left] (TSSL);
            (TSno) -- [funcdep right] (TSSNL);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                { (TTLSPlusNLS), (TTLS) } -- (TSno)
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTNLS) -- (TSno)
            };
            
            { [edges=projection left]
                (TTLS)  -- (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TTNLS) -- (TSSNL),
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (TStatus), (NLSCity), (TSname) },
            };
            
            % selection edges
            { [edges=selection left]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) },
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) },
                (TCity) -- (LSCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
                (TTLSPlusNLS) -- [bend right] (TSSC),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
            (TTLSPlusNLS) -- [projection right,bend right] (TSSC),
        };
    \end{tikzpicture}
    \caption{SIG for schema $\SC{2} = \{\LS, \NLS\}$.}
    \label{fig-sig-ls-nls}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\Type{\LS}$};  &[7.5mm]  \node (TTLS) {$\TT{\LS}$};  &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                        &                                       &                                   &   \node (TSname) {$\Type{\Sname}$};    \\
                                        &                                       &   \node (TSno) {$\Type{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\Type{City}$};   \\
                                        &                                       &                                   &                                       &                                           &   \node (TStatus) {$\Type{\Status}$};  \\
                                        &                                       &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TLS) -- (TTLS),
            };
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                (TTLS) -- (TSno)
            };
            
            { [edges=projection left]
                (TTLS)  -- (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (TStatus), (NLSCity), (TSname) },
            };
            
            % selection edges
            { [edges=selection left]
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                (TCity) -- (LSCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
        };
    \end{tikzpicture}
    \caption{SIG for schema $\SC{3} = \{\LS\}$.}
    \label{fig-sig-ls}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Schema $\SC{2}$}
\label{sec-sigs-s-ii}

\noindent The constraints appearing in $\SC{2}$ are \Constraint[\SC{2}]{\ref{constraint-key}}(a), \Constraint{\ref{constraint-key}}(b), \Constraint{\ref{constraint-key}}(c), \Constraint[\SC{2}]{\ref{constraint-union}}, \Constraint{\ref{constraint-notlondon}}--\Constraint{\ref{constraint-disjoint}}, \Constraint[\SC{2}]{\ref{constraint-relation-type-union}}--\Constraint[\SC{2}]{\ref{constraint-tuple-types}}, and \Constraint{\ref{constraint-implied-types-london}}--\Constraint{\ref{constraint-implied-types-TSSNL}}. This is essentially the same set of constraints as $\SC{1}$, but with different rewritings. This means that the SIG for $\SC{2}$ can be built in the same manner as for $\SC{1}$, with some nodes named differently. In particular, \Constraint[\SC{2}]{\ref{constraint-relation-type-union}} implies that there will be nodes $\TLSPlusNLS$ and $\TTLSPlusNLS$ instead of $\Type{S}$ and $\TT{S}$, respectively. There will also be $\Type{\LS}$ instead of $\Type{\LSsub}$, etc. The complete SIG for $\SC{2}$ is shown in Figure~\ref{fig-sig-ls-nls}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Schema $\SC{3}$}
\label{sec-sigs-s-iii}

\noindent The constraints appearing in $\SC{3}$ are \Constraint{\ref{constraint-key}}(b), \Constraint{\ref{constraint-notlondon}}, and \Constraint{\ref{constraint-implied-types-london}}--\Constraint{\ref{constraint-implied-types-TSSNL}}. The SIG will only include edges and nodes that relate either directly to $\LS$ or are independent of any relvar, such as the subtype node $\TSSNL$. The complete SIG for $\SC{3}$ is shown in Figure~\ref{fig-sig-ls} on the previous page.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Transforming the SIGs}
\label{sec-transforming}

Looking at the SIGs for $\SC{1}$ and $\SC{2}$, we can see almost immediately that both SIGs have the same node and edge structure; indeed the only difference is in the names of some of the nodes. However, we already know from the discussion in Sections~\ref{sec-constraints-s-i} and~\ref{sec-constraints-s-ii} that these differently named nodes are just rewritings that represent the same domains. We therefore have an immediate isomorphism between the two SIGs and can conclude that the information capacities of $\SC{1}$ and $\SC{2}$ are equivalent, i.e., $\Equivalent{\SC{1}}{\SC{2}}$.

The comparison between $\SC{1}$ and $\SC{3}$ is more interesting. There are some shared structures, but the two SIGs are clearly different. Since the SIG transformations discussed in Section~\ref{sec-sig-compare} only ever preserve or enhance information capacity, intuitively we should attempt to transform the ``smaller'' SIG for $\SC{3}$ into the ``larger'' SIG for $\SC{1}$. SIG transformations can generally be carried out in two broad phases (using the $\SC{3} \rightarrow \SC{1}$ transformation as an example):
\begin{description}

\item[Duplicate nodes] in $\SC{3}$ (producing $\SC{3}'$) that correspond to supertypes in $\SC{1}$, in order to create nodes corresponding to the extra nodes that $\SC{1}$ has over $\SC{3}$. The goal is to produce a node structure similar to $\SC{1}$. The duplicated nodes are connected to their original nodes by trivial selection edges. These transformations (if applicable) do not alter the information capacity of $\SC{3}'$ (i.e., $\Equivalent{\SC{3}}{\SC{3}'}$).
    
    \item[Copy and move edges] around the SIG to produce an edge structure similar to $\SC{1}$. The annotations of each moved or copied edge are determined by the path it is composed with, as discussed in Section~\ref{sec-sig-compare}. These transformations (if applicable) \emph{increase} the information capacity of $\SC{3}'$ (i.e., $\Dominates{\SC{3}'}{\SC{3}}$).
    
\end{description}

These two phases generally occur in order, but may be interspersed throughout with the following:
\begin{description}

\item[Delete duplicate nodes] in $\SC{3}'$ that are no longer needed, especially those that do not exist in $\SC{1}$. These transformations (if applicable) do not alter the information capacity of $\SC{3}'$ ($\equiv$).
    
    \item[Remove totality annotations] from trivial selection edges in $\SC{3}'$ to change them into non-trivial selection edges. This decouples duplicate nodes into subtypes and supertypes, ensuring that they represent different domains. These transformations (if applicable) \emph{increase} the information capacity of $\SC{3}'$ ($\preceq$).
    
\end{description}
For example, we might duplicate some nodes, remove some totality annotations, move or copy some edges, delete some no longer needed nodes, move or copy some more edges, then remove some more totality annotations.

Applying this process to $\SC{3}$, we first duplicate node $\Type{\LS}$ to $\Type{S}$ and node $\TT{\LS}$ to $\TT{S}$. To obtain a node structure similar to that of $\SC{1}$, we also need to further duplicate $\Type{S}$ to $\Type{\NLS}$ and $\TT{S}$ to $\TT{\NLS}$. The result of these duplications is shown in Figure~\ref{fig-transform-all-duplicates}. The transformed SIG now has the same node structure as that for $\SC{1}$.

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}[t]
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node [input keep] (TLS) {$\Type{\LS}$};     &[7mm]  \node [input keep] (TTLS) {$\TT{\LS}$};   &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                                        &                                                   &                                   &   \node (TSname) {$\Type{\Sname}$};    \\
            \node [output] (TLSPlusNLS) {$\Type{S}$};    &       \node [output] (TTLSPlusNLS) {$\TT{S}$};  &   \node (TSno) {$\Type{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\Type{City}$};   \\
                                                        &                                                   &                                   &                                       &                                           &   \node (TStatus) {$\Type{\Status}$};  \\
            \node [output] (TNLS) {$\Type{\NLS}$};       &       \node [output] (TTNLS) {$\TT{\NLS}$};     &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TLS) -- (TTLS),
            };
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                (TTLS) -- (TSno)
            };
            
            { [edges=projection left]
                (TTLS)  -- (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (TStatus), (NLSCity), (TSname) },
            };
            
            % selection edges
            { [edges=selection left]
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                (TCity) -- (LSCity),
            };
            
            { [edges={bijection,output},edge label'={\scriptsize$\RelRestrict$}]
                { (TLS), (TTLS) } -- { (TLSPlusNLS), (TTLSPlusNLS) },
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) },
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
            
            % transformation indicators
            { [edges={->,gray,dashed,bend left=45}]
                (TLS) -- [edge node={node[sloped,above]{\small\emph{copy}}}] (TLSPlusNLS),
                (TLSPlusNLS) -- [edge node={node[sloped,above,rotate=180]{\small\emph{copy}}}] (TNLS)
            };
        };
    \end{tikzpicture}
    \caption{SIG for transformed schema $\SC{3}'$ after duplicating \Type{\LS} and \TT{\LS} to produce analogues of the additional nodes in $\SC{1}$. Inputs to the transformations are coloured \textcolor{blue}{blue}, outputs are coloured \textcolor{red}{red}. [$\equiv$]}
    \label{fig-transform-all-duplicates}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%

The next phase is to copy and move edges. To produce the edge $\Type{S} \SurTotal \TT{S}$, we can first \emph{copy} $\Type{\LS} \SurTotal \TT{\LS}$ across the path $\TT{S} \TrivialSelection \TT{\LS}$ (giving $\Type{\LS} \SurTotal \TT{S}$), then \emph{move} it across $\Type{S} \TrivialSelection \Type{\LS}$. We can carry out a similar series of transformations to copy and move $\Type{S} \SurTotal \TT{S}$ to $\Type{\NLS} \SurTotal \TT{\NLS}$ (see Figure~\ref{fig-transform-edge-moves}).

We next copy and move the unique projection edge $\TT{\LS} \TrivialProjection \Type{\Sno}$ to produce the edges $\TT{S} \TrivialProjection \Type{\Sno}$ and $\TT{\NLS} \TrivialProjection \Type{\Sno}$. Similarly, we copy and move $\TT{\LS} \ProjectionEdge \TSSL$ to produce $\TT{S} \LabelledEdge{\sigedge{12}}{\RelProject} \TSSC$ and $\TT{\NLS} \LabelledEdge{\sigedge{02}}{\RelProject} \TSSNL$. Note that the first edge in the latter pair of transformations loses its totality annotation, and the second edge its surjectivity annotation, as a result of composition with the pre-existing non-bijective selection edges between the constructed tuple types running vertically down the centre of the SIG.

Finally, we carry out similar copy and move transformations to copy the key edge $\Type{\Sno} \KeyEdge \TSSC$ to $\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSL$, and $\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSNL$. Both new edges lose their totality annotations for the same reason as the projection edges above. The final result of all these edge transformations is shown in Figure~\ref{fig-transform-edge-moves}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}[t]
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\Type{\LS}$};      &[7mm]  \node (TTLS) {$\TT{\LS}$};        &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                            &                                           &                                   &   \node (TSname) {$\Type{\Sname}$};    \\
            \node (TLSPlusNLS) {$\Type{S}$}; &       \node (TTLSPlusNLS) {$\TT{S}$};   &   \node (TSno) {$\Type{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\Type{City}$};   \\
                                            &                                           &                                   &                                       &                                           &   \node (TStatus) {$\Type{\Status}$};  \\
            \node (TNLS) {$\Type{\NLS}$};    &       \node (TTNLS) {$\TT{\NLS}$};      &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TLS)        -- [input keep,edge node={node (c1a) {}}] (TTLS),
                (TLSPlusNLS) -- [output,edge node={node (c1b) {}}]     (TTLSPlusNLS),
                (TNLS)       -- [output,edge node={node (c1c) {}}]     (TTNLS),
            };
            
            % FDs
            (TSno) -- [funcdep left,arrows={-{crossbar}>},bend left,output] (TSSL);
            (TSno) -- [funcdep right,arrows={-{crossbar}>},output] (TSSNL);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                (TTLS)        -- [input keep] (TSno),
                (TTLSPlusNLS) -- [output] (TSno),
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTNLS)       -- [output] (TSno),
            };
            
            { [edges=projection left]
                (TTLS)  -- [input keep,edge node={node (c2a) {}}] (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TSSC)   -- { (TSname),  (TStatus) },
                (TSSL)   -- (TSname),
                (TSSNL)  -- { (TStatus), (NLSCity), (TSname) },
                (TTNLS)  -- [arrows={:->},output,edge node={node (c2c) {}}] (TSSNL),
            };

%%%%%%%%%%%%%%%%%%%%

If we compare Figure~\ref{fig-sig-s} with Figure~\ref{fig-transform-edge-moves}, we can see that the transformed SIG for $\SC{3}'$ now has the same topology as the SIG for $\SC{1}$. We still need to remove the totality annotations from the newly created trivial selection edges between the relation and tuple types on the left, but even when this is done, we do not achieve the desired isomorphism, as the edges $\TT{S} \LabelledEdge{\sigedge{12}}{\RelProject} \TSSC$, $\TT{\NLS} \LabelledEdge{\sigedge{02}}{\RelProject} \TSSNL$, $\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSL$, and $\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSNL$ all have different annotations from the corresponding edges in the SIG for $\SC{1}$. This clearly shows that the information capacities of $\SC{1}$ and $\SC{3}$ are incomparable, and that view updates based on this pair of schemas will be problematic.

If we were to create a fourth sub-schema $\SC{4} = \{\NLS\}$ and compare this with $\SC{1}$, the result would be similar to that for $\SC{3}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusion \& future work}
\label{sec-conclusion}

\noindent In this paper we have shown how to use the concept of relative information capacity and an extension of the schema intension graph (SIG) formalism to more formally model Date's notion of information equivalence with respect to relational view updates. Our analysis of Date's restriction view example \cite{Date.C-2013a-View} using these techniques is consistent with his findings, so the approach shows promise. The information capacities of $\SC{1} = \{S\}$ and $\SC{2} = \{\LS,\NLS\}$ are equivalent, so it should be possible to effectively propagate view updates between $\SC{1}$ and $\SC{2}$, regardless of the configuration of views and base relvars. Conversely, the information capacity of $\SC{3} = \{\LS\}$ is not comparable with that of $\SC{1}$, implying that there are updates that cannot be propagated between $\SC{1}$ and $\SC{3}$.

Our analysis reinforces the importance of explicitly identifying all constraints that can be propagated from the base schema $\SC{0}$ to its sub-schemas. Omission of constraints could produce a different SIG structure that might lead to an erroneous conclusion about a schema's information capacity relative to another.

There are several avenues for future work. First, the proof of concept discussed in this paper only deals with disjoint restriction views, which are arguably the simplest case of view updating. Further work needs to be done to characterise other types of view configuration, such as projection views, union views, and in particular, join views. This will enable us to test and refine the application of SIGs in this context.

The discussion in this paper focused only on the relational context, but relative information capacity and the SIG formalism are data model agnostic. It would therefore be interesting to investigate the application of these techniques to view updating contexts beyond the Relational Model (e.g., deductive databases).

It is unclear at present whether the omission of inclusion dependencies such as foreign keys is significant. If they are significant, the question then is how best to represent them in the SIG formalism. Adding such constraints will clearly change the structure of a SIG, but we suspect that this will not change the relationship between the relative information capacities of different schemas.

It is interesting to note that with the equivalent schemas $\SC{1}$ and $\SC{2}$, we were able to propagate all of the constraints from the base schema $\SC{0}$ (Section~\ref{sec-constraints-s-ii}), whereas with the incomparable schemas $\SC{1}$ and $\SC{3}$, only a subset of the constraints could be propagated (Section~\ref{sec-constraints-s-iii}). A reasonable conjecture is that the degree to which constraints can be propagated from $\SC{0}$ to its sub-schemas could be a direct indicator of information equivalence among the sub-schemas, but this needs to be further investigated.

Finally, we have recently discovered further work that explores and expands upon the concepts of information capacity and meaning within data and database schemas (e.g., \cite{Mantri.R-2010a-Data,Wang.X-2005a-SIG,Xu.K-2009a-Defining}). Some of these ideas may be applicable to our work, and need to be investigated in more detail.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\balance

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibliographystyle{abbrv}
\bibliography{APCCM2017_Stanger}

\end{document}

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\documentclass{beamer} \usepackage[no-math]{fontspec} \usepackage{xunicode} \usepackage{xltxtra} \usepackage{bm} \usepackage{tikz} \usepackage{graphicx} \usepackage{relalg} \usepackage{booktabs} \usepackage{mathtools} \usepackage{hyperref} % Beamer setup \usefonttheme{structurebold} \useinnertheme{rectangles} \usenavigationsymbolstemplate{} \setbeamertemplate{footline}[frame number] % TikZ setup \usetikzlibrary{positioning} \usetikzlibrary{graphs} \usetikzlibrary{decorations.pathreplacing} \usetikzlibrary{calc} \usetikzlibrary{arrows} % fontspec setup \defaultfontfeatures{Mapping=tex-text} \setmainfont{Minion Pro} \setsansfont[Scale=MatchUppercase,BoldFont={Open Sans}]{Open Sans Light} \setmonofont[Scale=MatchLowercase]{Letter Gothic 12 Pitch} % graphicx setup \graphicspath{{images/}} % custom macros % "empty" arrow tip \pgfarrowsdeclare{:}{:}{}{} % custom bar arrow tip, offset from end of line (use "empty" tip at line ends if no >) \tikzset{crossbar/.tip={\|[scale width=1.75,sep=0.25em]}} % various edge styles for TikZ \tikzset{ function/.style={arrows={->}}, injection/.style={arrows={<-}}, total/.style={arrows={:{crossbar}-}}, surjection/.style={arrows={-{crossbar}:}}, bijection/.style={arrows={<{crossbar}-{crossbar}>}}, projection/.style={arrows={:{crossbar}-{crossbar}>}}, projection left/.style={arrows={:{crossbar}-{crossbar}>},edge label={\scriptsize\(\RelProject\)}}, projection right/.style={arrows={:{crossbar}-{crossbar}>},edge label'={\scriptsize\(\RelProject\)}}, selection left/.style={arrows={<-{crossbar}>},edge label={\scriptsize\(\RelRestrict\)}}, selection right/.style={arrows={<-{crossbar}>},edge label'={\scriptsize\(\RelRestrict\)}}, funcdep left/.style={arrows={:{crossbar}-{crossbar}>},edge node={node[sloped,midway,above] {\scriptsize\emph{key}}}}, funcdep right/.style={arrows={:{crossbar}-{crossbar}>},edge node={node[sloped,midway,below] {\scriptsize\emph{key}}}}, surtotal/.style={arrows={:{crossbar}-{crossbar}:}}, input keep/.style={structure,thick}, input delete/.style={structure!40,thick,dashed}, output/.style={alert,thick}, output temp/.style={alert,thick,dashed}, path 1/.style={green!60!black,thick}, path 2/.style={orange,thick}, invisible/.style={opacity=0}, visible on/.style={alt=#1{}{invisible}}, input on/.style={alt=#1{input keep}{}}, output on/.style={alt=#1{output}{}}, alt/.code args={<#1>#2#3}{\alt<#1>{\pgfkeysalso{#2}}{\pgfkeysalso{#3}}} } % projection and selection edge annotations for TikZ \newcommand{\ProjectionAnnotation}[3][]{% \path (#2) to node[above,#1] {\scriptsize\(\RelProject\)} (#3);% } \newcommand{\SelectionAnnotation}[3][]{% \path (#2) to node[above,#1] {\scriptsize\(\RelRestrict\)} (#3);% } \newcounter{constraint} % misc \newcommand{\todo}[1]{\textbf{!!TODO!!} {[#1]}} % commonly used elements \newcommand{\identifier}[1]{\ensuremath{\mathit{#1}}} \newcommand{\LS}{\identifier{LS}} \newcommand{\NLS}{\identifier{NLS}} \newcommand{\LSsub}{\identifier{L}} \newcommand{\NLSsub}{\identifier{N}} \newcommand{\ST}{\identifier{ST}} \newcommand{\SC}{\identifier{SC}} \newcommand{\STsub}{\identifier{T}} \newcommand{\SCsub}{\identifier{C}} \newcommand{\TC}{\identifier{TC}} \newcommand{\TCsub}{\identifier{C}} \newcommand{\Sno}{\identifier{Sno}} \newcommand{\Sname}{\identifier{Sname}} \newcommand{\Status}{\identifier{Status}} \newcommand{\City}{\identifier{City}} \newcommand{\Type}[1]{\ensuremath{T_{#1}}} \newcommand{\TT}[1]{\ensuremath{T_{\{#1\}}}} \newcommand{\CityLondon}{\ensuremath{\{\City\colon\allowbreak\text{`\emph{London}'}\}}} \newcommand{\TCityMinusLondon}{\ensuremath{\Type{\City} \setminus \CityLondon}} \newcommand{\TSSC}{\ensuremath{\Type{\Sname} \times \Type{\Status} \times \Type{\City}}} \newcommand{\TSSL}{\ensuremath{\Type{\Sname} \times \Type{\Status} \times \CityLondon}} \newcommand{\TSSNL}{\ensuremath{\Type{\Sname} \times \Type{\Status} \times (\TCityMinusLondon)}} \newcommand{\TLSPlusNLS}{\ensuremath{\Type{\LS} + \Type{\NLS}}} \newcommand{\TTLSPlusNLS}{\ensuremath{\TT{\LS} + \TT{\NLS}}} \newcommand{\TLSPlusNLSsub}{\ensuremath{\Type{\LSsub} + \Type{\NLSsub}}} \newcommand{\TTLSPlusNLSsub}{\ensuremath{\TT{\LSsub} + \TT{\NLSsub}}} \newcommand{\StackTLSPlusNLS}{\ensuremath{\begin{array}{c}\Type{\LS}\,+ \\ \Type{\NLS}\end{array}}} \newcommand{\StackTTLSPlusNLS}{\ensuremath{\begin{array}{c}\TT{\LS}\,+ \\ \TT{\NLS}\end{array}}} \newcommand{\StackTLSPlusNLSsub}{\ensuremath{\begin{array}{c}\Type{\LSsub}\,+ \\ \Type{\NLSsub}\end{array}}} \newcommand{\StackTTLSPlusNLSsub}{\ensuremath{\begin{array}{c}\TT{\LSsub}\,+ \\ \TT{\NLSsub}\end{array}}} \newcommand{\StackTSSC}{\ensuremath{\begin{array}{c}\Type{\Sname}\,\times \\ \Type{\Status} \times \Type{\City}\end{array}}} \newcommand{\StackTSSL}{\ensuremath{\begin{array}{c}\Type{\Sname} \times \Type{\Status}\,\times \\ \CityLondon\end{array}}} \newcommand{\StackTSSNL}{\ensuremath{\begin{array}{c}\Type{\Sname} \times \Type{\Status}\,\times \\ (\TCityMinusLondon)\end{array}}} \newcommand{\StackTCityMinusLondon}{\ensuremath{\begin{array}{c}\Type{\City}\,\setminus \\ \CityLondon\end{array}}} \newcommand{\schema}[1]{\ensuremath{\mathcal{S}_{#1}}} % dominates \newcommand{\Dominates}[2]{\ensuremath{#2 \preceq #1}} \newcommand{\Equivalent}[2]{\ensuremath{#1 \equiv #2}} % SIG notation (in text) \newcommand{\Sedge}[1]{\ensuremath{\sigma_{\textrm{#1}}}} \newcommand{\SedgeP}[1]{\ensuremath{\sigma_{\textrm{#1}}^{'}}} \newcommand{\medmid}{\raise.125ex\hbox{\scalebox{1}[0.75]{$\mid$}}} % General SIG edges for use in formulas. % Adapted from: http://tex.stackexchange.com/questions/96330/adding-symbols-at-the-ends-of-a-horizontal-line \makeatletter \newlength{\@annotskipleft} \newlength{\@annotskipright} % #1 = left edge component % #2 = right edge component % #3 = left bar annotation % #4 = right bar annotation \newcommand\@sig@edge[4]{% \let\@middle\joinrel% \ifx#1\relbar% \@annotskipleft=.3em% % scrunch up the bare line so it's similar length to \long...arrow \ifx#2\relbar\def\@middle{\joinrel\joinrel\relbar\joinrel\joinrel}\fi% \else% % arrows need a little more clearance \@annotskipleft=.4em% \fi% \ifx#2\relbar\@annotskipright=.3em\else\@annotskipright=.4em\fi% \mathrel{\ooalign{$#1\@middle#2$\cr\hskip\@annotskipleft$#3$\hfil$#4$\hskip\@annotskipright\cr}}% } % 0 = nothing % 1 = bar % 2 = arrowhead % 3 = both \def\@sigedge#1#2{% \ifcase\numexpr#14+#2\relax% \@sig@edge{\relbar}{\relbar}{}{}\or % 00 = ----- \@sig@edge{\relbar}{\relbar}{}{\medmid}\or % 01 = ---+- \@sig@edge{\relbar}{\rightarrow}{}{}\or % 02 = ----> \@sig@edge{\relbar}{\rightarrow}{}{\medmid}\or % 03 = ---+> \@sig@edge{\relbar}{\relbar}{\medmid}{}\or % 10 = -+--- \@sig@edge{\relbar}{\relbar}{\medmid}{\medmid}\or % 11 = -+-+- \@sig@edge{\relbar}{\rightarrow}{\medmid}{}\or % 12 = -+--> \@sig@edge{\relbar}{\rightarrow}{\medmid}{\medmid}\or % 13 = -+-+> \@sig@edge{\leftarrow}{\relbar}{}{}\or % 20 = <---- \@sig@edge{\leftarrow}{\relbar}{}{\medmid}\or % 21 = <--+- \@sig@edge{\leftarrow}{\rightarrow}{}{}\or % 22 = <---> \@sig@edge{\leftarrow}{\rightarrow}{}{\medmid}\or % 23 = <--+> \@sig@edge{\leftarrow}{\relbar}{\medmid}{}\or % 30 = <+--- \@sig@edge{\leftarrow}{\relbar}{\medmid}{\medmid}\or % 31 = <+-+- \@sig@edge{\leftarrow}{\rightarrow}{\medmid}{}\or % 32 = <+--> \@sig@edge{\leftarrow}{\rightarrow}{\medmid}{\medmid} % 33 = <+-+> \fi% } \newcommand{\sigedge}[1]{\ensuremath{\@sigedge#1}} \makeatother \newcommand{\Edge}{\sigedge{00}} \newcommand{\Total}{\sigedge{10}} \newcommand{\Surjective}{\sigedge{01}} \newcommand{\SurTotal}{\sigedge{11}} \newcommand{\Functional}{\sigedge{02}} \newcommand{\Injective}{\sigedge{20}} \newcommand{\Bijective}{\sigedge{33}} % Hack to get the overset symbols to appear at the right height. % \smash removes the spacing around the operator, hence \mathop. \newcommand{\LabelledEdge}[2]{\mathop{\overset{\raisebox{0.3em}{\scriptsize\ensuremath{#2}}}{\smash[t]{#1}}}} \newcommand{\ProjectionEdge}{\LabelledEdge{\sigedge{13}}{\RelProject}} \newcommand{\SelectionEdge}{\LabelledEdge{\sigedge{23}}{\RelRestrict}} \newcommand{\TrivialProjection}{\ensuremath{\LabelledEdge{\Bijective}{\RelProject}}} \newcommand{\TrivialSelection}{\ensuremath{\LabelledEdge{\Bijective}{\RelRestrict}}} \newcommand{\KeyEdge}{\ensuremath{\LabelledEdge{\sigedge{13}}{\mathit{key}}}} % Constraints. \newcommand{\Constraint}[2][]{C\ensuremath{_{#2}\ifx&#1&\else^{#1}\fi}} \newenvironment{ConstraintList}[1][]{% \begin{list}{% \bfseries% \ifx&#1&% \Constraint{\ensuremath{\bm{\arabic{constraint}}}}% \else% \Constraint[\ensuremath{\bm{#1}}]{\ensuremath{\bm{\arabic{constraint}}}}% \fi% }% {\usecounter{constraint}}% }{\end{list}} % Draw a grid to aid TikZ picture drawing/debugging. \newcommand{\DrawGridTikZ}[2]{% \begin{scope}[color=lightgray] \draw[thin,step=1mm] (0.0,0.0) grid (#1,#2);% \draw[thick,step=1cm] (-0.1,-0.1) grid (#1+0.1,#2+0.1);% \pgftext[top,at={\pgfxy(0.0,-0.2)}]{\tiny 0}% \pgftext[right,at={\pgfxy(-0.2,0.0)}]{\tiny 0}% \foreach \x in {1,...,#1} {\pgftext[top,at={\pgfxy(\x,-0.2)}]{\tiny\x}}% \foreach \y in {1,...,#2} {\pgftext[right,at={\pgfxy(-0.2,\y)}]{\tiny\y}}% \end{scope} } % Sometimes we want to put a comment in tiny text on the next line, but the default line skip % will insert too much vertical space. Put a \tinyskip at the end of the line instead. \def\tinyskip{\\[-0.33\baselineskip]} % Bold-face text using the structure colour. \newcommand<>{\structurebf}[1]{\structure#2{\textbf{#1}}} % preamble \title{Characterising relational view updates \\ using relative information capacity} \author{\large Nigel Stanger \\[0.75cm] \hspace{0.5cm} \includegraphics[scale=0.2]{Business_School_LS_ID_crop} \hspace{0.57cm} \rule{1pt}{1.482cm} \hspace{0.46cm} \rmfamily\small \raisebox{0.4cm}{\shortstack[l]{Department of \\ Information Science}} } \date{February 1, 2017} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \thispagestyle{empty} \titlepage } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \centering \begin{tikzpicture} \node (book) {\includegraphics[height=8cm,keepaspectratio]{Date-ViewUpdatingandRelationalTheory-large.png}}; \node[anchor=north] (url) at (book.south) {\tiny\href{http://shop.oreilly.com/product/0636920028437.do}{http://shop.oreilly.com/product/0636920028437.do} (2013)}; \node (date) [below=0.8cm of book.east] {\includegraphics[height=2.2cm,keepaspectratio]{date.png}}; \only<2->{\fill[fill,yellow,semitransparent] (-0.25,-0.75) rectangle (2.55,-0.4);} % \DrawGridTikZ{7.0}{8.0} \end{tikzpicture} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{Some quick terminology} \begin{columns} \begin{column}{0.4\textwidth} \centering \includegraphics[width=0.95\columnwidth,keepaspectratio]{Date-SQLandRelationalTheory.png} \tiny\href{http://shop.oreilly.com/product/0636920022879.do}{http://shop.oreilly.com/product/0636920022879.do} \\ (second edition, 2011) \note<1>[item]{This is the second edition; the third edition is now available.} \end{column} \begin{column}{0.6\textwidth} \structurebf{Value vs.\ variable} \begin{itemize} \item a value is immutable (“2”) \item a variable contains a value \item[\(\Rightarrow\)] \emph{relation} value, variable (\emph{relvar}) \end{itemize} \medskip \structurebf{Relation \alert<2>{heading} \& \alert<3>{body}} \medskip {\scriptsize \begin{tabular}{c\|l\|l\|r\|l\|} \cline{2-5} \textbf{S} & \textbf{\alert<2>{Sno}} & \textbf{\alert<2>{Sname}} & \textbf{\alert<2>{Status}} & \textbf{\alert<2>{City}} \\ \cline{2-5} & \alert<3>{S1} & \alert<3>{Smith} & \alert<3>{20} & \alert<3>{London} \\ & \alert<3>{S2} & \alert<3>{Jones} & \alert<3>{30} & \alert<3>{Paris} \\ \cline{2-5} \end{tabular}} \bigskip \structurebf{Type (\(\bm{\Type{x}}\))} \begin{itemize} \item (finite) set of values that something (\(x\)) can take \end{itemize} \medskip \structurebf{Date doesn’t believe in nulls}\tinyskip {\tiny (but it doesn’t really matter here)} \end{column} \end{columns} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{More on types} \structurebf{Attribute type (\(\bm{\Type{A}}\))} \begin{itemize} \item set of all possible values that \(A\) can take \item[=] \(\{v_{1}\text{,}\, v_{2}\text{,}\, \dotsc\text{,}\, v_{m}\}\) {\tiny (for finite \(m\))} \end{itemize} \medskip \structurebf{Tuple type (\(\bm{\TT{R}}\)) of relvar \(\bm{R}\)} \begin{itemize} \item set of all possible tuple values with \(R\)’s heading \item[=] Cartesian product of \(R\)’s attribute types \item[=] \(\Type{A_{1}} \times \Type{A_{2}} \times \dotsb \times \Type{A_{n}}\) \end{itemize} \medskip \structurebf{Relation type (\(\bm{\Type{R}}\)) of relvar \(\bm{R}\)} \begin{itemize} \item set of all possible relation values with \(R\)’s heading \item[=] every possible combination of elements of \(\TT{R}\) \end{itemize} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{Date’s (hoary and clichéd) suppliers-and-parts schema} \structurebf{\(\bm{S\{\Sno\text{,}\, \Sname\text{,}\, \Status\text{,}\, \City\}}\)} \begin{quote} \upshape Supplier \(\Sno\) is under contract, is named \(\Sname\), has status \(\Status\), and is located in city \(\City\) \end{quote} \bigskip \structurebf{\(\bm{P\{\mathit{Pno}\text{,}\, \mathit{Pname}\text{,}\, \mathit{Colour}\text{,}\, \mathit{Weight}\text{,}\, \City\}}\)} \begin{quote} \upshape Part \(\mathit{Pno}\) is used in the enterprise, is named \(\mathit{Pname}\), has colour \(\mathit{Colour}\), and weight \(\mathit{Weight}\) \end{quote} \bigskip \structurebf{\(\bm{SP\{\Sno\text{,}\, \mathit{Pno}\text{,}\, \mathit{Qty}\}}\)} \begin{quote} \upshape Supplier \(\Sno\) supplies (ships) part \(\mathit{Pno}\) in quantity \(\mathit{Qty}\) \end{quote} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{The view update problem} \centering \Large \begin{tikzpicture}[every edge/.style={draw,semithick}] \node (vdb) {\(V(\mathit{DB})\)}; \node (db) [below=18mm of vdb] {\(\mathit{DB}\vphantom{(')}\)}; \node (vdbp) [right=4cm of vdb] {\(V(\mathit{DB}')\)}; \node (uvdb) [left=-1mm of vdbp] {\(U(V(\mathit{DB})) \stackrel{?}{=}\)}; \node (dbp) at (db -\| vdbp) {\(\mathit{DB}\smash[t]{'}\vphantom{()}\)}; \node (tudb) [left=-1mm of dbp] {\(T(U)(\mathit{DB}) =\)}; \graph{ { (db), (dbp) } -> [edge label={\(V\)},swap] { (vdb), (vdbp) }; (vdb) -> [edge node={node (u) [above=-0.6mm] {\(U\)}}] (uvdb); (db) -> [edge node={node (tu) [above=-0.6mm] {\(T(U)\)}}] (tudb); (u) -> [double equal sign distance=2pt,-implies,edge label={\(T\)}] (tu); }; \end{tikzpicture} \\ \tiny Keller (1985) \emph{Algorithms for translating view updates to database updates for views involving selections, projections, and joins} \note<1>{Keller: “The user specifies update \(U\) against the view of the database, \(V(\mathit{DB})\). The view update translator \(T\) supplies the database update \(T(U)\), which results in \(DB'\) when applied to the database. The new view state is \(V(\mathit{DB}')\). This translation has no side effects in the view if \(V(\mathit{DB}') = U(V(\mathit{DB}))\), that is, if the view has changed precisely in accordance with the user’s request.”} \bigskip \normalsize \begin{uncoverenv}<2-> \structurebf{Problem:} there are often multiple update translations \(T(U)\) that produce the effect of \(U\) but have quite different effects on \(\mathit{DB}\). \bigskip How do we choose? \end{uncoverenv} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{The view update problem} \begin{columns} \begin{column}[T]{0.45\textwidth} \footnotesize\color{gray!70} \begin{tabular}{\|l\|l\|r\|l\|} \multicolumn{4}{l}{\textbf{S}} \\ \hline \textbf{Sno} & \textbf{Sname} & \textbf{Status} & \textbf{City} \\ \hline S1 & Smith & 20 & London \\ S2 & Jones & 30 & Paris \\ S3 & Blake & 30 & Paris \\ \only<1-2,4->{S4} & \only<1-2,4->{Clark} & \only<1-2,4->{20} & \only<1-2,4,6>{London}\only<5>{\alert{???}}\only<7->{\alert{Paris}} \\ S5 & Adams & 30 & Athens \\ \hline \end{tabular} \end{column} \begin{column}[T]{0.45\textwidth} \footnotesize \begin{tabular}{\|l\|l\|r\|l\|} \multicolumn{4}{l}{\(\bm{\text{\textbf{LS}} = \RelRestrict_{\text{\textbf{City}} = \text{\textbf{`London'}}} \,\text{\textbf{S}}}\)} \\ \hline \textbf{Sno} & \textbf{Sname} & \textbf{Status} & \textbf{City} \\ \hline S1 & Smith & 20 & London \\ \only<1-2,4,6>{S4} & \only<1-2,4,6>{Clark}& \only<1-2,4,6>{20} & \only<1-2,4,6>{London} \\ \hline \end{tabular} \end{column} \end{columns} \bigskip \structurebf{Delete S4 from LS?} \begin{enumerate} \item<2-> Delete S4 from S. \item<4-> Change S4’s city to anything other than London. \end{enumerate} \bigskip \begin{uncoverenv}<6-> \structurebf{What if we instead \emph{change} S4’s city to Paris in LS?} \begin{itemize} \item<7->[\(\Rightarrow\)] changing S4’s city to Paris in S \item<7-> side effect: some changes behave like deletes! \end{itemize} \end{uncoverenv} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{The view update problem} \tiny \begin{columns} \begin{column}[T]{0.35\textwidth} \color{gray!70} \begin{tabular}{\|l\|l\|r\|l\|} \multicolumn{4}{l}{\textbf{S}} \\ \hline \textbf{Sno} & \textbf{Sname} & \textbf{Status} & \textbf{City} \\ \hline \only<1-6,8-9>{S1}\only<7>{\alert{S6}} & \only<1-9>{Smith} & \only<1-9>{20} & \only<1-9>{London} \\ S2 & Jones & 30 & Paris \\ S3 & Blake & 30 & Paris \\ S4 & Clark & 20 & London \\ S5 & Adams & 30 & Athens \\ \hline \end{tabular} \bigskip \begin{tabular}{\|l\|l\|r\|} \multicolumn{3}{l}{\textbf{SP}} \\ \hline \textbf{Sno} & \textbf{Pno} & \textbf{Qty} \\ \hline \only<1-2,4,6,8-9>{S1}\only<5>{\alert{S3}}\only<7>{\alert{S6}} & \only<1-2,4-9>{P1} & \only<1-2,4-9>{300} \\ S1 & P2 & 200 \\ S1 & P3 & 400 \\ S1 & P4 & 200 \\ S1 & P5 & 100 \\ S1 & P6 & 100 \\ S2 & P1 & 300 \\ S2 & P2 & 400 \\ S3 & P2 & 200 \\ S4 & P2 & 300 \\ S4 & P4 & 300 \\ S4 & P5 & 400 \\ \hline \end{tabular} \bigskip\medskip \color{black} \end{column} \begin{column}[T]{0.55\textwidth} \begin{tabular}{\|l\|l\|r\|l\|l\|r\|} \multicolumn{4}{l}{\(\bm{\mathsf{SSP} = \mathsf{S} \RelNaturalJoin \mathsf{SP}}\)} \\ \hline \textbf{Sno} & \textbf{Sname} & \textbf{Status} & \textbf{City} & \textbf{Pno} & \textbf{Qty} \\ \hline \only<1-2,4,6,8-9>{S1}\only<5>{\alert{S3}}\only<7>{\alert{S6}} & \only<1-2,4,6-9>{Smith}\only<5>{\alert{Blake}} & \only<1-2,4,6-9>{20}\only<5>{\alert{30}} & \only<1-2,4,6-9>{London}\only<5>{\alert{Paris}} & \only<1-2,4-9>{P1} & \only<1-2,4-9>{300} \\ \only<1-6,8-9>{S1}\only<7>{\alert{S6}} & \only<1-9>{Smith} & \only<1-9>{20} & \only<1-9>{London} & \only<1-9>{P2} & \only<1-9>{200} \\ \only<1-6,8-9>{S1}\only<7>{\alert{S6}} & \only<1-9>{Smith} & \only<1-9>{20} & \only<1-9>{London} & \only<1-9>{P3} & \only<1-9>{400} \\ \only<1-6,8-9>{S1}\only<7>{\alert{S6}} & \only<1-9>{Smith} & \only<1-9>{20} & \only<1-9>{London} & \only<1-9>{P4} & \only<1-9>{200} \\ \only<1-6,8-9>{S1}\only<7>{\alert{S6}} & \only<1-9>{Smith} & \only<1-9>{20} & \only<1-9>{London} & \only<1-9>{P5} & \only<1-9>{100} \\ \only<1-6,8-9>{S1}\only<7>{\alert{S6}} & \only<1-9>{Smith} & \only<1-9>{20} & \only<1-9>{London} & \only<1-9>{P6} & \only<1-9>{100} \\ S2 & Jones & 30 & Paris & P1 & 300 \\ S2 & Jones & 30 & Paris & P2 & 400 \\ S3 & Blake & 30 & Paris & P2 & 200 \\ S4 & Clark & 20 & London & P4 & 300 \\ S4 & Clark & 20 & London & P5 & 400 \\ S4 & Clark & 20 & London & P2 & 200 \\ \hline \end{tabular} \bigskip \scriptsize \structurebf{Delete first row of SSP?} \begin{enumerate} \item<2-> Delete \(\text{SP}\{\text{S1}\text{,}\, \text{P1}\text{,}\, \text{300}\}\). \item<4-\|alert@8> Change supplier number in \(\text{SP}\{\text{S1}\text{,}\, \text{P1}\text{,}\, \text{300}\}\). \item<6-\|alert@8> Change S1’s supplier number in S. \item<9-> Delete S1 from S. {\tiny (nuclear option)} \end{enumerate} \uncover<8>{\tiny\alert{\beamerbutton{technically correct}}} \uncover<11>{\structurebf{Changes are even more problematic}} \end{column} \end{columns} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{Date's solution} \begin{itemize} \item Explicitly specify \emph{all} constraints in advance.\tinyskip {\tiny (``metadata approach'' with similar issues to many earlier attempts)} \item Automatically generate \emph{compensatory rules} from constraints.\tinyskip {\tiny (kind of like declarative triggers)} \item No problem if original and view schemas are \emph{information equivalent}. {\tiny (no hidden information)} \begin{itemize} \item[] ``…if and only if the constraints that apply to [schema \(\schema{1}\)] and [schema \(\schema{2}\)] are such that every proposition that can be represented by [\(\schema{1}\)] can be represented by [\(\schema{2}\)] and vice versa''\\ \hfill{\tiny Date (2013) \emph{View Updating \& Relational Theory}} \end{itemize} \medskip \item[BUT:] Still potential for ambiguity when views hide information.\tinyskip {\tiny(i.e., no longer information equivalent)} \medskip \item Date’s approach involves “\emph{…a clear element of pragma…}”.\tinyskip {\tiny (essentially he argues for “common sense” ambiguity resolution)} \end{itemize} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{Relative information capacity} \framesubtitle{} \begin{itemize} \item “Information content” of a schema. {\tiny (including constraints)} \item Determines set of all valid instances of a schema. \item Can only be compared, not quantitatively measured. \begin{description} \item[equivalence] \(\Equivalent{\schema{1}}{\schema{2}}\) \item[dominance] \(\Dominates{\schema{2}}{\schema{1}}\) \end{description} \item Mappings between schemas can preserve: \begin{description} \item[information] information capacity increases (\(\Rightarrow\Dominates{\schema{2}}{\schema{1}}\)) \item[equivalence] information capacity stays the same (\(\Rightarrow\Equivalent{\schema{1}}{\schema{2}}\)) \end{description} \end{itemize} \flushright\tiny Hull (1986) \emph{Relative information capacity of simple relational database schemata} \\ Miller (1993) \emph{The use of information capacity in schema integration and translation} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{Schema intension graphs (SIGs)} \structurebf{Nodes = typed domains (sets of values)} \begin{itemize} \item simple \item constructed (\(\times\), \(+\)) \end{itemize} \medskip \structurebf{Edges = associations between domains} {\tiny (binary relations)} \begin{itemize} \item Simple constraints: \begin{itemize} \item total (\(A \Total B\)): defined for all \(a \in A\) \item surjective (\(A \Surjective B\)): defined for all \(b \in B\) \item functional (\(A \Functional B\)): \(a\) maps to at most one \(b\) \item injective (\(A \Injective B\)): \(b\) maps to at most one \(a\) \item bijective (\(A \Bijective B\)): all four \end{itemize} \item Selection (\(A \SelectionEdge B\)): \(B\) is a subset of \(A\). {\tiny (\(\Rightarrow B\) is a specialisation or subtype of \(A\))} \item Projection (\(A \ProjectionEdge B\)): \(B\) is a projection of \(A\). {\tiny (\(\Rightarrow A\) is a constructed tuple type)} \end{itemize} \flushright\tiny Miller (1994) \emph{Schema equivalence in heterogeneous systems: Bridging theory and practice} \\ Miller (1994) \emph{Schema intension graphs: A formal model for the study of schema equivalence} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame<1-2>[label=frame.SIG.example] { \frametitle{Example SIG for relvar \(\bm{S\{\Sno\text{,}\, \Sname\text{,}\, \Status\text{,}\, \City\}}\)} \framesubtitle{\uncover<3>{\hyperlink{frame.SIG.Sone}{\beamerreturnbutton{Back}}}} \centering\Large \begin{tikzpicture}[level distance=18mm] \node (TS) {\(\Type{S}\)}; \node (TTS) [right=1.5cm of TS] {\(\TT{S}\)} [sibling distance=4.5cm] child[arrows={<{crossbar}-{crossbar}>}] {node (TSno) {\(\Type{\Sno}\)}} child[arrows={:{crossbar}-{crossbar}>}] {node (TSSC) {\(\TSSC\)} [sibling distance=22mm] child[arrows={:{crossbar}-{crossbar}>}] {node (TSname) {\(\Type{\Sname}\)}} child[arrows={:{crossbar}-{crossbar}>}] {node (TStatus) {\(\Type{\Status}\)}} child[arrows={:{crossbar}-{crossbar}>}] {node (TCity) {\(\Type{\City}\)}}}; \node (TTSexpansion) [right=-2mm of TTS,visible on=<2>,alert] {\tiny\(= \Type{\Sno} \times \Type{\Sname} \times \Type{\Status} \times \Type{\City}\)}; % TS to T{S} \draw[arrows={:{crossbar}-{crossbar}:}] (TS) to (TTS); % FD \draw[arrows={:{crossbar}-{crossbar}>}] (TSno) to node[below=-2pt] {\scriptsize\emph{key}} (TSSC); % projection edges \ProjectionAnnotation[above left=-2pt and -4pt]{TTS}{TSno} \ProjectionAnnotation[above right=-2pt and -4pt]{TTS}{TSSC} \ProjectionAnnotation[above left=-2pt and -4pt]{TSSC}{TSname} \ProjectionAnnotation[right=-2pt]{TSSC}{TStatus} \ProjectionAnnotation[above right=-2pt and -4pt]{TSSC}{TCity} \end{tikzpicture} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}[fragile] \frametitle{SIGs for relational schemas} \centering \begin{tabular}{lll} \toprule \textbf{Relational} & \textbf{SIG} & \textbf{Notation} \\ \midrule attribute type & simple domain & \(\Type{A}\) \\ tuple type & constructed domain (\(\times\)) & \(\TT{R}\) or \(\Type{A} \times \Type{B} \times \dotsc\) \\ relation type & simple domain\footnote{Or possibly a complex constructed type involving union?} & \(\Type{R}\) \\ \midrule key\footnote{Implied by functional dependency.} & func., total, surj.\ edge & \(\Type{A} \KeyEdge \Type{B}\) \\ foreign key\footnote{Inclusion dependency.} & — & — \\ \midrule restrict (\(\RelRestrict\)) & selection edge & \(A \SelectionEdge B\) or \(A \TrivialSelection B\)\footnote{Trivial case.} \\ project (\(\RelProject\)) & projection edge & \(A \ProjectionEdge B\) or \(A \TrivialProjection B\) \\ \bottomrule \end{tabular} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{Comparing and transforming SIGs} \begin{itemize} \item Isomorphism between SIGs for \(\schema{1}\) and \(\schema{2}\) (both structure and semantics) indicates information equivalence: \begin{itemize} \item \(\Equivalent{\schema{1}}{\schema{2}}\) if we can make \(\schema{1}\) isomorphic to \(\schema{2}\) using equivalence preserving transformations {\tiny (or vice versa)} \item \(\Dominates{\schema{2}}{\schema{1}}\) if we can make \(\schema{1}\) isomorphic to \(\schema{2}\) using information preserving transformations {\tiny (or vice versa)} \item \(\schema{1}\) \& \(\schema{2}\) non-comparable if we can’t make them isomorphic \end{itemize} \medskip \item Transforming a SIG effectively transforms the schema. \begin{itemize} \item delete annotation, \(\RelRestrict\), \(\RelProject\) (\(\preceq\)) \item move and/or copy edge (\(\preceq\)) \item create/delete duplicate node: \(A \Rightarrow A \TrivialSelection A'\) (\(\equiv\)) \item delete bijective (trivial) selection edge: \(A \TrivialSelection A' \Rightarrow A \quad A'\) (\(\preceq\)) \item create recursive bijective selection edge: \(A \Rightarrow\)\smash{\tikz[baseline={(A.base)},every loop/.style={out=-25,in=25,bijection}]{\node (A) {\(A\)}; \path (A) edge [loop right] node {\scriptsize\(\RelRestrict\)} ();}}(\(\equiv\)) \end{itemize} \end{itemize} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{Gotta catch ’em all…} \framesubtitle{(constraints, that is)} \begin{columns} \begin{column}[T]{0.5\textwidth} \structurebf{Consider {\tiny (disjoint restriction views, Date Ch. 4)}} \small \begin{itemize} \item \(S\{\Sno\text{,}\,\Sname\text{,}\,\Status\text{,}\,\City\}\) \item view \(\LS = \RelRestrict_{\City = \text{`\emph{London}'}}S\) \item view \(\NLS = \RelRestrict_{\City \ne \text{`\emph{London}'}}S\) \item Base schema \(\schema{0} = \{S\text{,}\, \LS\text{,}\, \NLS\}\) \item Sub-schemas \(\schema{1} = \{S\}\), \(\schema{2} = \{\LS\text{,}\, \NLS\}\), \(\schema{3} = \{LS\}\) \item Date says \((\schema{1}\text{,}\, \schema{2})\) are informa- tion equivalent, \((\schema{1}\text{,}\, \schema{3})\) aren’t.\tinyskip {\tiny (due to information hiding)} \end{itemize} \alert{To construct truly comparable SIGs, we need to propagate all the constraints that we can from \(\schema{0}\) to \(\schema{1}\), \(\schema{2}\), and \(\schema{3}\).} \end{column} \begin{column}[T]{0.5\textwidth} \begin{uncoverenv}<2-> \structurebf{Constraints in \(\bm{\schema{0}}\)} \tiny \begin{ConstraintList} \item \(\{\Sno\} \rightarrow \{\Sname\text{,}\, \Status\text{,}\, \City\}\) holds in (a) \(S\); (b) \(\LS\); and (c) \(\NLS\)\quad [key] \item \(\RelProject_{\{\Sno\}}\LS \subseteq \RelProject_{\{\Sno\}}S\)\quad [FK \(\LS \rightarrow S\)] \item \(\RelProject_{\{\Sno\}}\NLS \subseteq \RelProject_{\{\Sno\}}S\)\quad [FK \(\NLS \rightarrow S\)] \item \(S = \LS \RelUnion \NLS\) \item \(\RelRestrict_{\City \ne \text{`\emph{London}'}}\LS = \emptyset\) \item \(\RelRestrict_{\City = \text{`\emph{London}'}}\NLS = \emptyset\) \item \(\LS \RelIntersect \NLS = \emptyset\) \item (a) \(\Type{S} = \Type{\LS} \RelUnion \Type{\NLS}\); (b) \(\TT{S} = \TT{\LS} \RelUnion \TT{\NLS}\) \item (a) \(\Type{\LS} \subset \Type{S}\); (b) \(\TT{\LS} \subset \TT{S}\) \item (a) \(\Type{\NLS} \subset \Type{S}\); (b) \(\TT{\NLS} \subset \TT{S}\) \item \(\CityLondon \subset \Type{\City}\) \item \((\TCityMinusLondon) \subset \Type{\City}\) \item \(\TSSL \subset \TSSC\) \item \(\TSSNL \subset \TSSC\) \end{ConstraintList} \end{uncoverenv} \end{column} \end{columns} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{Propagating constraints to sub-schemas} \framesubtitle{(ignoring \(\bm{\text{C}_{2}}\) and \(\bm{\text{C}_{3}}\) for now)} \small \structurebf{Directly propagate constraints that reference no relvars} \begin{itemize} \item[] \(\text{C}_{11}\), \(\text{C}_{12}\), \(\text{C}_{13}\), \(\text{C}_{14}\) \end{itemize} \medskip \structurebf{Directly propagate constraints that reference sub-schema relvars only} \begin{itemize} \item[\(\schema{1}\)] \(\text{C}_{1}\)(a) \item[\(\schema{2}\)] \(\text{C}_{1}\)(b), \(\text{C}_{1}\)(c), \(\text{C}_{5}\), \(\text{C}_{6}\), \(\text{C}_{7}\) \item[\(\schema{3}\)] \(\text{C}_{1}\)(b), \(\text{C}_{5}\) \end{itemize} \medskip \structurebf{Rewrite remaining constraints in terms of sub-schema relvars {\tiny (if possible)}} \begin{itemize} \item[\(\schema{1}\)] Replace \(\LS\) with \(\RelRestrict_{\City = \text{`\emph{London}'}}\,S\) and \(\NLS\) with \(\RelRestrict_{\City \ne \text{`\emph{London}'}}\,S\) in \(\text{C}_{1}\)(b), \(\text{C}_{1}\)(c), \(\text{C}_{4}\), \(\text{C}_{5}\), \(\text{C}_{6}\), \(\text{C}_{7}\), \(\text{C}_{8}\), \(\text{C}_{9}\), \(\text{C}_{10}\). \item[\(\schema{2}\)] Replace \(S\) with \(\LS \RelUnion \NLS\) in \(\text{C}_{1}\)(a), \(\text{C}_{4}\), \(\text{C}_{8}\), \(\text{C}_{9}\), \(\text{C}_{10}\). \item[\(\schema{3}\)] Can’t rewrite any remaining constraints. \end{itemize} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{Propagating constraints to sub-schemas} \structurebf{Example} \medskip \begin{tabular}{lll} \textcolor{structure}{\(\schema{0}\)} & \(\text{C}_{4}\) & \(S = \LS \RelUnion \NLS\) \\[4pt] & & \Large\quad\(\Downarrow\) \\[4pt] \textcolor{structure}{\(\schema{1}\)} & \(\text{C}_{4}^{\schema{1}}\) & \(S = (\RelRestrict_{\City = \text{`\emph{London}'}}\,S) \RelUnion (\RelRestrict_{\City \ne \text{`\emph{London}'}}\,S)\) \\[8pt] \textcolor{structure}{\(\schema{2}\)} & \(\text{C}_{4}^{\schema{2}}\) & \(\LS \RelUnion \NLS = \LS \RelUnion \NLS\) \\[8pt] \textcolor{structure}{\(\schema{3}\)} & — & (can’t be propagated) \\ \end{tabular} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame<1>[label=frame.SIG.Sone] { \frametitle{The SIGs: \(\bm{\schema{1} = \{S\}}\)} \framesubtitle{\alt<1>{\hyperlink{frame.SIG.example<3>}{\beamergotobutton{Compare}}}{\hyperlink{frame.SIG.transform<9>}{\beamerreturnbutton{Back}}}} \centering \begin{tikzpicture}[ampersand replacement=\&,transform canvas={scale=0.8}] \matrix[row sep=0.5cm] { \node (TLS) {\(\mathrlap{\,\Type{\LSsub}}\hphantom{\Type{\LS}}\)}; \&[7mm] \node (TTLS) {\(\mathrlap{\,\TT{\LSsub}}\hphantom{\TT{\LS}}\)}; \&[7mm] \&[4mm] \&[-4mm] \node (TSSL) {\(\StackTSSL\)}; \& \& \node (LSCity) {\(\CityLondon\)}; \\ \& \& \& \node (TSname) {\(\Type{\Sname}\)}; \\ \node (TLSPlusNLS) {\(\Type{S}\)}; \& \node (TTLSPlusNLS) {\(\TT{S}\)}; \& \node (TSno) {\(\Type{\Sno}\)}; \& \& \node (TSSC) {\(\StackTSSC\)}; \& \& \node (TCity) {\(\Type{City}\)}; \\ \& \& \& \& \& \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TNLS) {\(\mathrlap{\,\:\Type{\NLSsub}}\hphantom{\Type{\NLS}}\)}; \& \node (TTNLS) {\(\mathrlap{\,\:\TT{\NLSsub}}\hphantom{\TT{\NLS}}\)}; \& \& \& \node (TSSNL) {\(\StackTSSNL\)}; \& \& \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) }, }; % FDs (TSno) -- [funcdep left,bend left] (TSSL); (TSno) -- [funcdep right] (TSSNL); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTLSPlusNLS), (TTLS) } -- (TSno) }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTNLS) -- (TSno) }; { [edges=projection left] (TTLS) -- (TSSL), (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TTNLS) -- (TSSNL), (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (NLSCity), (TStatus) }, (TSSNL) -- [bend left=10] (TSname), }; % selection edges { [edges=selection left] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) }, (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, (TCity) -- (LSCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right] (TSSC), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,bend right] (TSSC), }; \node[below=5mm of NLSCity,alert] (caption) {\scriptsize\shortstack[r]{\textbf{Note:} \(\LSsub = \RelRestrict_{\City = \text{`\emph{London}'}}\,S\) \\ \(\NLSsub = \RelRestrict_{\City \ne \text{`\emph{London}'}}\,S\)}}; \end{tikzpicture} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{The SIGs: \(\bm{\schema{2} = \{\LS\text{,}\, \NLS\}}\)} \framesubtitle{\invisible{\beamergotobutton{Compare}}} \centering \begin{tikzpicture}[ampersand replacement=\&,transform canvas={scale=0.8}] \matrix[row sep=0.5cm] { \node (TLS) {\(\Type{\LS}\)}; \&[7mm] \node (TTLS) {\(\TT{\LS}\)}; \&[7mm] \&[4mm] \&[-4mm] \node (TSSL) {\(\StackTSSL\)}; \& \& \node (LSCity) {\(\CityLondon\)}; \\ \& \& \& \node (TSname) {\(\Type{\Sname}\)}; \\ \node (TLSPlusNLS) {\(\StackTLSPlusNLS\)}; \& \node (TTLSPlusNLS) {\(\StackTTLSPlusNLS\)}; \& \node (TSno) {\(\Type{\Sno}\)}; \& \& \node (TSSC) {\(\StackTSSC\)}; \& \& \node (TCity) {\(\Type{City}\)}; \\ \& \& \& \& \& \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TNLS) {\(\Type{\NLS}\)}; \& \node (TTNLS) {\(\TT{\NLS}\)}; \& \& \& \node (TSSNL) {\(\StackTSSNL\)}; \& \& \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) }, }; % FDs (TSno) -- [funcdep left,bend left] (TSSL); (TSno) -- [funcdep right] (TSSNL); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTLSPlusNLS), (TTLS) } -- (TSno) }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTNLS) -- (TSno) }; { [edges=projection left] (TTLS) -- (TSSL), (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TTNLS) -- (TSSNL), (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (NLSCity), (TStatus) }, (TSSNL) -- [bend left=10] (TSname), }; % selection edges { [edges=selection left] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) }, (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, (TCity) -- (LSCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right] (TSSC), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,bend right] (TSSC), }; \node[below=4cm of current page.south west,structure] (caption) {\Large This is already isomorphic to the SIG for \(\schema{1}\), so \(\Equivalent{\schema{1}}{\schema{2}}\).}; \end{tikzpicture} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{The SIGs: \(\bm{\schema{3} = \{\LS\}}\)} \framesubtitle{\invisible{\beamergotobutton{Compare}}} \centering \begin{tikzpicture}[ampersand replacement=\&,transform canvas={scale=0.8}] \matrix[row sep=0.5cm] { \node (TLS) {\(\Type{\LS}\)}; \&[7mm] \node (TTLS) {\(\TT{\LS}\)}; \&[7mm] \&[4mm] \&[-4mm] \node (TSSL) {\(\StackTSSL\)}; \& \& \node (LSCity) {\(\CityLondon\)}; \\ \& \& \& \node (TSname) {\(\Type{\Sname}\)}; \\ \node[invisible] (TLSPlusNLS) {\(\Type{S}\)}; \& \node[invisible] (TTLSPlusNLS) {\(\TT{S}\)}; \& \node (TSno) {\(\Type{\Sno}\)}; \& \& \node (TSSC) {\(\StackTSSC\)}; \& \& \node (TCity) {\(\Type{City}\)}; \\ \& \& \& \& \& \node (TStatus) {\(\Type{\Status}\)}; \\ \node[invisible] (TNLS) {\(\Type{\NLS}\)}; \& \node[invisible] (TTNLS) {\(\TT{\NLS}\)}; \& \& \& \node (TSSNL) {\(\StackTSSNL\)}; \& \& \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TLS) -- (TTLS), (TLSPlusNLS) --[invisible] (TTLSPlusNLS), (TNLS) --[invisible] (TTNLS), }; % FDs (TSno) -- [funcdep left,bend left,invisible] (TSSL); (TSno) -- [funcdep right,invisible] (TSSNL); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTLS) -- (TSno) }; { [edges=projection left] (TTLS) -- (TSSL), (TSSL) -- { (TStatus), (LSCity) }, (TTLSPlusNLS) --[invisible] (TSno), }; { [edges=projection right] (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (NLSCity), (TStatus) }, (TSSNL) -- [bend left=10] (TSname), (TTNLS) --[invisible] (TSno), (TTNLS) -- [arrows={:->},invisible] (TSSNL), }; % selection edges { [edges=selection left] (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] (TCity) -- (LSCity), }; { [edges=bijection,edge label={\scriptsize\(\RelRestrict\)}] { (TLS), (TTLS) } -- [invisible] { (TLSPlusNLS), (TTLSPlusNLS) }, { (TLSPlusNLS), (TTLSPlusNLS) } -- [invisible] { (TNLS), (TTNLS) }, }; { [edges=selection right] { (TLSPlusNLS), (TTLSPlusNLS) } --[invisible] { (TLS), (TTLS) }, }; { [edges=selection left] { (TLSPlusNLS), (TTLSPlusNLS) } --[invisible] { (TNLS), (TTNLS) }, }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right,invisible] (TSSC), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,arrows={:{crossbar}->},bend right,invisible] (TSSC); }; \node[below=4cm of current page.south west,structure] (caption) {\Large Similarities with \(\schema{1}\) but not identical, so we need to transform…}; \end{tikzpicture} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame[label=frame.SIG.transform] { \frametitle{Transforming \(\bm{\schema{3} \Rightarrow \schema{1}}\)} \framesubtitle{\alt<1-8>{\textcolor{structure}{blue} = input to transformation, \textcolor{alert}{red} = output\invisible{\beamergotobutton{Compare}}}{\hyperlink{frame.SIG.Sone<2>}{\beamergotobutton{Compare}}}} \centering \begin{tikzpicture}[ampersand replacement=\&,transform canvas={scale=0.8}] \matrix[row sep=0.5cm] { \node[input on=<2>] (TLS) {\(\Type{\LS}\)}; \&[7mm] \node[input on=<2>] (TTLS) {\(\TT{\LS}\)}; \&[7mm] \&[4mm] \&[-4mm] \node (TSSL) {\(\StackTSSL\)}; \& \& \node (LSCity) {\(\CityLondon\)}; \\ \& \& \& \node (TSname) {\(\Type{\Sname}\)}; \\ \node[visible on=<2->,output on=<2>,input on=<3>] (TLSPlusNLS) {\(\Type{S}\)}; \& \node[visible on=<2->,output on=<2>,input on=<3>] (TTLSPlusNLS) {\(\TT{S}\)}; \& \node (TSno) {\(\Type{\Sno}\)}; \& \& \node (TSSC) {\(\StackTSSC\)}; \& \& \node (TCity) {\(\Type{City}\)}; \\ \& \& \& \& \& \node (TStatus) {\(\Type{\Status}\)}; \\ \node[visible on=<3->,output on=<3>] (TNLS) {\(\Type{\NLS}\)}; \& \node[visible on=<3->,output on=<3>] (TTNLS) {\(\TT{\NLS}\)}; \& \& \& \node (TSSNL) {\(\StackTSSNL\)}; \& \& \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TLS) --[input on=<4>] (TTLS), (TLSPlusNLS) --[visible on=<4->,output on=<4>] (TTLSPlusNLS), (TNLS) --[visible on=<4->,output on=<4>] (TTNLS), }; % FDs (TSno) -- [funcdep left,arrows={:-{crossbar}>},bend left,visible on=<7->,output on=<7>] (TSSL); (TSno) -- [funcdep right,arrows={:-{crossbar}>},visible on=<7->,output on=<7>] (TSSNL); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTLS) --[input on=<5>] (TSno), (TTLSPlusNLS) --[visible on=<5->,output on=<5>] (TSno) }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTNLS) --[visible on=<5->,output on=<5>] (TSno) }; { [edges=projection left] (TTLS) --[input on=<6>] (TSSL), (TSSL) -- { (TStatus), (LSCity) } }; { [edges=projection right] (TTNLS) -- [arrows={:->},visible on=<6->,output on=<6>] (TSSNL), (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (NLSCity), (TStatus) }, (TSSNL) -- [bend left=10] (TSname), }; % selection edges { [edges=selection left] (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] (TCity) -- (LSCity), }; { [edges=bijection,edge label'={\scriptsize\(\RelRestrict\)}] { (TLS), (TTLS) } -- [visible on=<2-7>,output on=<2>] { (TLSPlusNLS), (TTLSPlusNLS) }, { (TLSPlusNLS), (TTLSPlusNLS) } -- [visible on=<3-7>,output on=<3>] { (TNLS), (TTNLS) }, }; { [edges=selection left] { (TLSPlusNLS), (TTLSPlusNLS) } --[visible on=<8->] { (TLS), (TTLS) }, }; { [edges=selection right] { (TLSPlusNLS), (TTLSPlusNLS) } --[visible on=<8->] { (TNLS), (TTNLS) }, }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right,visible on=<6->] (TSSC), }; (TSno) -- [funcdep right,input on=<7>] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,arrows={:{crossbar}->},bend right,visible on=<6->,output on=<6>] (TSSC); }; \begin{scope}[circle,visible on=<8>,output on=<8>] \node[draw] (c1) at (TLSPlusNLS) [above=11.5pt] {}; \node[draw] (c2) at (TLSPlusNLS) [below=11.5pt] {}; \node[draw] (c3) at (c1 -\| TTLSPlusNLS) {}; \node[draw] (c4) at (c2 -\| TTLSPlusNLS) {}; \end{scope} \node[below=4cm of current page.south west,structure] (caption) { \Large \only<1>{Initial state} \only<2>{Duplicate \(\Type{\LS}\) and \(\TT{\LS}\) [\(\equiv\)]} \only<3>{Duplicate \(\Type{S}\) and \(\TT{S}\) [\(\equiv\)]} \only<4>{Copy and move \(\Type{\LS} \SurTotal \TT{\LS}\) edge (\(\times 2\)) [\(\preceq\)]} \only<5>{Copy and move \(\TT{\LS} \ProjectionEdge \Type{\Sno}\) edge (\(\times 2\)) [\(\preceq\)]} \only<6>{Copy and move \(\TT{\LS} \ProjectionEdge \TSSL\) edge (\(\times 2\)) [\(\preceq\)]} \only<7>{Copy and move \(\Type{\Sno} \KeyEdge \TSSC\) edge (\(\times 2\)) [\(\preceq\)]} \only<8>{Remove totality annotations [\(\preceq\)]} \only<9>{Final state: \emph{not} isomorphic to SIG for \(\schema{1} \Rightarrow\) non-comparable.} }; \end{tikzpicture} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{So what does this mean?} \structurebf{\(\bm{\schema{1}}\) and \(\bm{\schema{2}}\) are information equivalent} \begin{itemize} \item[\(\Rightarrow\)] all view updates between them should be OK \end{itemize} \bigskip \structurebf{\(\bm{\schema{1}}\) and \(\bm{\schema{3}}\) aren’t information equivalent} \begin{itemize} \item[\(\Rightarrow\)] some view updates between them will cause problems \end{itemize} \bigskip \structurebf{This agrees with what Date found} {\tiny (yay!)} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{More examples} \framesubtitle{(Nonloss projection views, Date Ch. 5)} \(S\{\Sno, \Status, \City\}\); views \(\ST = \RelProject_{\{\Sno, \Status\}}S\), \(\SC = \RelProject_{\{\Sno, \City\}}S\) \medskip\bigskip \begin{columns} \begin{column}{0.33\textwidth} \centering \scalebox{0.5}{% \begin{tikzpicture} \matrix[row sep=1.5cm,ampersand replacement=\&] { \node (TST) {\(\Type{\STsub}\)}; \&[7mm] \node (TTST) {\(\TT{\STsub}\)}; \&[10mm] \&[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TSTPlusSC) {\(\Type{S}\)}; \& \node (TTSTPlusSC) {\(\TT{S}\)}; \& \node (TSno) {\(\Type{\Sno}\)}; \& \node (TSSC) {\(\Type{\Status} \times \Type{\City}\)}; \\ \node (TSC) {\(\Type{\SCsub}\)}; \& \node (TTSC) {\(\TT{\SCsub}\)}; \& \& \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TST), (TSC), (TSTPlusSC) } -- { (TTST), (TTSC), (TTSTPlusSC) }, }; % FDs (TSno) -- [funcdep left] (TStatus); (TSno) -- [funcdep right] (TCity); (TSno) -- [funcdep right] (TSSC); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTSTPlusSC), (TTST) } -- (TSno), }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTSC) -- (TSno), }; { [edges=projection left] { (TSTPlusSC), (TTSTPlusSC) } -- { (TST), (TTST) }, (TSSC) -- (TCity), (TTST) -- (TStatus), }; { [edges=projection right] { (TSTPlusSC), (TTSTPlusSC) } -- { (TSC), (TTSC) }, (TSSC) -- (TStatus), (TTSC) -- (TCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TTSTPlusSC) -- [bend right] (TSSC), }; (TTSTPlusSC) -- [projection right, bend right] (TSSC), }; \end{tikzpicture}% } \\ \(\schema{1} = \{S\}\) \end{column} \begin{column}{0.33\textwidth} \centering \scalebox{0.5}{% \begin{tikzpicture} \matrix[row sep=1.5cm,ampersand replacement=\&] { \node (TST) {\(\Type{\ST}\)}; \&[7mm] \node (TTST) {\(\TT{\ST}\)}; \&[10mm] \&[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TSTPlusSC) {\(\Type{J}\)}; \& \node (TTSTPlusSC) {\(\TT{J}\)}; \& \node (TSno) {\(\Type{\Sno}\)}; \& \node (TSSC) {\(\Type{\Status} \times \Type{\City}\)}; \\ \node (TSC) {\(\Type{\SC}\)}; \& \node (TTSC) {\(\TT{\SC}\)}; \& \& \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TST), (TSC), (TSTPlusSC) } -- { (TTST), (TTSC), (TTSTPlusSC) }, }; % FDs (TSno) -- [funcdep left] (TStatus); (TSno) -- [funcdep right] (TCity); (TSno) -- [funcdep right] (TSSC); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTSTPlusSC), (TTST) } -- (TSno), }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTSC) -- (TSno), }; { [edges=projection left] { (TSTPlusSC), (TTSTPlusSC) } -- { (TST), (TTST) }, (TSSC) -- (TCity), (TTST) -- (TStatus), }; { [edges=projection right] { (TSTPlusSC), (TTSTPlusSC) } -- { (TSC), (TTSC) }, (TSSC) -- (TStatus), (TTSC) -- (TCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TTSTPlusSC) -- [bend right] (TSSC), }; (TTSTPlusSC) -- [projection right, bend right] (TSSC), }; \end{tikzpicture}% } \\ \(\schema{2} = \{\ST, \SC\}\) \end{column} \begin{column}{0.33\textwidth} \centering \scalebox{0.5}{% \begin{tikzpicture} \matrix[row sep=1.5cm,ampersand replacement=\&] { \node (TST) {\(\Type{\ST}\)}; \&[7mm] \node (TTST) {\(\TT{\ST}\)}; \&[10mm] \&[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \& \& \node (TSno) {\(\Type{\Sno}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TST) -- (TTST), }; % FDs (TSno) -- [funcdep left] (TStatus); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTST) -- (TSno), }; { [edges=projection left] (TTST) -- (TStatus), }; }; \end{tikzpicture}% } \\ \(\schema{3} = \{\ST\}\) \end{column} \end{columns} \medskip\medskip \begin{tabbing} \structurebf{Results:} \=\(\Equivalent{\schema{1}}{\schema{2}}\) {\tiny (as expected)} \\ \>\(\schema{1}\) and \(\schema{3}\) incomparable {\tiny (as expected)} \end{tabbing} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{More examples} \framesubtitle{(Lossy projection views, Date Ch. 5)} \small \(S\{\Sno, \Status, \City\}\); views \(\ST = \RelProject_{\{\Sno, \Status\}}S\), \(\TC = \RelProject_{\{\Status, \City\}}S\) \medskip \begin{columns} \begin{column}{0.5\textwidth} \centering \scalebox{0.4}{% \begin{tikzpicture} \matrix[row sep=1.5cm,ampersand replacement=\&] { \node (TST) {\(\Type{\STsub}\)}; \&[7mm] \node (TTST) {\(\TT{\STsub}\)}; \&[10mm] \&[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TSTPlusTC) {\(\Type{S}\)}; \& \node (TTSTPlusTC) {\(\TT{S}\)}; \& \node (TSno) {\(\Type{\Sno}\)}; \& \node (TSSC) {\(\Type{\Status} \times \Type{\City}\)}; \\ \node (TTC) {\(\Type{\TCsub}\)}; \& \node (TTTC) {\(\TT{\TCsub}\)}; \& \& \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TST), (TTC), (TSTPlusTC) } -- { (TTST), (TTTC), (TTSTPlusTC) }, }; % FDs { [edges=funcdep right] (TSno) -- { (TCity), (TSSC) }, (TSSC) -- [bend right] (TStatus), }; { [edges=funcdep left] (TSno) -- (TStatus), (TSSC) -- [bend left] (TCity), }; % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTSTPlusTC), (TTST) } -- (TSno), }; { [edges=projection left] { (TSTPlusTC), (TTSTPlusTC) } -- { (TST), (TTST) }, (TTST) -- (TStatus), (TSSC) -- [bend right] (TCity), }; { [edges=projection right] { (TSTPlusTC), (TTSTPlusTC) } -- { (TTC), (TTTC) }, (TSSC) -- [bend left] (TStatus), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TTSTPlusTC) -- [bend right] (TSSC), (TTTC) -- [bend right=10] (TSSC), }; (TTSTPlusTC) -- [projection right, bend right] (TSSC), (TTTC) -- [edges=bijection,edge label'={\scriptsize\(\RelProject\)}, bend right=10] (TSSC), }; \end{tikzpicture} } \\ \(\schema{1} = \{S\}\) \\[\baselineskip] \scalebox{0.4}{% \begin{tikzpicture} \matrix[row sep=1.5cm,ampersand replacement=\&] { \node (TST) {\(\Type{\ST}\)}; \&[7mm] \node (TTST) {\(\TT{\ST}\)}; \&[10mm] \&[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \& \& \node (TSno) {\(\Type{\Sno}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TST) -- (TTST), }; % FDs (TSno) -- [funcdep left] (TStatus); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTST) -- (TSno), }; { [edges=projection left] (TTST) -- (TStatus), }; }; \end{tikzpicture} } \\ \(\schema{3} = \{\ST\}\) \end{column} \begin{column}{0.5\textwidth} \centering \scalebox{0.4}{% \begin{tikzpicture} \matrix[row sep=1.5cm,ampersand replacement=\&] { \node (TST) {\(\Type{\ST}\)}; \&[7mm] \node (TTST) {\(\TT{\ST}\)}; \&[10mm] \&[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \& \& \node (TSno) {\(\Type{\Sno}\)}; \& \node (TSSC) {\(\Type{\Status} \times \Type{\City}\)}; \\ \node (TTC) {\(\Type{\TC}\)}; \& \node (TTTC) {\(\TT{\TC}\)}; \& \& \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TST), (TTC) } -- { (TTST), (TTTC) }, }; % FDs { [edges=funcdep right] (TSSC) -- [bend right] (TStatus), }; { [edges=funcdep left] (TSno) -- (TStatus), (TSSC) -- [bend left] (TCity), }; % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTST) } -- (TSno), }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTTC) -- (TSSC), }; { [edges=projection left] (TTST) -- (TStatus), (TSSC) -- [bend right] (TCity), }; { [edges=projection right] (TSSC) -- [bend left] (TStatus), }; }; \end{tikzpicture} } \\ \(\schema{2} = \{\ST, \TC\}\) \\[\baselineskip] \scalebox{0.4}{% \begin{tikzpicture} \matrix[row sep=1.5cm,ampersand replacement=\&] { \&[7mm] \&[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TTC) {\(\Type{\TC}\)}; \& \node (TTTC) {\(\TT{\TC}\)}; \& \node (TSSC) {\(\Type{\Status} \times \Type{\City}\)}; \\ \& \& \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TTC) } -- { (TTTC) }, }; % FDs { [edges=funcdep right] (TSSC) -- [bend right] (TStatus), }; { [edges=funcdep left] (TSSC) -- [bend left] (TCity), }; % projection edges { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTTC) -- (TSSC), }; { [edges=projection left] (TSSC) -- [bend right] (TCity), }; { [edges=projection right] (TSSC) -- [bend left] (TStatus), }; }; \end{tikzpicture} } \\ \(\schema{4} = \{\TC\}\) \end{column} \end{columns} \bigskip \structurebf{Results:} \(\schema{1}\) isn't comparable with any of \(\schema{2}\), \(\schema{3}\), or \(\schema{4}\) {\tiny (as expected)} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{More examples} \framesubtitle{(Lossy one to one join views, Date Ch. 6)} \(\ST\{\Sno, \Status\}\), \(\SC\{\Sno, \City\}\); view \(S = \ST \RelNaturalJoin \SC\) \medskip\bigskip \begin{columns} \begin{column}{0.33\textwidth} \centering \scalebox{0.5}{% \begin{tikzpicture} \matrix[row sep=1.5cm,ampersand replacement=\&] { \&[7mm] \&[10mm] \&[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TSTPlusSC) {\(\Type{S}\)}; \& \node (TTSTPlusSC) {\(\TT{S}\)}; \& \node (TSno) {\(\Type{\Sno}\)}; \& \node (TSSC) {\(\Type{\Status} \times \Type{\City}\)}; \\ \& \& \& \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TSTPlusSC) -- (TTSTPlusSC), }; % FDs (TSno) -- [funcdep left] (TStatus); (TSno) -- [funcdep right] (TCity); (TSno) -- [funcdep right] (TSSC); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTSTPlusSC) } -- (TSno), }; { [edges=projection left] (TSSC) -- (TCity), }; { [edges=projection right] (TSSC) -- (TStatus), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TTSTPlusSC) -- [bend right] (TSSC), }; (TTSTPlusSC) -- [projection right, bend right] (TSSC), }; \end{tikzpicture} } \\ \(\schema{1} = \{S\}\) \end{column} \begin{column}{0.33\textwidth} \centering \scalebox{0.5}{% \begin{tikzpicture} \matrix[row sep=1.5cm,ampersand replacement=\&] { \node (TST) {\(\Type{\ST}\)}; \&[7mm] \node (TTST) {\(\TT{\ST}\)}; \&[10mm] \&[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \& \& \node (TSno) {\(\Type{\Sno}\)}; \& \\ \node (TSC) {\(\Type{\SC}\)}; \& \node (TTSC) {\(\TT{\SC}\)}; \& \& \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TST), (TSC) } -- { (TTST), (TTSC) }, }; % FDs (TSno) -- [funcdep left] (TStatus); (TSno) -- [funcdep right] (TCity); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTST) } -- (TSno), }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTSC) -- (TSno), }; { [edges=projection left] (TTST) -- (TStatus), }; { [edges=projection right] (TTSC) -- (TCity), }; }; \end{tikzpicture} } \\ \(\schema{2} = \{\ST, \SC\}\) \end{column} \begin{column}{0.33\textwidth} \centering \scalebox{0.5}{% \begin{tikzpicture} \matrix[row sep=1.5cm,ampersand replacement=\&] { \node (TST) {\(\Type{\ST}\)}; \&[7mm] \node (TTST) {\(\TT{\ST}\)}; \&[10mm] \&[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \& \& \node (TSno) {\(\Type{\Sno}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TST) -- (TTST), }; % FDs (TSno) -- [funcdep left] (TStatus); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTST) -- (TSno), }; { [edges=projection left] (TTST) -- (TStatus), }; }; \end{tikzpicture}% } \\ \(\schema{3} = \{\ST\}\) \end{column} \end{columns} \bigskip\medskip \structurebf{Results:} \(\schema{1}\) isn't comparable with either \(\schema{2}\) or \(\schema{3}\) {\tiny (as expected)} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \frametitle{Now what?} \begin{itemize} \item Is this a useful technique? {\tiny (working on analysing more complex examples)} \item Are inclusion dependencies important? \\ If so, how to represent them? \item Do we need the whole transformation process? Determining whether or not all constraints can be propagated from the base schema may already be enough. \item Could it help provide a more formal basis for Date’s “pragma”? \item Might it reveal some interesting new insights about the view update problem? \end{itemize} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \frame { \centering\Huge \structurebf{Questions?} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \againframe<3>{frame.SIG.example} \againframe<2>{frame.SIG.Sone} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}

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APCCM_2017/Presentation/View_Updating.tex 0 → 100644

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% preamble
\title{Characterising relational view updates \\ using relative information capacity}
\author{\large Nigel Stanger \\[0.75cm]
    \hspace*{0.5cm}
    \includegraphics[scale=0.2]{Business_School_LS_ID_crop}
    \hspace*{0.57cm}
    \rule{1pt}{1.482cm}
    \hspace*{0.46cm}
    \rmfamily\small
    \raisebox{0.4cm}{\shortstack[l]{Department of \\ Information Science}}
}
\date{February 1, 2017}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \thispagestyle{empty}
    \titlepage
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \centering
    \begin{tikzpicture}
        \node (book) {\includegraphics[height=8cm,keepaspectratio]{Date-ViewUpdatingandRelationalTheory-large.png}};
        \node[anchor=north] (url) at (book.south) {\tiny\href{http://shop.oreilly.com/product/0636920028437.do}{http://shop.oreilly.com/product/0636920028437.do} (2013)};
        \node (date) [below=0.8cm of book.east] {\includegraphics[height=2.2cm,keepaspectratio]{date.png}};
        \only<2->{\fill[fill,yellow,semitransparent] (-0.25,-0.75) rectangle (2.55,-0.4);}
%         \DrawGridTikZ{7.0}{8.0}
    \end{tikzpicture}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{Some quick terminology}
    
    \begin{columns}
        \begin{column}{0.4\textwidth}
            \centering
            \includegraphics[width=0.95\columnwidth,keepaspectratio]{Date-SQLandRelationalTheory.png}
            
            \tiny\href{http://shop.oreilly.com/product/0636920022879.do}{http://shop.oreilly.com/product/0636920022879.do} \\
            (second edition, 2011)
            
            \note<1>[item]{This is the second edition; the third edition is now available.}
        \end{column}
        \begin{column}{0.6\textwidth}
            \structurebf{Value vs.\ variable}
            \begin{itemize}
                \item a value is immutable (“2”)
                \item a variable contains a value
                \item[$\Rightarrow$] \emph{relation} value, variable (\emph{relvar})
            \end{itemize}
            \medskip
            
            \structurebf{Relation \alert<2>{heading} \& \alert<3>{body}}
            \medskip
            
            {\scriptsize
            \begin{tabular}{c|l|l|r|l|}
                \cline{2-5}
                \textbf{S}  &   \textbf{\alert<2>{Sno}} &   \textbf{\alert<2>{Sname}}   &   \textbf{\alert<2>{Status}}  &   \textbf{\alert<2>{City}}    \\
                \cline{2-5}
                            &   \alert<3>{S1}           &   \alert<3>{Smith}            &   \alert<3>{20}               &   \alert<3>{London}   \\
                            &   \alert<3>{S2}           &   \alert<3>{Jones}            &   \alert<3>{30}               &   \alert<3>{Paris}    \\
                \cline{2-5}
            \end{tabular}}
            \bigskip
            
            \structurebf{Type ($\bm{\Type{x}}$)}
            \begin{itemize}
                \item (finite) set of values that something ($x$) can take
            \end{itemize}
            \medskip
            
            \structurebf{Date doesn’t believe in nulls}\tinyskip
            {\tiny (but it doesn’t really matter here)}
        \end{column}
    \end{columns}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{More on types}
    
    \structurebf{Attribute type ($\bm{\Type{A}}$)}
    \begin{itemize}
        \item set of all possible values that $A$ can take
        \item[=] $\{v_{1}\text{,}\, v_{2}\text{,}\, \dotsc\text{,}\, v_{m}\}$ {\tiny (for finite $m$)}
    \end{itemize}
    \medskip
    
    \structurebf{Tuple type ($\bm{\TT{R}}$) of relvar $\bm{R}$}
    \begin{itemize}
        \item set of all possible tuple values with $R$’s heading
        \item[=] Cartesian product of $R$’s attribute types
        \item[=] $\Type{A_{1}} \times \Type{A_{2}} \times \dotsb \times \Type{A_{n}}$
    \end{itemize}
    \medskip
    
    \structurebf{Relation type ($\bm{\Type{R}}$) of relvar $\bm{R}$}
    \begin{itemize}
        \item set of all possible relation values with $R$’s heading
        \item[=] every possible combination of elements of $\TT{R}$
    \end{itemize}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{Date’s (hoary and clichéd) suppliers-and-parts schema}
    
    \structurebf{$\bm{S\{\Sno\text{,}\, \Sname\text{,}\, \Status\text{,}\, \City\}}$}
    
    \begin{quote}
        \upshape
        Supplier $\Sno$ is under contract, is named $\Sname$, has status $\Status$, and is located in city $\City$
    \end{quote}
    \bigskip
    
    \structurebf{$\bm{P\{\mathit{Pno}\text{,}\, \mathit{Pname}\text{,}\, \mathit{Colour}\text{,}\, \mathit{Weight}\text{,}\, \City\}}$}
    
    \begin{quote}
        \upshape
        Part $\mathit{Pno}$ is used in the enterprise, is named $\mathit{Pname}$, has colour $\mathit{Colour}$, and weight $\mathit{Weight}$
    \end{quote}
    \bigskip
    
    \structurebf{$\bm{SP\{\Sno\text{,}\, \mathit{Pno}\text{,}\, \mathit{Qty}\}}$}
    
    \begin{quote}
        \upshape
        Supplier $\Sno$ supplies (ships) part $\mathit{Pno}$ in quantity $\mathit{Qty}$
    \end{quote}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{The view update problem}

\centering
    \Large
    \begin{tikzpicture}[every edge/.style={draw,semithick}]
        \node (vdb)                         {$V(\mathit{DB})$};
        \node (db)   [below=18mm of vdb]    {$\mathit{DB}\vphantom{(')}$};

\node (vdbp) [right=4cm of vdb]     {$V(\mathit{DB}')$};
        \node (uvdb) [left=-1mm of vdbp]    {$U(V(\mathit{DB})) \stackrel{?}{=}$};

\node (dbp)  at (db -| vdbp)        {$\mathit{DB}\smash[t]{'}\vphantom{()}$};
        \node (tudb) [left=-1mm of dbp]     {$T(U)(\mathit{DB}) =$};

\graph{
            { (db), (dbp) } -> [edge label={$V$},swap] { (vdb), (vdbp) };

(vdb) -> [edge node={node (u)  [above=-0.6mm] {$U$}}]    (uvdb);
            (db)  -> [edge node={node (tu) [above=-0.6mm] {$T(U)$}}] (tudb);

(u)   -> [double equal sign distance=2pt,-implies,edge label={$T$}] (tu);
        };
    \end{tikzpicture}
    \\
    \tiny Keller (1985) \emph{Algorithms for translating view updates to database updates for views involving selections, projections, and joins}
    
    \note<1>{Keller: “The user specifies update $U$ against the view of the database, $V(\mathit{DB})$. The view update translator $T$ supplies the database update $T(U)$, which results in $DB'$ when applied to the database. The new view state is $V(\mathit{DB}')$. This translation has no side effects in the view if $V(\mathit{DB}') = U(V(\mathit{DB}))$, that is, if the view has changed precisely in accordance with the user’s request.”}
    
    \bigskip
    \normalsize
    
    \begin{uncoverenv}<2->
        \structurebf{Problem:} there are often multiple update translations $T(U)$ that produce the effect of $U$ but have quite different effects on $\mathit{DB}$.
        \bigskip
    
        How do we choose?
    \end{uncoverenv}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{The view update problem}
    
    \begin{columns}
        \begin{column}[T]{0.45\textwidth}
            \footnotesize\color{gray!70}
            \begin{tabular}{|l|l|r|l|}
                \multicolumn{4}{l}{\textbf{S}}  \\
                \hline
                \textbf{Sno}        &   \textbf{Sname}      &   \textbf{Status}     &   \textbf{City}   \\
                \hline
                S1                  & Smith                 & 20                    & London \\
                S2                  & Jones                 & 30                    & Paris \\
                S3                  & Blake                 & 30                    & Paris \\
                \only<1-2,4->{S4}  & \only<1-2,4->{Clark} & \only<1-2,4->{20}    & \only<1-2,4,6>{London}\only<5>{\alert{???}}\only<7->{\alert{Paris}} \\
                S5                  & Adams                 & 30                    & Athens \\
                \hline
            \end{tabular}
        \end{column}
        \begin{column}[T]{0.45\textwidth}
            \footnotesize
            \begin{tabular}{|l|l|r|l|}
                \multicolumn{4}{l}{$\bm{\text{\textbf{LS}} = \RelRestrict_{\text{\textbf{City}} = \text{\textbf{`London'}}} \,\text{\textbf{S}}}$}  \\
                \hline
                \textbf{Sno}    &   \textbf{Sname}  &   \textbf{Status} &   \textbf{City}   \\
                \hline
                S1              & Smith             & 20                & London \\
                \only<1-2,4,6>{S4} & \only<1-2,4,6>{Clark}& \only<1-2,4,6>{20}   & \only<1-2,4,6>{London} \\
                \hline
            \end{tabular}
        \end{column}
    \end{columns}
    \bigskip
    
    \structurebf{Delete S4 from LS?}
    
    \begin{enumerate}
        \item<2-> Delete S4 from S.
        \item<4-> Change S4’s city to anything other than London.
    \end{enumerate}
    \bigskip
    
    \begin{uncoverenv}<6->
        \structurebf{What if we instead \emph{change} S4’s city to Paris in LS?}
        \begin{itemize}
            \item<7->[$\Rightarrow$] changing S4’s city to Paris in S
            \item<7-> side effect: some changes behave like deletes!
        \end{itemize}
    \end{uncoverenv}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{The view update problem}
    
    \tiny
    \begin{columns}
        \begin{column}[T]{0.35\textwidth}
            \color{gray!70}
            \begin{tabular}{|l|l|r|l|}
                \multicolumn{4}{l}{\textbf{S}}  \\
                \hline
                \textbf{Sno}        &   \textbf{Sname}      &   \textbf{Status}     &   \textbf{City}   \\
                \hline
                \only<1-6,8-9>{S1}\only<7>{\alert{S6}} & \only<1-9>{Smith} & \only<1-9>{20} & \only<1-9>{London} \\
                S2                  & Jones                 & 30                    & Paris \\
                S3                  & Blake                 & 30                    & Paris \\
                S4                  & Clark                 & 20                    & London \\
                S5                  & Adams                 & 30                    & Athens \\
                \hline
            \end{tabular}
            \bigskip
            
            \begin{tabular}{|l|l|r|}
                \multicolumn{3}{l}{\textbf{SP}}  \\
                \hline
                \textbf{Sno}        &   \textbf{Pno}      &   \textbf{Qty}   \\
                \hline
                \only<1-2,4,6,8-9>{S1}\only<5>{\alert{S3}}\only<7>{\alert{S6}} & \only<1-2,4-9>{P1} & \only<1-2,4-9>{300} \\
                S1 & P2 & 200 \\
                S1 & P3 & 400 \\
                S1 & P4 & 200 \\
                S1 & P5 & 100 \\
                S1 & P6 & 100 \\
                S2 & P1 & 300 \\
                S2 & P2 & 400 \\
                S3 & P2 & 200 \\
                S4 & P2 & 300 \\
                S4 & P4 & 300 \\
                S4 & P5 & 400 \\
                \hline
            \end{tabular}
            \bigskip\medskip
            
            \color{black}
        \end{column}
        \begin{column}[T]{0.55\textwidth}
            \begin{tabular}{|l|l|r|l|l|r|}
                \multicolumn{4}{l}{$\bm{\mathsf{SSP} = \mathsf{S} \RelNaturalJoin \mathsf{SP}}$}  \\
                \hline
                \textbf{Sno}        &   \textbf{Sname}      &   \textbf{Status}     &   \textbf{City}   &   \textbf{Pno}    &   \textbf{Qty}   \\
                \hline
                \only<1-2,4,6,8-9>{S1}\only<5>{\alert{S3}}\only<7>{\alert{S6}} & \only<1-2,4,6-9>{Smith}\only<5>{\alert{Blake}} & \only<1-2,4,6-9>{20}\only<5>{\alert{30}} & \only<1-2,4,6-9>{London}\only<5>{\alert{Paris}} & \only<1-2,4-9>{P1} & \only<1-2,4-9>{300} \\
                \only<1-6,8-9>{S1}\only<7>{\alert{S6}} & \only<1-9>{Smith} & \only<1-9>{20} & \only<1-9>{London} & \only<1-9>{P2} & \only<1-9>{200} \\
                \only<1-6,8-9>{S1}\only<7>{\alert{S6}} & \only<1-9>{Smith} & \only<1-9>{20} & \only<1-9>{London} & \only<1-9>{P3} & \only<1-9>{400} \\
                \only<1-6,8-9>{S1}\only<7>{\alert{S6}} & \only<1-9>{Smith} & \only<1-9>{20} & \only<1-9>{London} & \only<1-9>{P4} & \only<1-9>{200} \\
                \only<1-6,8-9>{S1}\only<7>{\alert{S6}} & \only<1-9>{Smith} & \only<1-9>{20} & \only<1-9>{London} & \only<1-9>{P5} & \only<1-9>{100} \\
                \only<1-6,8-9>{S1}\only<7>{\alert{S6}} & \only<1-9>{Smith} & \only<1-9>{20} & \only<1-9>{London} & \only<1-9>{P6} & \only<1-9>{100} \\
                S2 & Jones & 30 & Paris & P1 & 300 \\
                S2 & Jones & 30 & Paris & P2 & 400 \\
                S3 & Blake & 30 & Paris & P2 & 200 \\
                S4 & Clark & 20 & London & P4 & 300 \\
                S4 & Clark & 20 & London & P5 & 400 \\
                S4 & Clark & 20 & London & P2 & 200 \\
                \hline
            \end{tabular}
            \bigskip
            
            \scriptsize
            \structurebf{Delete first row of SSP?}
            
            \begin{enumerate}
                \item<2-> Delete $\text{SP}\{\text{S1}\text{,}\, \text{P1}\text{,}\, \text{300}\}$.
                \item<4-|alert@8> Change supplier number in $\text{SP}\{\text{S1}\text{,}\, \text{P1}\text{,}\, \text{300}\}$.
                \item<6-|alert@8> Change S1’s supplier number in S.
                \item<9-> Delete S1 from S. {\tiny (nuclear option)}
            \end{enumerate}
            \uncover<8>{\tiny\alert{\beamerbutton{technically correct}}}
            
            \uncover<11>{\structurebf{Changes are even more problematic}}
        \end{column}
    \end{columns}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{Date's solution}
    
    \begin{itemize}
    
        \item Explicitly specify \emph{all} constraints in advance.\tinyskip
        {\tiny (``metadata approach'' with similar issues to many earlier attempts)}
        
        \item Automatically generate \emph{compensatory rules} from constraints.\tinyskip
        {\tiny (kind of like declarative triggers)}
        
        \item No problem if original and view schemas are \emph{information equivalent}.
        {\tiny (no hidden information)}
        \begin{itemize}
        
            \item[] ``…if and only if the constraints that apply to [schema $\schema{1}$] and [schema $\schema{2}$] are such that every proposition that can be represented by [$\schema{1}$] can be represented by [$\schema{2}$] and vice versa''\\
            \hfill{\tiny Date (2013) \emph{View Updating \& Relational Theory}}
        
        \end{itemize}
        \medskip
        
        \item[BUT:] Still potential for ambiguity when views hide information.\tinyskip
        {\tiny(i.e., no longer information equivalent)}
        \medskip
                
        \item Date’s approach involves “\emph{…a clear element of pragma…}”.\tinyskip
        {\tiny (essentially he argues for “common sense” ambiguity resolution)}
    
    \end{itemize}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{Relative information capacity}
    \framesubtitle{}
    
    \begin{itemize}
    
        \item “Information content” of a schema. {\tiny (including constraints)}
        
        \item Determines set of all valid instances of a schema.
        
        \item Can only be compared, not quantitatively measured.
        \begin{description}
            \item[equivalence] $\Equivalent{\schema{1}}{\schema{2}}$
            \item[dominance] $\Dominates{\schema{2}}{\schema{1}}$
        \end{description}
        
        \item Mappings between schemas can preserve:
        \begin{description}
            \item[information] information capacity increases ($\Rightarrow\Dominates{\schema{2}}{\schema{1}}$)
            \item[equivalence] information capacity stays the same ($\Rightarrow\Equivalent{\schema{1}}{\schema{2}}$)
        \end{description}
    
    \end{itemize}
    
    \flushright\tiny
    Hull (1986) \emph{Relative information capacity of simple relational database schemata} \\
    Miller (1993) \emph{The use of information capacity in schema integration and translation}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{Schema intension graphs (SIGs)}
    
    \structurebf{Nodes = typed domains (sets of values)}
    \begin{itemize}
        \item simple
        \item constructed ($\times$, $+$)
    \end{itemize}
    \medskip
    
    \structurebf{Edges = associations between domains} {\tiny (binary relations)}
    \begin{itemize}
        \item Simple constraints:
        \begin{itemize}
            \item total ($A \Total B$): defined for all $a \in A$
            \item surjective ($A \Surjective B$): defined for all $b \in B$
            \item functional ($A \Functional B$): $a$ maps to at most one $b$
            \item injective ($A \Injective B$): $b$ maps to at most one $a$
            \item bijective ($A \Bijective B$): all four
        \end{itemize}
        \item Selection ($A \SelectionEdge B$): $B$ is a subset of $A$. {\tiny ($\Rightarrow B$ is a specialisation or subtype of $A$)}
        \item Projection ($A \ProjectionEdge B$): $B$ is a projection of $A$. {\tiny ($\Rightarrow A$ is a constructed tuple type)}
    \end{itemize}

\flushright\tiny
    Miller (1994) \emph{Schema equivalence in heterogeneous systems: Bridging theory and practice} \\
    Miller (1994) \emph{Schema intension graphs: A formal model for the study of schema equivalence}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame<1-2>[label=frame.SIG.example]
{
    \frametitle{Example SIG for relvar $\bm{S\{\Sno\text{,}\, \Sname\text{,}\, \Status\text{,}\, \City\}}$}
    \framesubtitle{\uncover<3>{\hyperlink{frame.SIG.Sone}{\beamerreturnbutton{Back}}}}
    
    \centering\Large
    \begin{tikzpicture}[level distance=18mm]
        \node (TS) {$\Type{S}$};
        \node (TTS) [right=1.5cm of TS] {$\TT{S}$}
            [sibling distance=4.5cm]
            child[arrows={<{crossbar}-{crossbar}>}] {node (TSno) {$\Type{\Sno}$}}
            child[arrows={:{crossbar}-{crossbar}>}] {node (TSSC) {$\TSSC$}
                [sibling distance=22mm]
                child[arrows={:{crossbar}-{crossbar}>}] {node (TSname) {$\Type{\Sname}$}}
                child[arrows={:{crossbar}-{crossbar}>}] {node (TStatus) {$\Type{\Status}$}}
                child[arrows={:{crossbar}-{crossbar}>}] {node (TCity) {$\Type{\City}$}}};
        
        \node (TTSexpansion) [right=-2mm of TTS,visible on=<2>,alert] {\tiny$= \Type{\Sno} \times \Type{\Sname} \times \Type{\Status} \times \Type{\City}$};
        
        % TS to T{S}
        \draw[arrows={:{crossbar}-{crossbar}:}] (TS) to (TTS);
        
        % FD
        \draw[arrows={:{crossbar}-{crossbar}>}] (TSno) to node[below=-2pt] {\scriptsize\emph{key}} (TSSC);
        
        % projection edges
        \ProjectionAnnotation[above left=-2pt and -4pt]{TTS}{TSno}
        \ProjectionAnnotation[above right=-2pt and -4pt]{TTS}{TSSC}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}[fragile]
    \frametitle{SIGs for relational schemas}
    
    \centering
    \begin{tabular}{lll}
        \toprule
        \textbf{Relational} &   \textbf{SIG}                    &   \textbf{Notation}   \\
        \midrule
        attribute type      &   simple domain                   &   $\Type{A}$    \\
        tuple type          &   constructed domain ($\times$) &   $\TT{R}$ or $\Type{A} \times \Type{B} \times \dotsc$  \\
        relation type       &   simple domain\footnote{Or possibly a complex constructed type involving union?}               &   $\Type{R}$    \\
        \midrule
        key\footnote{Implied by functional dependency.}       &   func., total, surj.\ edge                 &   $\Type{A} \KeyEdge \Type{B}$    \\
        foreign key\footnote{Inclusion dependency.}         &   —                               &   —   \\
        \midrule
        restrict ($\RelRestrict$)  &   selection edge  &   $A \SelectionEdge B$ or $A \TrivialSelection B$\footnote{Trivial case.} \\
        project ($\RelProject$)  &   projection edge  &   $A \ProjectionEdge B$ or $A \TrivialProjection B$ \\
        \bottomrule
    \end{tabular}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{Comparing and transforming SIGs}
    
    \begin{itemize}
    
        \item Isomorphism between SIGs for $\schema{1}$ and $\schema{2}$ (both structure and semantics) indicates information equivalence:
        \begin{itemize}
            \item $\Equivalent{\schema{1}}{\schema{2}}$ if we can make $\schema{1}$ isomorphic to $\schema{2}$ using equivalence preserving transformations {\tiny (or vice versa)}
            \item $\Dominates{\schema{2}}{\schema{1}}$ if we can make $\schema{1}$ isomorphic to $\schema{2}$ using information preserving transformations {\tiny (or vice versa)}
            \item $\schema{1}$ \& $\schema{2}$ non-comparable if we can’t make them isomorphic
        \end{itemize}
        \medskip
        
        \item Transforming a SIG effectively transforms the schema.
        \begin{itemize}
            \item delete annotation, $\RelRestrict$, $\RelProject$ ($\preceq$)
            \item move and/or copy edge ($\preceq$)
            \item create/delete duplicate node: $A \Rightarrow A \TrivialSelection A'$ ($\equiv$)
            \item delete bijective (trivial) selection edge: $A \TrivialSelection A' \Rightarrow A \quad A'$ ($\preceq$)
            \item create recursive bijective selection edge: $A \Rightarrow$\smash{\tikz[baseline={(A.base)},every loop/.style={out=-25,in=25,bijection}]{\node (A) {$A$}; \path (A) edge [loop right] node {\scriptsize$\RelRestrict$} ();}}($\equiv$)
        \end{itemize}
        
    \end{itemize}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{Gotta catch ’em all…}
    \framesubtitle{(constraints, that is)}
    
    \begin{columns}
        \begin{column}[T]{0.5\textwidth}
            \structurebf{Consider {\tiny (disjoint restriction views, Date Ch. 4)}}
            
            \small
            \begin{itemize}
                \item $S\{\Sno\text{,}\,\Sname\text{,}\,\Status\text{,}\,\City\}$
                \item view $\LS = \RelRestrict_{\City = \text{`\emph{London}'}}S$
                \item view $\NLS = \RelRestrict_{\City \ne \text{`\emph{London}'}}S$
                \item Base schema $\schema{0} = \{S\text{,}\, \LS\text{,}\, \NLS\}$
                \item Sub-schemas $\schema{1} = \{S\}$, $\schema{2} = \{\LS\text{,}\, \NLS\}$, $\schema{3} = \{LS\}$
                \item Date says $(\schema{1}\text{,}\, \schema{2})$ are informa- tion equivalent, $(\schema{1}\text{,}\, \schema{3})$ aren’t.\tinyskip
                {\tiny (due to information hiding)}
            \end{itemize}
            
            \alert{To construct truly comparable SIGs, we need to propagate all the constraints that we can from $\schema{0}$ to $\schema{1}$, $\schema{2}$, and $\schema{3}$.}
        \end{column}
        \begin{column}[T]{0.5\textwidth}
            \begin{uncoverenv}<2->
                \structurebf{Constraints in $\bm{\schema{0}}$}
            
                \tiny
                \begin{ConstraintList}
                    \item $\{\Sno\} \rightarrow \{\Sname\text{,}\, \Status\text{,}\, \City\}$ holds in (a) $S$; (b) $\LS$; and (c) $\NLS$\quad [key]
    
                    \item $\RelProject_{\{\Sno\}}\LS \subseteq \RelProject_{\{\Sno\}}S$\quad [FK $\LS \rightarrow S$]
    
                    \item $\RelProject_{\{\Sno\}}\NLS \subseteq \RelProject_{\{\Sno\}}S$\quad [FK $\NLS \rightarrow S$]
    
                    \item $S = \LS \RelUnion \NLS$
    
                    \item $\RelRestrict_{\City \ne \text{`\emph{London}'}}\LS = \emptyset$
    
                    \item $\RelRestrict_{\City = \text{`\emph{London}'}}\NLS = \emptyset$
    
                    \item $\LS \RelIntersect \NLS = \emptyset$
    
                    \item (a) $\Type{S} = \Type{\LS} \RelUnion \Type{\NLS}$; (b) $\TT{S} = \TT{\LS} \RelUnion \TT{\NLS}$
    
                    \item (a) $\Type{\LS} \subset \Type{S}$; (b) $\TT{\LS} \subset \TT{S}$
    
                    \item (a) $\Type{\NLS} \subset \Type{S}$; (b) $\TT{\NLS} \subset \TT{S}$
    
                    \item $\CityLondon \subset \Type{\City}$
    
                    \item $(\TCityMinusLondon) \subset \Type{\City}$
    
                    \item $\TSSL \subset \TSSC$
    
                    \item $\TSSNL \subset \TSSC$
                \end{ConstraintList}
            \end{uncoverenv}
        \end{column}
    \end{columns}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{Propagating constraints to sub-schemas}
    \framesubtitle{(ignoring $\bm{\text{C}_{2}}$ and $\bm{\text{C}_{3}}$ for now)}
    
    \small
    
    \structurebf{Directly propagate constraints that reference no relvars}
    \begin{itemize}
        \item[] $\text{C}_{11}$, $\text{C}_{12}$, $\text{C}_{13}$, $\text{C}_{14}$
    \end{itemize}
    \medskip
    
    \structurebf{Directly propagate constraints that reference sub-schema relvars only}
    \begin{itemize}
        \item[$\schema{1}$] $\text{C}_{1}$(a)
        \item[$\schema{2}$] $\text{C}_{1}$(b), $\text{C}_{1}$(c), $\text{C}_{5}$, $\text{C}_{6}$, $\text{C}_{7}$
        \item[$\schema{3}$] $\text{C}_{1}$(b), $\text{C}_{5}$
    \end{itemize}
    \medskip
    
    \structurebf{Rewrite remaining constraints in terms of sub-schema relvars {\tiny (if possible)}}
    
    \begin{itemize}
        \item[$\schema{1}$] Replace $\LS$ with $\RelRestrict_{\City = \text{`\emph{London}'}}\,S$ and $\NLS$ with $\RelRestrict_{\City \ne \text{`\emph{London}'}}\,S$ in $\text{C}_{1}$(b), $\text{C}_{1}$(c), $\text{C}_{4}$, $\text{C}_{5}$, $\text{C}_{6}$, $\text{C}_{7}$, $\text{C}_{8}$, $\text{C}_{9}$, $\text{C}_{10}$.
        
        \item[$\schema{2}$] Replace $S$ with $\LS \RelUnion \NLS$ in $\text{C}_{1}$(a), $\text{C}_{4}$, $\text{C}_{8}$, $\text{C}_{9}$, $\text{C}_{10}$.
        
        \item[$\schema{3}$] Can’t rewrite any remaining constraints.
    \end{itemize}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{Propagating constraints to sub-schemas}
    
    \structurebf{Example}
    \medskip
    
    \begin{tabular}{lll}
        \textcolor{structure}{$\schema{0}$}    &   $\text{C}_{4}$            &   $S = \LS \RelUnion \NLS$    \\[4pt]
                                        &                               &   \Large\quad$\Downarrow$  \\[4pt]
        \textcolor{structure}{$\schema{1}$}    &   $\text{C}_{4}^{\schema{1}}$   &   $S = (\RelRestrict_{\City = \text{`\emph{London}'}}\,S) \RelUnion (\RelRestrict_{\City \ne \text{`\emph{London}'}}\,S)$    \\[8pt]
        \textcolor{structure}{$\schema{2}$}    &   $\text{C}_{4}^{\schema{2}}$   &   $\LS \RelUnion \NLS = \LS \RelUnion \NLS$ \\[8pt]
        \textcolor{structure}{$\schema{3}$}    &  —                            &   (can’t be propagated)   \\
    \end{tabular}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame<1>[label=frame.SIG.Sone]
{
    \frametitle{The SIGs: $\bm{\schema{1} = \{S\}}$}
    \framesubtitle{\alt<1>{\hyperlink{frame.SIG.example<3>}{\beamergotobutton{Compare}}}{\hyperlink{frame.SIG.transform<9>}{\beamerreturnbutton{Back}}}}
    
    \centering
    \begin{tikzpicture}[ampersand replacement=\&,transform canvas={scale=0.8}]
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\mathrlap{\,\Type{\LSsub}}\hphantom{\Type{\LS}}$};    \&[7mm]  \node (TTLS) {$\mathrlap{\,\TT{\LSsub}}\hphantom{\TT{\LS}}$};     \&[7mm]                              \&[4mm]                                  \&[-4mm] \node (TSSL) {$\StackTSSL$};      \&                                       \&   \node (LSCity) {$\CityLondon$};   \\
                                                \&                                           \&                                   \&   \node (TSname) {$\Type{\Sname}$};    \\
            \node (TLSPlusNLS) {$\Type{S}$};  \&       \node (TTLSPlusNLS) {$\TT{S}$};   \&   \node (TSno) {$\Type{\Sno}$}; \&                                       \&       \node (TSSC) {$\StackTSSC$};      \&                                       \&   \node (TCity) {$\Type{City}$};   \\
                                                \&                                           \&                                   \&                                       \&                                           \&   \node (TStatus) {$\Type{\Status}$};  \\
            \node (TNLS) {$\mathrlap{\,\:\Type{\NLSsub}}\hphantom{\Type{\NLS}}$};  \&       \node (TTNLS) {$\mathrlap{\,\:\TT{\NLSsub}}\hphantom{\TT{\NLS}}$};   \&                                   \&                                       \&       \node (TSSNL) {$\StackTSSNL$};    \&                                       \&   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) },
            };
            
            % FDs
            (TSno) -- [funcdep left,bend left] (TSSL);
            (TSno) -- [funcdep right] (TSSNL);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                { (TTLSPlusNLS), (TTLS) } -- (TSno)
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTNLS) -- (TSno)
            };
            
            { [edges=projection left]
                (TTLS)  -- (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TTNLS) -- (TSSNL),
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (NLSCity), (TStatus) },
                (TSSNL) -- [bend left=10] (TSname),
            };
            
            % selection edges
            { [edges=selection left]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) },
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) },
                (TCity) -- (LSCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
                (TTLSPlusNLS) -- [bend right] (TSSC),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
            (TTLSPlusNLS) -- [projection right,bend right] (TSSC),
        };
        
        \node[below=5mm of NLSCity,alert] (caption) {\scriptsize\shortstack[r]{\textbf{Note:} $\LSsub = \RelRestrict_{\City = \text{`\emph{London}'}}\,S$ \\ $\NLSsub = \RelRestrict_{\City \ne \text{`\emph{London}'}}\,S$}};
    \end{tikzpicture}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{The SIGs: $\bm{\schema{2} = \{\LS\text{,}\, \NLS\}}$}
    \framesubtitle{\invisible{\beamergotobutton{Compare}}}
    
    \centering
    \begin{tikzpicture}[ampersand replacement=\&,transform canvas={scale=0.8}]
        \matrix[row sep=0.5cm]
        {
            \node (TLS) {$\Type{\LS}$};                  \&[7mm]  \node (TTLS) {$\TT{\LS}$};                    \&[7mm]                              \&[4mm]                                  \&[-4mm] \node (TSSL) {$\StackTSSL$};      \&                                       \&   \node (LSCity) {$\CityLondon$};   \\
                                                        \&                                                       \&                                   \&   \node (TSname) {$\Type{\Sname}$};    \\
            \node (TLSPlusNLS) {$\StackTLSPlusNLS$};  \&       \node (TTLSPlusNLS) {$\StackTTLSPlusNLS$};    \&   \node (TSno) {$\Type{\Sno}$};    \&                                       \&       \node (TSSC) {$\StackTSSC$};      \&                                       \&   \node (TCity) {$\Type{City}$};   \\
                                                        \&                                                       \&                                   \&                                       \&                                           \&   \node (TStatus) {$\Type{\Status}$};  \\
            \node (TNLS) {$\Type{\NLS}$};                \&       \node (TTNLS) {$\TT{\NLS}$};                  \&                                   \&                                       \&       \node (TSSNL) {$\StackTSSNL$};    \&                                       \&   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) },
            };
            
            % FDs
            (TSno) -- [funcdep left,bend left] (TSSL);
            (TSno) -- [funcdep right] (TSSNL);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                { (TTLSPlusNLS), (TTLS) } -- (TSno)
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTNLS) -- (TSno)
            };
            
            { [edges=projection left]
                (TTLS)  -- (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TTNLS) -- (TSSNL),
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (NLSCity), (TStatus) },
                (TSSNL) -- [bend left=10] (TSname),
            };
            
            % selection edges
            { [edges=selection left]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) },
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) },
                (TCity) -- (LSCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
                (TTLSPlusNLS) -- [bend right] (TSSC),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
            (TTLSPlusNLS) -- [projection right,bend right] (TSSC),
        };
        
        \node[below=4cm of current page.south west,structure] (caption) {\Large This is already isomorphic to the SIG for $\schema{1}$, so $\Equivalent{\schema{1}}{\schema{2}}$.};
    \end{tikzpicture}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{The SIGs: $\bm{\schema{3} = \{\LS\}}$}
    \framesubtitle{\invisible{\beamergotobutton{Compare}}}
    
    \centering
    \begin{tikzpicture}[ampersand replacement=\&,transform canvas={scale=0.8}]
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\Type{\LS}$};      \&[7mm]  \node (TTLS) {$\TT{\LS}$};        \&[7mm]                              \&[4mm]                                  \&[-4mm] \node (TSSL) {$\StackTSSL$};      \&                                       \&   \node (LSCity) {$\CityLondon$};   \\
                                            \&                                           \&                                   \&   \node (TSname) {$\Type{\Sname}$};    \\
            \node[invisible] (TLSPlusNLS) {$\Type{S}$}; \&       \node[invisible] (TTLSPlusNLS) {$\TT{S}$};   \&   \node (TSno) {$\Type{\Sno}$};    \&                                       \&       \node (TSSC) {$\StackTSSC$};      \&                                       \&   \node (TCity) {$\Type{City}$};   \\
                                            \&                                           \&                                   \&                                       \&                                           \&   \node (TStatus) {$\Type{\Status}$};  \\
            \node[invisible] (TNLS) {$\Type{\NLS}$};    \&       \node[invisible] (TTNLS) {$\TT{\NLS}$};      \&                                   \&                                       \&       \node (TSSNL) {$\StackTSSNL$};    \&                                       \&   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TLS)        -- (TTLS),
                (TLSPlusNLS) --[invisible] (TTLSPlusNLS),
                (TNLS)       --[invisible] (TTNLS),
            };
            
            % FDs
            (TSno) -- [funcdep left,bend left,invisible] (TSSL);
            (TSno) -- [funcdep right,invisible] (TSSNL);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                (TTLS) -- (TSno)
            };
            
            { [edges=projection left]
                (TTLS)  -- (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
                (TTLSPlusNLS) --[invisible] (TSno),
            };
            
            { [edges=projection right]
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (NLSCity), (TStatus) },
                (TSSNL) -- [bend left=10] (TSname),
                (TTNLS)  --[invisible] (TSno),
                (TTNLS)  -- [arrows={:->},invisible] (TSSNL),
            };
            
            % selection edges
            { [edges=selection left]
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                (TCity) -- (LSCity),
            };
            
            { [edges=bijection,edge label={\scriptsize$\RelRestrict$}]
                { (TLS), (TTLS) }               -- [invisible] { (TLSPlusNLS), (TTLSPlusNLS) },
                { (TLSPlusNLS), (TTLSPlusNLS) } -- [invisible] { (TNLS), (TTNLS) },
            };
            
            { [edges=selection right]
                { (TLSPlusNLS), (TTLSPlusNLS) } --[invisible] { (TLS), (TTLS) },
            };
            
            { [edges=selection left]
                { (TLSPlusNLS), (TTLSPlusNLS) } --[invisible] { (TNLS), (TTNLS) },
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
                (TTLSPlusNLS) -- [bend right,invisible] (TSSC),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
            (TTLSPlusNLS) -- [projection right,arrows={:{crossbar}->},bend right,invisible] (TSSC);
        };
        
        \node[below=4cm of current page.south west,structure] (caption) {\Large Similarities with $\schema{1}$ but not identical, so we need to transform…};
    \end{tikzpicture}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame[label=frame.SIG.transform]
{
    \frametitle{Transforming $\bm{\schema{3} \Rightarrow \schema{1}}$}
    \framesubtitle{\alt<1-8>{\textcolor{structure}{blue} = input to transformation, \textcolor{alert}{red} = output\invisible{\beamergotobutton{Compare}}}{\hyperlink{frame.SIG.Sone<2>}{\beamergotobutton{Compare}}}}
    
    \centering
    \begin{tikzpicture}[ampersand replacement=\&,transform canvas={scale=0.8}]
        \matrix[row sep=0.5cm]
        { 
            \node[input on=<2>] (TLS) {$\Type{\LS}$};      \&[7mm]  \node[input on=<2>] (TTLS) {$\TT{\LS}$};        \&[7mm]                              \&[4mm]                                  \&[-4mm] \node (TSSL) {$\StackTSSL$};      \&                                       \&   \node (LSCity) {$\CityLondon$};   \\
                                            \&                                           \&                                   \&   \node (TSname) {$\Type{\Sname}$};    \\
            \node[visible on=<2->,output on=<2>,input on=<3>] (TLSPlusNLS) {$\Type{S}$}; \&       \node[visible on=<2->,output on=<2>,input on=<3>] (TTLSPlusNLS) {$\TT{S}$};   \&   \node (TSno) {$\Type{\Sno}$};    \&                                       \&       \node (TSSC) {$\StackTSSC$};      \&                                       \&   \node (TCity) {$\Type{City}$};   \\
                                            \&                                           \&                                   \&                                       \&                                           \&   \node (TStatus) {$\Type{\Status}$};  \\
            \node[visible on=<3->,output on=<3>] (TNLS) {$\Type{\NLS}$};    \&       \node[visible on=<3->,output on=<3>] (TTNLS) {$\TT{\NLS}$};      \&                                   \&                                       \&       \node (TSSNL) {$\StackTSSNL$};    \&                                       \&   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TLS)        --[input on=<4>] (TTLS),
                (TLSPlusNLS) --[visible on=<4->,output on=<4>] (TTLSPlusNLS),
                (TNLS)       --[visible on=<4->,output on=<4>] (TTNLS),
            };
            
            % FDs
            (TSno) -- [funcdep left,arrows={:-{crossbar}>},bend left,visible on=<7->,output on=<7>] (TSSL);
            (TSno) -- [funcdep right,arrows={:-{crossbar}>},visible on=<7->,output on=<7>] (TSSNL);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                (TTLS)  --[input on=<5>] (TSno),
                (TTLSPlusNLS) --[visible on=<5->,output on=<5>] (TSno)
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTNLS)  --[visible on=<5->,output on=<5>] (TSno)
            };
            
            { [edges=projection left]
                (TTLS)  --[input on=<6>] (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) }
            };
            
            { [edges=projection right]
                (TTNLS)  -- [arrows={:->},visible on=<6->,output on=<6>] (TSSNL),
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (NLSCity), (TStatus) },
                (TSSNL) -- [bend left=10] (TSname),
            };

% selection edges
            { [edges=selection left]
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                (TCity) -- (LSCity),
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelRestrict$}]
                { (TLS), (TTLS) }               -- [visible on=<2-7>,output on=<2>] { (TLSPlusNLS), (TTLSPlusNLS) },
                { (TLSPlusNLS), (TTLSPlusNLS) } -- [visible on=<3-7>,output on=<3>] { (TNLS), (TTNLS) },
            };
            
            { [edges=selection left]
                { (TLSPlusNLS), (TTLSPlusNLS) } --[visible on=<8->] { (TLS), (TTLS) },
            };
            
            { [edges=selection right]
                { (TLSPlusNLS), (TTLSPlusNLS) } --[visible on=<8->] { (TNLS), (TTNLS) },
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
                (TTLSPlusNLS) -- [bend right,visible on=<6->] (TSSC),
            };
            (TSno) -- [funcdep right,input on=<7>] (TSSC);
            (TSSC) -- [projection left] (TCity);
            (TTLSPlusNLS) -- [projection right,arrows={:{crossbar}->},bend right,visible on=<6->,output on=<6>] (TSSC);
        };
        
        \begin{scope}[circle,visible on=<8>,output on=<8>]
            \node[draw] (c1) at (TLSPlusNLS) [above=11.5pt] {};
            \node[draw] (c2) at (TLSPlusNLS) [below=11.5pt] {};
            \node[draw] (c3) at (c1 -| TTLSPlusNLS) {};
            \node[draw] (c4) at (c2 -| TTLSPlusNLS) {};
        \end{scope}
        
        \node[below=4cm of current page.south west,structure] (caption) {
            \Large
            \only<1>{Initial state}
            \only<2>{Duplicate $\Type{\LS}$ and $\TT{\LS}$ [$\equiv$]}
            \only<3>{Duplicate $\Type{S}$ and $\TT{S}$ [$\equiv$]}
            \only<4>{Copy and move $\Type{\LS} \SurTotal \TT{\LS}$ edge ($\times 2$) [$\preceq$]}
            \only<5>{Copy and move $\TT{\LS} \ProjectionEdge \Type{\Sno}$ edge ($\times 2$) [$\preceq$]}
            \only<6>{Copy and move $\TT{\LS} \ProjectionEdge \TSSL$ edge ($\times 2$) [$\preceq$]}
            \only<7>{Copy and move $\Type{\Sno} \KeyEdge \TSSC$ edge ($\times 2$) [$\preceq$]}
            \only<8>{Remove totality annotations [$\preceq$]}
            \only<9>{Final state: \emph{not} isomorphic to SIG for $\schema{1} \Rightarrow$ non-comparable.}
        };
    \end{tikzpicture}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{So what does this mean?}
    
    \structurebf{$\bm{\schema{1}}$ and $\bm{\schema{2}}$ are information equivalent}
    
    \begin{itemize}
    
        \item[$\Rightarrow$] all view updates between them should be OK
    
    \end{itemize}
    \bigskip
    
    \structurebf{$\bm{\schema{1}}$ and $\bm{\schema{3}}$ aren’t information equivalent}
    
    \begin{itemize}
    
        \item[$\Rightarrow$] some view updates between them will cause problems
    
    \end{itemize}
    \bigskip
    
    \structurebf{This agrees with what Date found} {\tiny (yay!)}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{More examples}
    \framesubtitle{(Nonloss projection views, Date Ch. 5)}
    
    $S\{\Sno, \Status, \City\}$; views $\ST = \RelProject_{\{\Sno, \Status\}}S$, $\SC = \RelProject_{\{\Sno, \City\}}S$
    \medskip\bigskip
    
    \begin{columns}
        \begin{column}{0.33\textwidth}
            \centering
            \scalebox{0.5}{%
                \begin{tikzpicture}
                    \matrix[row sep=1.5cm,ampersand replacement=\&]
                    { 
                        \node (TST) {$\Type{\STsub}$};    \&[7mm]  \node (TTST) {$\TT{\STsub}$};     \&[10mm]                             \&[10mm] \node (TStatus) {$\Type{\Status}$};              \\
                        \node (TSTPlusSC) {$\Type{S}$};   \&       \node (TTSTPlusSC) {$\TT{S}$};    \&   \node (TSno) {$\Type{\Sno}$}; \& \node (TSSC) {$\Type{\Status} \times \Type{\City}$};  \\
                        \node (TSC) {$\Type{\SCsub}$};    \&       \node (TTSC) {$\TT{\SCsub}$};     \&                                   \& \node (TCity) {$\Type{\City}$};                       \\
                    };
        
                    \graph{
                        % relation types to tuple types
                        { [edges=surtotal]
                            { (TST), (TSC), (TSTPlusSC) } -- { (TTST), (TTSC), (TTSTPlusSC) },
                        };
            
                        % FDs
                        (TSno) -- [funcdep left] (TStatus);
                        (TSno) -- [funcdep right] (TCity);
                        (TSno) -- [funcdep right] (TSSC);
            
                        % projection edges
                        { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                            { (TTSTPlusSC), (TTST) } -- (TSno),
                        };
            
                        { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                            (TTSC) -- (TSno),
                        };
            
                        { [edges=projection left]
                            { (TSTPlusSC), (TTSTPlusSC) } -- { (TST), (TTST) },
                            (TSSC)  -- (TCity),
                            (TTST)  -- (TStatus),
                        };
            
                        { [edges=projection right]
                            { (TSTPlusSC), (TTSTPlusSC) } -- { (TSC), (TTSC) },
                            (TSSC)       -- (TStatus),
                            (TTSC) -- (TCity),
                        };
            
                        % insert "gaps" into crossed edges
                        { [edges={white,line width=1.5mm}]
                            (TTSTPlusSC) -- [bend right] (TSSC),
                        };
                        (TTSTPlusSC) -- [projection right, bend right] (TSSC),
                    };
                \end{tikzpicture}%
            } \\
            $\schema{1} = \{S\}$
        \end{column}
        \begin{column}{0.33\textwidth}
            \centering
            \scalebox{0.5}{%
                \begin{tikzpicture}
                    \matrix[row sep=1.5cm,ampersand replacement=\&]
                    { 
                        \node (TST) {$\Type{\ST}$};       \&[7mm]  \node (TTST) {$\TT{\ST}$};        \&[10mm]                             \&[10mm] \node (TStatus) {$\Type{\Status}$};              \\
                        \node (TSTPlusSC) {$\Type{J}$};   \&       \node (TTSTPlusSC) {$\TT{J}$};    \&   \node (TSno) {$\Type{\Sno}$}; \& \node (TSSC) {$\Type{\Status} \times \Type{\City}$};  \\
                        \node (TSC) {$\Type{\SC}$};       \&       \node (TTSC) {$\TT{\SC}$};        \&                                   \& \node (TCity) {$\Type{\City}$};                       \\
                    };
        
                    \graph{
                        % relation types to tuple types
                        { [edges=surtotal]
                            { (TST), (TSC), (TSTPlusSC) } -- { (TTST), (TTSC), (TTSTPlusSC) },
                        };
            
                        % FDs
                        (TSno) -- [funcdep left] (TStatus);
                        (TSno) -- [funcdep right] (TCity);
                        (TSno) -- [funcdep right] (TSSC);
            
                        % projection edges
                        { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                            { (TTSTPlusSC), (TTST) } -- (TSno),
                        };
            
                        { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                            (TTSC) -- (TSno),
                        };
            
                        { [edges=projection left]
                            { (TSTPlusSC), (TTSTPlusSC) } -- { (TST), (TTST) },
                            (TSSC)  -- (TCity),
                            (TTST)  -- (TStatus),
                        };
            
                        { [edges=projection right]
                            { (TSTPlusSC), (TTSTPlusSC) } -- { (TSC), (TTSC) },
                            (TSSC)       -- (TStatus),
                            (TTSC) -- (TCity),
                        };
            
                        % insert "gaps" into crossed edges
                        { [edges={white,line width=1.5mm}]
                            (TTSTPlusSC) -- [bend right] (TSSC),
                        };
                        (TTSTPlusSC) -- [projection right, bend right] (TSSC),
                    };
                \end{tikzpicture}%
            } \\
            $\schema{2} = \{\ST, \SC\}$
        \end{column}
        \begin{column}{0.33\textwidth}
            \centering
            \scalebox{0.5}{%
                \begin{tikzpicture}
                    \matrix[row sep=1.5cm,ampersand replacement=\&]
                    { 
                        \node (TST) {$\Type{\ST}$};   \&[7mm]  \node (TTST) {$\TT{\ST}$};    \&[10mm]                             \&[10mm] \node (TStatus) {$\Type{\Status}$};    \\
                                                        \&                                       \&   \node (TSno) {$\Type{\Sno}$}; \\
                    };
        
                    \graph{
                        % relation types to tuple types
                        { [edges=surtotal]
                            (TST) -- (TTST),
                        };
            
                        % FDs
                        (TSno) -- [funcdep left] (TStatus);
            
                        % projection edges
                        { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                            (TTST) -- (TSno),
                        };
            
                        { [edges=projection left]
                            (TTST)  -- (TStatus),
                        };
                    };
                \end{tikzpicture}%
            } \\
            $\schema{3} = \{\ST\}$
        \end{column}
    \end{columns}
    \medskip\medskip
    
    \begin{tabbing}
        \structurebf{Results:} \=$\Equivalent{\schema{1}}{\schema{2}}$ {\tiny (as expected)}    \\
                               \>$\schema{1}$ and $\schema{3}$ incomparable {\tiny (as expected)}
    \end{tabbing}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{More examples}
    \framesubtitle{(Lossy projection views, Date Ch. 5)}
    
    \small
    $S\{\Sno, \Status, \City\}$; views $\ST = \RelProject_{\{\Sno, \Status\}}S$, $\TC = \RelProject_{\{\Status, \City\}}S$
    \medskip
    
    \begin{columns}
        \begin{column}{0.5\textwidth}
            \centering
            \scalebox{0.4}{%
                \begin{tikzpicture}
                    \matrix[row sep=1.5cm,ampersand replacement=\&]
                    { 
                        \node (TST) {$\Type{\STsub}$};    \&[7mm]  \node (TTST) {$\TT{\STsub}$};     \&[10mm]                             \&[10mm] \node (TStatus) {$\Type{\Status}$};              \\
                        \node (TSTPlusTC) {$\Type{S}$};   \&       \node (TTSTPlusTC) {$\TT{S}$};    \&   \node (TSno) {$\Type{\Sno}$}; \& \node (TSSC) {$\Type{\Status} \times \Type{\City}$};  \\
                        \node (TTC) {$\Type{\TCsub}$};    \&       \node (TTTC) {$\TT{\TCsub}$};     \&                                   \& \node (TCity) {$\Type{\City}$};                       \\
                    };
        
                    \graph{
                        % relation types to tuple types
                        { [edges=surtotal]
                            { (TST), (TTC), (TSTPlusTC) } -- { (TTST), (TTTC), (TTSTPlusTC) },
                        };
            
                        % FDs
                        { [edges=funcdep right]
                            (TSno) -- { (TCity),  (TSSC) },
                            (TSSC) -- [bend right] (TStatus),
                        };
                        { [edges=funcdep left]
                            (TSno) -- (TStatus),
                            (TSSC) -- [bend left] (TCity),
                        };
            
                        % projection edges
                        { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                            { (TTSTPlusTC), (TTST) } -- (TSno),
                        };
            
                        { [edges=projection left]
                            { (TSTPlusTC), (TTSTPlusTC) } -- { (TST), (TTST) },
                            (TTST)  -- (TStatus),
                            (TSSC) -- [bend right] (TCity),
                        };
            
                        { [edges=projection right]
                            { (TSTPlusTC), (TTSTPlusTC) } -- { (TTC), (TTTC) },
                            (TSSC) -- [bend left] (TStatus),
                        };
            
                        % insert "gaps" into crossed edges
                        { [edges={white,line width=1.5mm}]
                            (TTSTPlusTC) -- [bend right] (TSSC),
                            (TTTC) -- [bend right=10] (TSSC),
                        };
                        (TTSTPlusTC) -- [projection right, bend right] (TSSC),
                        (TTTC) -- [edges=bijection,edge label'={\scriptsize$\RelProject$}, bend right=10] (TSSC),
                    };
                \end{tikzpicture}
            } \\
            $\schema{1} = \{S\}$  \\[\baselineskip]
            \scalebox{0.4}{%
                \begin{tikzpicture}
                    \matrix[row sep=1.5cm,ampersand replacement=\&]
                    { 
                        \node (TST) {$\Type{\ST}$};   \&[7mm]  \node (TTST) {$\TT{\ST}$};    \&[10mm]                             \&[10mm] \node (TStatus) {$\Type{\Status}$};    \\
                                                        \&                                       \&   \node (TSno) {$\Type{\Sno}$}; \\
                    };
        
                    \graph{
                        % relation types to tuple types
                        { [edges=surtotal]
                            (TST) -- (TTST),
                        };
            
                        % FDs
                        (TSno) -- [funcdep left] (TStatus);
            
                        % projection edges
                        { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                            (TTST) -- (TSno),
                        };
            
                        { [edges=projection left]
                            (TTST)  -- (TStatus),
                        };
                    };
                \end{tikzpicture}
            } \\
            $\schema{3} = \{\ST\}$
        \end{column}
        \begin{column}{0.5\textwidth}
            \centering
            \scalebox{0.4}{%
                \begin{tikzpicture}
                    \matrix[row sep=1.5cm,ampersand replacement=\&]
                    { 
                        \node (TST) {$\Type{\ST}$};    \&[7mm]  \node (TTST) {$\TT{\ST}$};     \&[10mm]                             \&[10mm] \node (TStatus) {$\Type{\Status}$};              \\
                                                         \&                                        \&   \node (TSno) {$\Type{\Sno}$}; \& \node (TSSC) {$\Type{\Status} \times \Type{\City}$};  \\
                        \node (TTC) {$\Type{\TC}$};    \&       \node (TTTC) {$\TT{\TC}$};     \&                                   \& \node (TCity) {$\Type{\City}$};                       \\
                    };
        
                    \graph{
                        % relation types to tuple types
                        { [edges=surtotal]
                            { (TST), (TTC) } -- { (TTST), (TTTC) },
                        };
            
                        % FDs
                        { [edges=funcdep right]
                            (TSSC) -- [bend right] (TStatus),
                        };
                        { [edges=funcdep left]
                            (TSno) -- (TStatus),
                            (TSSC) -- [bend left] (TCity),
                        };
            
                        % projection edges
                        { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                            { (TTST) } -- (TSno),
                        };
            
                        { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                            (TTTC) -- (TSSC),
                        };
            
                        { [edges=projection left]
                            (TTST)  -- (TStatus),
                            (TSSC) -- [bend right] (TCity),
                        };
            
                        { [edges=projection right]
                            (TSSC) -- [bend left] (TStatus),
                        };
                    };
                \end{tikzpicture}
            } \\
            $\schema{2} = \{\ST, \TC\}$   \\[\baselineskip]
            \scalebox{0.4}{%
                \begin{tikzpicture}
                    \matrix[row sep=1.5cm,ampersand replacement=\&]
                    { 
                                                            \&[7mm]                                      \&[10mm] \node (TStatus) {$\Type{\Status}$};              \\
                        \node (TTC) {$\Type{\TC}$};       \&       \node (TTTC) {$\TT{\TC}$};        \& \node (TSSC) {$\Type{\Status} \times \Type{\City}$};  \\
                                                            \&                                           \& \node (TCity) {$\Type{\City}$};                       \\
                    };
        
                    \graph{
                        % relation types to tuple types
                        { [edges=surtotal]
                            { (TTC) } -- { (TTTC) },
                        };
            
                        % FDs
                        { [edges=funcdep right]
                            (TSSC) -- [bend right] (TStatus),
                        };
                        { [edges=funcdep left]
                            (TSSC) -- [bend left] (TCity),
                        };
            
                        % projection edges
                        { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                            (TTTC) -- (TSSC),
                        };
            
                        { [edges=projection left]
                            (TSSC) -- [bend right] (TCity),
                        };
            
                        { [edges=projection right]
                            (TSSC) -- [bend left] (TStatus),
                        };
                    };
                \end{tikzpicture}
            } \\
            $\schema{4} = \{\TC\}$
        \end{column}
    \end{columns}
    \bigskip
    
    \structurebf{Results:} $\schema{1}$ isn't comparable with any of $\schema{2}$, $\schema{3}$, or $\schema{4}$ {\tiny (as expected)}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{More examples}
    \framesubtitle{(Lossy one to one join views, Date Ch. 6)}
    
    $\ST\{\Sno, \Status\}$, $\SC\{\Sno, \City\}$; view $S = \ST \RelNaturalJoin \SC$
    \medskip\bigskip
    
    \begin{columns}
        \begin{column}{0.33\textwidth}
            \centering
            \scalebox{0.5}{%
                \begin{tikzpicture}
                    \matrix[row sep=1.5cm,ampersand replacement=\&]
                    { 
                                                            \&[7mm]                                      \&[10mm]                             \&[10mm] \node (TStatus) {$\Type{\Status}$};              \\
                        \node (TSTPlusSC) {$\Type{S}$};   \&       \node (TTSTPlusSC) {$\TT{S}$};    \&   \node (TSno) {$\Type{\Sno}$}; \& \node (TSSC) {$\Type{\Status} \times \Type{\City}$};  \\
                                                            \&                                           \&                                   \& \node (TCity) {$\Type{\City}$};                       \\
                    };
        
                    \graph{
                        % relation types to tuple types
                        { [edges=surtotal]
                            (TSTPlusSC) -- (TTSTPlusSC),
                        };
            
                        % FDs
                        (TSno) -- [funcdep left] (TStatus);
                        (TSno) -- [funcdep right] (TCity);
                        (TSno) -- [funcdep right] (TSSC);
            
                        % projection edges
                        { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                            { (TTSTPlusSC) } -- (TSno),
                        };
            
                        { [edges=projection left]
                            (TSSC) -- (TCity),
                        };
            
                        { [edges=projection right]
                            (TSSC) -- (TStatus),
                        };
            
                        % insert "gaps" into crossed edges
                        { [edges={white,line width=1.5mm}]
                            (TTSTPlusSC) -- [bend right] (TSSC),
                        };
                        (TTSTPlusSC) -- [projection right, bend right] (TSSC),
                    };
                \end{tikzpicture}
            } \\
            $\schema{1} = \{S\}$
        \end{column}
        \begin{column}{0.33\textwidth}
            \centering
            \scalebox{0.5}{%
                \begin{tikzpicture}
                    \matrix[row sep=1.5cm,ampersand replacement=\&]
                    { 
                        \node (TST) {$\Type{\ST}$};       \&[7mm]  \node (TTST) {$\TT{\ST}$};        \&[10mm]                             \&[10mm] \node (TStatus) {$\Type{\Status}$};              \\
                                                            \&                                           \&   \node (TSno) {$\Type{\Sno}$}; \&   \\
                        \node (TSC) {$\Type{\SC}$};       \&       \node (TTSC) {$\TT{\SC}$};        \&                                   \& \node (TCity) {$\Type{\City}$};                       \\
                    };
        
                    \graph{
                        % relation types to tuple types
                        { [edges=surtotal]
                            { (TST), (TSC) } -- { (TTST), (TTSC) },
                        };
            
                        % FDs
                        (TSno) -- [funcdep left] (TStatus);
                        (TSno) -- [funcdep right] (TCity);
            
                        % projection edges
                        { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                            { (TTST) } -- (TSno),
                        };
            
                        { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                            (TTSC) -- (TSno),
                        };
            
                        { [edges=projection left]
                            (TTST)  -- (TStatus),
                        };
            
                        { [edges=projection right]
                            (TTSC) -- (TCity),
                        };
                    };
                \end{tikzpicture}
            } \\
            $\schema{2} = \{\ST, \SC\}$
        \end{column}
        \begin{column}{0.33\textwidth}
            \centering
            \scalebox{0.5}{%
                \begin{tikzpicture}
                    \matrix[row sep=1.5cm,ampersand replacement=\&]
                    { 
                        \node (TST) {$\Type{\ST}$};   \&[7mm]  \node (TTST) {$\TT{\ST}$};    \&[10mm]                             \&[10mm] \node (TStatus) {$\Type{\Status}$};    \\
                                                        \&                                       \&   \node (TSno) {$\Type{\Sno}$}; \\
                    };
        
                    \graph{
                        % relation types to tuple types
                        { [edges=surtotal]
                            (TST) -- (TTST),
                        };
            
                        % FDs
                        (TSno) -- [funcdep left] (TStatus);
            
                        % projection edges
                        { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                            (TTST) -- (TSno),
                        };
            
                        { [edges=projection left]
                            (TTST)  -- (TStatus),
                        };
                    };
                \end{tikzpicture}%
            } \\
            $\schema{3} = \{\ST\}$
        \end{column}
    \end{columns}
    \bigskip\medskip
    
    \structurebf{Results:} $\schema{1}$ isn't comparable with either $\schema{2}$ or $\schema{3}$ {\tiny (as expected)}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \frametitle{Now what?}
    
    \begin{itemize}
    
        \item Is this a useful technique? {\tiny (working on analysing more complex examples)}
        
        \item Are inclusion dependencies important? \\
        If so, how to represent them?
        
        \item Do we need the whole transformation process? Determining whether or not all constraints can be propagated from the base schema may already be enough.
        
        \item Could it help provide a more formal basis for Date’s “pragma”?
        
        \item Might it reveal some interesting new insights about the view update problem?
        
    \end{itemize}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\frame
{
    \centering\Huge
    \structurebf{Questions?}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\againframe<3>{frame.SIG.example}

\againframe<2>{frame.SIG.Sone}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

Ignore Space Show notes View APCCM_2017/Presentation/examples.tex 0 → 100644
% APCCM2017_Stanger.tex % Author: Nigel Stanger % % Built using Tex Live 2016 distribution. % All packages except perhaps PGF should be in the standard TeX Live packages. % REQUIRES PGF v3.0 or later for diagrams (custom arrow tips). % REQUIRES TeX Gyre fonts (newtxmath package). % NOTE: the last two figures use a small amount of colour. % % Revisions: 22 Aug 2016 Initial submission. % 07 Nov 2016 Camera-ready version. %% %% MAX 10 PAGES (all material) %% \documentclass{article} \usepackage{newtxtext} \usepackage{newtxmath} \usepackage{bm} \usepackage{subfig} \usepackage{booktabs} \usepackage[margin=1in]{geometry} \usepackage{tikz} \usetikzlibrary{positioning} \usetikzlibrary{graphs} \usetikzlibrary{decorations.pathreplacing} \usetikzlibrary{calc} \usetikzlibrary{arrows} % Relational algebra operators \newcommand{\RelRestrict}{\ensuremath{\sigma}} \newcommand{\RelProject}{\ensuremath{\pi}} \newcommand{\RelUnion}{\ensuremath{\cup}} \newcommand{\RelIntersect}{\ensuremath{\cap}} \newcommand{\RelJoin}{\ensuremath{\Join}} % "empty" arrow tip \pgfarrowsdeclare{:}{:}{}{} % custom bar arrow tip, offset from end of line (use "empty" tip at line ends if no >) \tikzset{crossbar/.tip={\|[scale width=1.75,sep=0.25em]}} % various edge styles for SIG diagrams in TikZ \tikzset{ function/.style={arrows={->}}, injection/.style={arrows={<-}}, total/.style={arrows={:{crossbar}-}}, surjection/.style={arrows={-{crossbar}:}}, bijection/.style={arrows={<{crossbar}-{crossbar}>}}, projection/.style={arrows={:{crossbar}-{crossbar}>}}, projection left/.style={arrows={:{crossbar}-{crossbar}>},edge label={\scriptsize\(\RelProject\)}}, projection right/.style={arrows={:{crossbar}-{crossbar}>},edge label'={\scriptsize\(\RelProject\)}}, selection left/.style={arrows={<-{crossbar}>},edge label={\scriptsize\(\RelRestrict\)}}, selection right/.style={arrows={<-{crossbar}>},edge label'={\scriptsize\(\RelRestrict\)}}, funcdep left/.style={arrows={:{crossbar}-{crossbar}>},edge node={node[sloped,midway,above] {\scriptsize\emph{key}}}}, funcdep right/.style={arrows={:{crossbar}-{crossbar}>},edge node={node[sloped,midway,below] {\scriptsize\emph{key}}}}, surtotal/.style={arrows={:{crossbar}-{crossbar}:}}, input keep/.style={blue,thick}, input delete/.style={blue!40,thick,dashed}, output/.style={red,thick}, output temp/.style={red,thick,dashed}, path 1/.style={green!60!black,thick}, path 2/.style={orange,thick}, } % projection and selection edge annotations for TikZ \newcommand{\ProjectionAnnotation}[3][]{% \path (#2) to node[above,#1] {\scriptsize\(\RelProject\)} (#3);% } \newcommand{\SelectionAnnotation}[3][]{% \path (#2) to node[above,#1] {\scriptsize\(\RelRestrict\)} (#3);% } \newcounter{constraint} % misc % \newcommand{\todo}[1]{\textbf{!!TODO!!} {[#1]}} % commonly used elements \newcommand{\identifier}[1]{\ensuremath{\mathit{#1}}} \newcommand{\LS}{\identifier{LS}} \newcommand{\NLS}{\identifier{NLS}} \newcommand{\LSsub}{\identifier{L}} \newcommand{\NLSsub}{\identifier{N}} \newcommand{\ST}{\identifier{ST}} \newcommand{\SC}{\identifier{SC}} \newcommand{\STsub}{\identifier{T}} \newcommand{\SCsub}{\identifier{C}} \newcommand{\TC}{\identifier{TC}} \newcommand{\TCsub}{\identifier{C}} \newcommand{\Sno}{\identifier{Sno}} \newcommand{\Sname}{\identifier{Sname}} \newcommand{\Status}{\identifier{Status}} \newcommand{\City}{\identifier{City}} \newcommand{\Type}[1]{\ensuremath{T_{#1}}} \newcommand{\TT}[1]{\ensuremath{T_{\{#1\}}}} \newcommand{\CityLondon}{\ensuremath{\{\City\colon\allowbreak\text{`\emph{London}'}\}}} \newcommand{\TCityMinusLondon}{\ensuremath{\Type{\City} \setminus \CityLondon}} \newcommand{\TSSC}{\ensuremath{\Type{\Sname} \times \Type{\Status} \times \Type{\City}}} \newcommand{\TSSL}{\ensuremath{\Type{\Sname} \times \Type{\Status} \times \CityLondon}} \newcommand{\TSSNL}{\ensuremath{\Type{\Sname} \times \Type{\Status} \times (\TCityMinusLondon)}} \newcommand{\TLSPlusNLS}{\ensuremath{\Type{\LS} + \Type{\NLS}}} \newcommand{\TTLSPlusNLS}{\ensuremath{\TT{\LS} + \TT{\NLS}}} \newcommand{\TLSPlusNLSsub}{\ensuremath{\Type{\LSsub} + \Type{\NLSsub}}} \newcommand{\TTLSPlusNLSsub}{\ensuremath{\TT{\LSsub} + \TT{\NLSsub}}} \newcommand{\StackTLSPlusNLS}{\ensuremath{\begin{array}{c}\Type{\LS}\,+ \\ \Type{\NLS}\end{array}}} \newcommand{\StackTTLSPlusNLS}{\ensuremath{\begin{array}{c}\TT{\LS}\,+ \\ \TT{\NLS}\end{array}}} \newcommand{\StackTLSPlusNLSsub}{\ensuremath{\begin{array}{c}\Type{\LSsub}\,+ \\ \Type{\NLSsub}\end{array}}} \newcommand{\StackTTLSPlusNLSsub}{\ensuremath{\begin{array}{c}\TT{\LSsub}\,+ \\ \TT{\NLSsub}\end{array}}} \newcommand{\StackTSSC}{\ensuremath{\begin{array}{c}\Type{\Sname}\,\times \\ \Type{\Status} \times \Type{\City}\end{array}}} \newcommand{\StackTSSL}{\ensuremath{\begin{array}{c}\Type{\Sname} \times \Type{\Status}\,\times \\ \CityLondon\end{array}}} \newcommand{\StackTSSNL}{\ensuremath{\begin{array}{c}\Type{\Sname} \times \Type{\Status}\,\times \\ (\TCityMinusLondon)\end{array}}} \newcommand{\StackTCityMinusLondon}{\ensuremath{\begin{array}{c}\Type{\City}\,\setminus \\ \CityLondon\end{array}}} \newcommand{\schema}[1]{\ensuremath{\mathcal{S}_{#1}}} % dominates \newcommand{\Dominates}[2]{\ensuremath{#2 \preceq #1}} \newcommand{\Equivalent}[2]{\ensuremath{#1 \equiv #2}} % SIG notation (in text) \newcommand{\Sedge}[1]{\ensuremath{\sigma_{\textrm{#1}}}} \newcommand{\SedgeP}[1]{\ensuremath{\sigma_{\textrm{#1}}^{'}}} \newcommand{\medmid}{\raise.125ex\hbox{\scalebox{1}[0.75]{$\mid$}}} % General SIG edges for use in formulas. % Adapted from: http://tex.stackexchange.com/questions/96330/adding-symbols-at-the-ends-of-a-horizontal-line \makeatletter \newlength{\@annotskipleft} \newlength{\@annotskipright} % #1 = left edge component % #2 = right edge component % #3 = left bar annotation % #4 = right bar annotation \newcommand\@sig@edge[4]{% \let\@middle\joinrel% \ifx#1\relbar% \@annotskipleft=.3em% % scrunch up the bare line so it's similar length to \long...arrow \ifx#2\relbar\def\@middle{\joinrel\joinrel\relbar\joinrel\joinrel}\fi% \else% % arrows need a little more clearance \@annotskipleft=.4em% \fi% \ifx#2\relbar\@annotskipright=.3em\else\@annotskipright=.4em\fi% \mathrel{\ooalign{$#1\@middle#2$\cr\hskip\@annotskipleft$#3$\hfil$#4$\hskip\@annotskipright\cr}}% } % 0 = nothing % 1 = bar % 2 = arrowhead % 3 = both \def\@sigedge#1#2{% \ifcase\numexpr#14+#2\relax% \@sig@edge{\relbar}{\relbar}{}{}\or % 00 = ----- \@sig@edge{\relbar}{\relbar}{}{\medmid}\or % 01 = ---+- \@sig@edge{\relbar}{\rightarrow}{}{}\or % 02 = ----> \@sig@edge{\relbar}{\rightarrow}{}{\medmid}\or % 03 = ---+> \@sig@edge{\relbar}{\relbar}{\medmid}{}\or % 10 = -+--- \@sig@edge{\relbar}{\relbar}{\medmid}{\medmid}\or % 11 = -+-+- \@sig@edge{\relbar}{\rightarrow}{\medmid}{}\or % 12 = -+--> \@sig@edge{\relbar}{\rightarrow}{\medmid}{\medmid}\or % 13 = -+-+> \@sig@edge{\leftarrow}{\relbar}{}{}\or % 20 = <---- \@sig@edge{\leftarrow}{\relbar}{}{\medmid}\or % 21 = <--+- \@sig@edge{\leftarrow}{\rightarrow}{}{}\or % 22 = <---> \@sig@edge{\leftarrow}{\rightarrow}{}{\medmid}\or % 23 = <--+> \@sig@edge{\leftarrow}{\relbar}{\medmid}{}\or % 30 = <+--- \@sig@edge{\leftarrow}{\relbar}{\medmid}{\medmid}\or % 31 = <+-+- \@sig@edge{\leftarrow}{\rightarrow}{\medmid}{}\or % 32 = <+--> \@sig@edge{\leftarrow}{\rightarrow}{\medmid}{\medmid} % 33 = <+-+> \fi% } \newcommand{\sigedge}[1]{\ensuremath{\@sigedge#1}} \makeatother \newcommand{\Edge}{\sigedge{00}} \newcommand{\Total}{\sigedge{10}} \newcommand{\Surjective}{\sigedge{01}} \newcommand{\SurTotal}{\sigedge{11}} \newcommand{\Functional}{\sigedge{02}} \newcommand{\Injective}{\sigedge{20}} \newcommand{\Bijective}{\sigedge{33}} % Hack to get the overset symbols to appear at the right height. % \smash removes the spacing around the operator, hence \mathop. \newcommand{\LabelledEdge}[2]{\mathop{\overset{\raisebox{0.3em}{\scriptsize\ensuremath{#2}}}{\smash[t]{#1}}}} \newcommand{\ProjectionEdge}{\LabelledEdge{\sigedge{13}}{\RelProject}} \newcommand{\SelectionEdge}{\LabelledEdge{\sigedge{23}}{\RelRestrict}} \newcommand{\TrivialProjection}{\ensuremath{\LabelledEdge{\Bijective}{\RelProject}}} \newcommand{\TrivialSelection}{\ensuremath{\LabelledEdge{\Bijective}{\RelRestrict}}} \newcommand{\KeyEdge}{\ensuremath{\LabelledEdge{\sigedge{13}}{\mathit{key}}}} % Constraints. \newcommand{\Constraint}[2][]{C\ensuremath{_{#2}\ifx&#1&\else^{#1}\fi}} \newenvironment{ConstraintList}[1][]{% \begin{list}{% \bfseries% \ifx&#1&% \Constraint{\ensuremath{\bm{\arabic{constraint}}}}% \else% \Constraint[\ensuremath{\bm{#1}}]{\ensuremath{\bm{\arabic{constraint}}}}% \fi% }% {\usecounter{constraint}}% }{\end{list}} \newcommand{\eqnnumber}{\refstepcounter{equation}(\theequation)} \hyphenation{co-pied sche-ma} \pagestyle{empty} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} % OUTLINE (20 mins max) % Quick overview of view update problem and Date's effort % - problem: Date's approach appears little different from earlier metadata-heavy approaches % - problem: effectively punts on the hardest and most interesting cases % - problem: operational definition of information equivalence % - not focusing on view update problem, but rather how it can be characterised % Quick overview of information capacity % - relative measure, not quantitative % - designed for relational schemas, so ideal % - basic concepts of SIG transformation and isomorphism % - my notation extensions % Examples: % - quickly go through restriction view example in more detail % - other examples (summary): projection views, join views? % Future work? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Disjoint restriction views (Date, Chapter 4)} \label{sec-disjoint-restriction} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Relvars} \begin{itemize} \item base relvar \(S\{\Sno, \Sname, \Status, \City\}\) \item view \(\LS = \RelRestrict_{\City = \mathit{'London'}}S\) [London suppliers] \eqnnumber\label{eqn-ls} \item view \(\NLS = \RelRestrict_{\City \ne \mathit{'London'}}S\) [non-London suppliers] \eqnnumber\label{eqn-nls} \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Tuple types} \begin{align} \TT{S} &= \Type{\Sno} \times \Type{\Sname} \times \Type{\Status} \times \Type{\City} \nonumber \\ \TT{\LS} &= \Type{\Sno} \times \Type{\Sname} \times \Type{\Status} \times \CityLondon \nonumber \\ \TT{\NLS} &= \Type{\Sno} \times \Type{\Sname} \times \Type{\Status} \times (\TCityMinusLondon) \nonumber \end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Base schema \(\bm{\schema{0} = \{S, \LS, \NLS\}}\)} \begin{ConstraintList} \item\label{constraint-restrict-key} \Sno\ is a key for each of \(S\), \(\LS\), and \(\NLS\), so the functional dependency \(\{\Sno\} \rightarrow \{\Sname, \Status, \City\}\) also holds in each of (a) \(S\); (b) \(\LS\); and (c) \(\NLS\). % not sure how relevant the FK constraints are? % how to represent? FK is a subset of the original key, so selection edge between S.Sno and each of LS.Sno and NLS.Sno % or perhaps a selection edge from Sno to itself? (labelled FK) % Notation: \Type{\Sno}, \Type{\LS.\Sno}, \Type{\NLS.\Sno}? \item\label{constraint-restrict-lsfk} \(\RelProject_{\{\Sno\}}\LS \subseteq \RelProject_{\{\Sno\}}S\) [foreign key from \(\LS\) to \(S\)] \item\label{constraint-restrict-nlsfk} \(\RelProject_{\{\Sno\}}\NLS \subseteq \RelProject_{\{\Sno\}}S\) [foreign key from \(\NLS\) to \(S\)] \item\label{constraint-restrict-union} \(S = \LS \RelUnion \NLS\) [from (\ref{eqn-ls}) and (\ref{eqn-nls}) above] \item\label{constraint-restrict-notlondon} \(\RelRestrict_{\City \ne \mathit{'London'}}\LS = \emptyset\) [from (\ref{eqn-ls}) above] \item\label{constraint-restrict-london} \(\RelRestrict_{\City = \mathit{'London'}}\NLS = \emptyset\) [from (\ref{eqn-nls}) above] \item\label{constraint-restrict-disjoint} \(\LS \RelIntersect \NLS = \emptyset\) [from \Constraint{\ref{constraint-restrict-notlondon}} and \Constraint{\ref{constraint-restrict-london}}] \item\label{constraint-restrict-relation-type-union} (a) \(\Type{S} = \Type{\LS} \RelUnion \Type{\NLS}\); (b) \(\TT{S} = \TT{\LS} \RelUnion \TT{\NLS}\) [from \Constraint{\ref{constraint-restrict-union}}] \item\label{constraint-restrict-relation-types} (a) \(\Type{\LS} \subset \Type{S}\); (b) \(\TT{\LS} \subset \TT{S}\) [from (\ref{eqn-ls}) above] \item\label{constraint-restrict-tuple-types} (a) \(\Type{\NLS} \subset \Type{S}\); (b) \(\TT{\NLS} \subset \TT{S}\) [from (\ref{eqn-nls}) above] \item\label{constraint-restrict-implied-types-london} \(\CityLondon \subset \Type{\City}\) \item\label{constraint-restrict-implied-types-nonlondon} \((\TCityMinusLondon) \subset \Type{\City}\) \item\label{constraint-restrict-implied-types-TSSL} \(\TSSL \subset \TSSC\) \item\label{constraint-restrict-implied-types-TSSNL} \(\TSSNL \subset \TSSC\) \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-schema \(\bm{\schema{1} = \{S\}}\)} \noindent A user granted access only to \(\schema{1}\) can see only relvar \(S\). Constraint \Constraint{\ref{constraint-restrict-key}}(a) can be propagated directly from \(\schema{0}\) into \(\schema{1}\) as it is already expressed in terms of \(S\) only. Constraints \Constraint{\ref{constraint-restrict-implied-types-london}}--\Constraint{\ref{constraint-restrict-implied-types-TSSNL}} do not include any named relvars in \(\schema{0}\), and thus can also be propagated directly from \(\schema{0}\) into \(\schema{1}\). Constraints \Constraint{\ref{constraint-restrict-key}}(b), \Constraint{\ref{constraint-restrict-key}}(c), and \Constraint{\ref{constraint-restrict-union}}--\Constraint{\ref{constraint-restrict-tuple-types}} cannot be propagated directly from \(\schema{0}\) into \(\schema{1}\), but can all be rewritten in terms of \(S\) only, by substituting the definitions of \(\LS\) (equation~\ref{eqn-ls}) and \(\NLS\) (equation~\ref{eqn-nls}). To simplify expressions, we shall use \(\LSsub\) to denote \(\RelRestrict_{\City = \mathit{'London'}}S\) (\ref{eqn-ls}) and \(\NLSsub\) to denote \(\RelRestrict_{\City \ne \mathit{'London'}}S\) (\ref{eqn-nls}). (This also clearly distinguishes the substitutions from the \(\LS\) and \(\NLS\) relvars in \(\schema{0}\), \(\schema{2}\), and \(\schema{3}\).) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-i-restrict} \begin{ConstraintList}[\schema{1}] \item \Sno\ is a key for each of \(\LSsub\) and \(\NLSsub\), therefore the functional dependency \(\{\Sno\} \rightarrow \{\Sname, \Status, \City\}\) also holds in each of (b) \(\LSsub\); and (c) \(\NLSsub\). [The fact that we are dealing with two disjoint subsets of \(\City\) does not change this.] \setcounter{constraint}{3} \item \(S = \LSsub \RelUnion \NLSsub\) [or even just \(S = S\)] \item \(\RelRestrict_{\City \ne \mathit{'London'}}\LSsub = \emptyset\) \item \(\RelRestrict_{\City = \mathit{'London'}}\NLSsub = \emptyset\) \item \(\LSsub \RelIntersect \NLSsub = \emptyset\) \item (a) \(\Type{S} = \Type{\LSsub} \RelUnion \Type{\NLSsub}\); (b) \(\TT{S} = \TT{\LSsub} \RelUnion \TT{\NLSsub}\) \newline {[or even just \(\Type{S} = \Type{S}\) and \(\TT{S} = \TT{S}\)]} \item (a) \(\Type{\LSsub} \subset \Type{S}\); (b) \(\TT{\LSsub} \subset \TT{S}\) \item (a) \(\Type{\NLSsub} \subset \Type{S}\); (b) \(\TT{\NLSsub} \subset \TT{S}\) \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-i-restrict} \noindent The constraints appearing in \(\schema{1}\) are \Constraint{\ref{constraint-restrict-key}}(a), \Constraint[\schema{1}]{\ref{constraint-restrict-key}}(b), \Constraint[\schema{1}]{\ref{constraint-restrict-key}}(c), \Constraint[\schema{1}]{\ref{constraint-restrict-union}}--\Constraint[\schema{1}]{\ref{constraint-restrict-tuple-types}}, and \Constraint{\ref{constraint-restrict-implied-types-london}}--\Constraint{\ref{constraint-restrict-implied-types-TSSNL}}. The constraints within the schema also imply the existence of several subtype nodes (e.g., the two subtypes of \(\Type{City}\) implied by \Constraint{\ref{constraint-restrict-implied-types-london}} and \Constraint{\ref{constraint-restrict-implied-types-nonlondon}}), along with corresponding selection and projection edges. \Constraint[\schema{1}]{\ref{constraint-restrict-union}}, \Constraint[\schema{1}]{\ref{constraint-restrict-relation-type-union}}(a), and \Constraint[\schema{1}]{\ref{constraint-restrict-relation-type-union}}(b) are effectively tautologies that add no new information, and can thus be safely ignored. The completed SIG for \(\schema{1}\) is shown in Figure~\ref{fig-sig-s-i-restrict}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node (TLS) {\(\Type{\LSsub}\)}; &[7mm] \node (TTLS) {\(\TT{\LSsub}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\Type{\Sname}\)}; \\ \node (TLSPlusNLS) {\(\Type{S}\)}; & \node (TTLSPlusNLS) {\(\TT{S}\)}; & \node (TSno) {\(\Type{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\Type{City}\)}; \\ & & & & & \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TNLS) {\(\Type{\NLSsub}\)}; & \node (TTNLS) {\(\TT{\NLSsub}\)}; & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) }, }; % FDs (TSno) -- [funcdep left,bend left] (TSSL); (TSno) -- [funcdep right] (TSSNL); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTLSPlusNLS), (TTLS) } -- (TSno) }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTNLS) -- (TSno) }; { [edges=projection left] (TTLS) -- (TSSL), (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TTNLS) -- (TSSNL), (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (NLSCity), (TStatus) }, (TSSNL) -- [bend left=10] (TSname), }; % selection edges { [edges=selection left] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) }, (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, (TCity) -- (LSCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right] (TSSC), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,bend right] (TSSC), }; \end{tikzpicture} \caption{SIG for schema \(\schema{1} = \{S\}\).} \label{fig-sig-s-i-restrict} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-schema \(\bm{\schema{2} = \{\LS, \NLS\}}\)} \noindent A user granted access only to \(\schema{2}\) can see only relvars \(\LS\) and \(\NLS\). Constraints \Constraint{\ref{constraint-restrict-key}}(b), \Constraint{\ref{constraint-restrict-key}}(c), and \Constraint{\ref{constraint-restrict-notlondon}}--\Constraint{\ref{constraint-restrict-disjoint}} can be propagated directly from \(\schema{0}\) into \(\schema{2}\) as they are defined only in terms of \(\LS\) and \(\NLS\). \Constraint{\ref{constraint-restrict-implied-types-london}}--\Constraint{\ref{constraint-restrict-implied-types-TSSNL}} can be propagated directly into \(\schema{2}\) as they are not expressed in terms of any named relvars in \(\schema{0}\). \Constraint{\ref{constraint-restrict-key}}(a), \Constraint{\ref{constraint-restrict-union}}, \Constraint{\ref{constraint-restrict-relation-type-union}}, \Constraint{\ref{constraint-restrict-relation-types}}, and \Constraint{\ref{constraint-restrict-tuple-types}} can be rewritten in terms of \(\LS\) and \(\NLS\) only. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-ii-restrict} \begin{ConstraintList}[\schema{2}] \item \Sno\ is a key for \(\LS \RelUnion \NLS\), therefore the functional dependency \(\{\Sno\} \rightarrow \{\Sname, \Status, \City\}\) also holds in (a) \(\LS \RelUnion \NLS\). \setcounter{constraint}{3} \item \(\LS \RelUnion \NLS = \LS \RelUnion \NLS\) \setcounter{constraint}{7} \item (a) \(\Type{\LS} \RelUnion \Type{\NLS} = \Type{\LS} \RelUnion \Type{\NLS}\); \newline (b) \(\TT{\LS} \RelUnion \TT{\NLS} = \TT{\LS} \RelUnion \TT{\NLS}\) \item (a) \(\Type{\LS} \subset \Type{\LS} \RelUnion \Type{\NLS}\); (b) \(\TT{\LS} \subset \TT{\LS} \RelUnion \TT{\NLS}\) \item (a) \(\Type{\NLS} \subset \Type{\LS} \RelUnion \Type{\NLS}\); (b) \(\TT{\NLS} \subset \TT{\LS} \RelUnion \TT{\NLS}\) \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-ii-restrict} \noindent The constraints appearing in \(\schema{2}\) are \Constraint[\schema{2}]{\ref{constraint-restrict-key}}(a), \Constraint{\ref{constraint-restrict-key}}(b), \Constraint{\ref{constraint-restrict-key}}(c), \Constraint[\schema{2}]{\ref{constraint-restrict-union}}, \Constraint{\ref{constraint-restrict-notlondon}}--\Constraint{\ref{constraint-restrict-disjoint}}, \Constraint[\schema{2}]{\ref{constraint-restrict-relation-type-union}}--\Constraint[\schema{2}]{\ref{constraint-restrict-tuple-types}}, and \Constraint{\ref{constraint-restrict-implied-types-london}}--\Constraint{\ref{constraint-restrict-implied-types-TSSNL}}. This is essentially the same set of constraints as \(\schema{1}\), but with different rewritings. This means that the SIG for \(\schema{2}\) can be built in the same manner as for \(\schema{1}\), with some nodes named differently. In particular, \Constraint[\schema{2}]{\ref{constraint-restrict-relation-type-union}} implies that there will be nodes \(\TLSPlusNLS\) and \(\TTLSPlusNLS\) instead of \(\Type{S}\) and \(\TT{S}\), respectively. There will also be \(\Type{\LS}\) instead of \(\Type{\LSsub}\), etc. The complete SIG for \(\schema{2}\) is shown in Figure~\ref{fig-sig-s-ii-restrict}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node (TLS) {\(\Type{\LS}\)}; &[7mm] \node (TTLS) {\(\TT{\LS}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\Type{\Sname}\)}; \\ \node (TLSPlusNLS) {\(\StackTLSPlusNLS\)}; & \node (TTLSPlusNLS) {\(\StackTTLSPlusNLS\)}; & \node (TSno) {\(\Type{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\Type{City}\)}; \\ & & & & & \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TNLS) {\(\Type{\NLS}\)}; & \node (TTNLS) {\(\TT{\NLS}\)}; & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) }, }; % FDs (TSno) -- [funcdep left,bend left] (TSSL); (TSno) -- [funcdep right] (TSSNL); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTLSPlusNLS), (TTLS) } -- (TSno) }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTNLS) -- (TSno) }; { [edges=projection left] (TTLS) -- (TSSL), (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TTNLS) -- (TSSNL), (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (NLSCity), (TStatus) }, (TSSNL) -- [bend left=10] (TSname), }; % selection edges { [edges=selection left] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) }, (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, (TCity) -- (LSCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right] (TSSC), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,bend right] (TSSC), }; \end{tikzpicture} \caption{SIG for schema \(\schema{2} = \{\LS, \NLS\}\).} \label{fig-sig-s-ii-restrict} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-chema \(\bm{\schema{3} = \{LS\}}\)} \noindent A user granted access only to \(\schema{3}\) can see only relvar \(\LS\). Constraints \Constraint{\ref{constraint-restrict-key}}(b) and \Constraint{\ref{constraint-restrict-notlondon}} can be propagated directly into \(\schema{3}\), as they are defined in terms of \(\LS\) only. \Constraint{\ref{constraint-restrict-implied-types-london}}--\Constraint{\ref{constraint-restrict-implied-types-TSSNL}} can also be propagated directly into \(\schema{3}\) as previously discussed. The remaining constraints cannot be rewritten in terms of \(\LS\) only, and so cannot be propagated into \(\schema{3}\). As we were able to propagate all constraints into \(\schema{1}\), this indicates a qualitative difference between \(\schema{1}\) and \(\schema{3}\). It therefore seems likely that the relative information capacities of \(\schema{1}\) and \(\schema{3}\) will also differ. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-iii-restrict} \noindent N/A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-iii-restrict} \noindent The constraints appearing in \(\schema{3}\) are \Constraint{\ref{constraint-restrict-key}}(b), \Constraint{\ref{constraint-restrict-notlondon}}, and \Constraint{\ref{constraint-restrict-implied-types-london}}--\Constraint{\ref{constraint-restrict-implied-types-TSSNL}}. The SIG will only include edges and nodes that relate either directly to \(\LS\) or are independent of any relvar, such as the subtype node \(\TSSNL\). The complete SIG for \(\schema{3}\) is shown in Figure~\ref{fig-sig-s-iii-restrict} on the previous page. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node (TLS) {\(\Type{\LS}\)}; &[7.5mm] \node (TTLS) {\(\TT{\LS}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\Type{\Sname}\)}; \\ & & \node (TSno) {\(\Type{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\Type{City}\)}; \\ & & & & & \node (TStatus) {\(\Type{\Status}\)}; \\ & & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TLS) -- (TTLS), }; % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTLS) -- (TSno) }; { [edges=projection left] (TTLS) -- (TSSL), (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (NLSCity), (TStatus) }, (TSSNL) -- [bend left=10] (TSname), }; % selection edges { [edges=selection left] (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] (TCity) -- (LSCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); }; \end{tikzpicture} \caption{SIG for schema \(\schema{3} = \{\LS\}\).} \label{fig-sig-s-iii-restrict} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Transformations} \label{sec-transforming-restrict} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{\(\bm{\schema{1}, \schema{2}}\)} \noindent Looking at the SIGs for \(\schema{1}\) and \(\schema{2}\), we can see almost immediately that both SIGs have the same node and edge structure; indeed the only difference is in the names of some of the nodes. However, we already know from the discussion in Sections~\ref{sec-constraints-s-i-restrict} and~\ref{sec-constraints-s-ii-restrict} that these differently named nodes are just rewritings that represent the same domains. We therefore have an immediate isomorphism between the two SIGs and can conclude that the information capacities of \(\schema{1}\) and \(\schema{2}\) are equivalent, i.e., \(\Equivalent{\schema{1}}{\schema{2}}\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{\(\bm{\schema{1}, \schema{3}}\)} There are some shared structures, but the two SIGs are clearly different. Since SIG transformations only ever preserve or enhance information capacity, intuitively we should attempt to transform the ``smaller'' SIG for \(\schema{3}\) into the ``larger'' SIG for \(\schema{1}\). First, duplicate node \(\Type{\LS}\) to \(\Type{S}\) and node \(\TT{\LS}\) to \(\TT{S}\). To obtain a node structure similar to that of \(\schema{1}\), we also need to further duplicate \(\Type{S}\) to \(\Type{\NLS}\) and \(\TT{S}\) to \(\TT{\NLS}\). The result of these duplications is shown in Figure~\ref{fig-transform-all-duplicates}. The transformed SIG now has the same node structure as that for \(\schema{1}\). %%%%%%%%%%%%%%%%%%%% \begin{figure}[t] \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node [input keep] (TLS) {\(\Type{\LS}\)}; &[7mm] \node [input keep] (TTLS) {\(\TT{\LS}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\Type{\Sname}\)}; \\ \node [output] (TLSPlusNLS) {\(\Type{S}\)}; & \node [output] (TTLSPlusNLS) {\(\TT{S}\)}; & \node (TSno) {\(\Type{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\Type{City}\)}; \\ & & & & & \node (TStatus) {\(\Type{\Status}\)}; \\ \node [output] (TNLS) {\(\Type{\NLS}\)}; & \node [output] (TTNLS) {\(\TT{\NLS}\)}; & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TLS) -- (TTLS), }; % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTLS) -- (TSno) }; { [edges=projection left] (TTLS) -- (TSSL), (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (NLSCity), (TStatus) }, (TSSNL) -- [bend left=10] (TSname), }; % selection edges { [edges=selection left] (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] (TCity) -- (LSCity), }; { [edges={bijection,output},edge label'={\scriptsize\(\RelRestrict\)}] { (TLS), (TTLS) } -- { (TLSPlusNLS), (TTLSPlusNLS) }, { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), }; (TSno) -- [funcdep right] (TSSC); (TSSC) -- [projection left] (TCity); % transformation indicators { [edges={->,gray,dashed,bend left=45}] (TLS) -- [edge node={node[sloped,above]{\small\emph{copy}}}] (TLSPlusNLS), (TLSPlusNLS) -- [edge node={node[sloped,above,rotate=180]{\small\emph{copy}}}] (TNLS) }; }; \end{tikzpicture} \caption{SIG for transformed schema \(\schema{3}'\) after duplicating \Type{\LS} and \TT{\LS} to produce analogues of the additional nodes in \(\schema{1}\). Inputs to the transformations are coloured \textcolor{blue}{blue}, outputs are coloured \textcolor{red}{red}. [\(\equiv\)]} \label{fig-transform-all-duplicates} \end{figure} %%%%%%%%%%%%%%%%%%%% The next phase is to copy and move edges. To produce the edge \(\Type{S} \SurTotal \TT{S}\), we can first \emph{copy} \(\Type{\LS} \SurTotal \TT{\LS}\) across the path \(\TT{S} \TrivialSelection \TT{\LS}\) (giving \(\Type{\LS} \SurTotal \TT{S}\)), then \emph{move} it across \(\Type{S} \TrivialSelection \Type{\LS}\). We can carry out a similar series of transformations to copy and move \(\Type{S} \SurTotal \TT{S}\) to \(\Type{\NLS} \SurTotal \TT{\NLS}\) (see Figure~\ref{fig-transform-edge-moves}). We next copy and move the unique projection edge \(\TT{\LS} \TrivialProjection \Type{\Sno}\) to produce the edges \(\TT{S} \TrivialProjection \Type{\Sno}\) and \(\TT{\NLS} \TrivialProjection \Type{\Sno}\). Similarly, we copy and move \(\TT{\LS} \ProjectionEdge \TSSL\) to produce \(\TT{S} \LabelledEdge{\sigedge{12}}{\RelProject} \TSSC\) and \(\TT{\NLS} \LabelledEdge{\sigedge{02}}{\RelProject} \TSSNL\). Note that the first edge in the latter pair of transformations loses its totality annotation, and the second edge its surjectivity annotation, as a result of composition with the pre-existing non-bijective selection edges between the constructed tuple types running vertically down the centre of the SIG. Finally, we carry out similar copy and move transformations to copy the key edge \(\Type{\Sno} \KeyEdge \TSSC\) to \(\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSL\), and \(\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSNL\). Both new edges lose their totality annotations for the same reason as the projection edges above. The final result of all these edge transformations is shown in Figure~\ref{fig-transform-edge-moves}. %%%%%%%%%%%%%%%%%%%% \begin{figure}[t] \centering \begin{tikzpicture} \matrix[row sep=0.5cm] { \node (TLS) {\(\Type{\LS}\)}; &[7mm] \node (TTLS) {\(\TT{\LS}\)}; &[7mm] &[4mm] &[-4mm] \node (TSSL) {\(\StackTSSL\)}; & & \node (LSCity) {\(\CityLondon\)}; \\ & & & \node (TSname) {\(\Type{\Sname}\)}; \\ \node (TLSPlusNLS) {\(\Type{S}\)}; & \node (TTLSPlusNLS) {\(\TT{S}\)}; & \node (TSno) {\(\Type{\Sno}\)}; & & \node (TSSC) {\(\StackTSSC\)}; & & \node (TCity) {\(\Type{City}\)}; \\ & & & & & \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TNLS) {\(\Type{\NLS}\)}; & \node (TTNLS) {\(\TT{\NLS}\)}; & & & \node (TSSNL) {\(\StackTSSNL\)}; & & \node (NLSCity) {\(\StackTCityMinusLondon\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TLS) -- [input keep,edge node={node (c1a) {}}] (TTLS), (TLSPlusNLS) -- [output,edge node={node (c1b) {}}] (TTLSPlusNLS), (TNLS) -- [output,edge node={node (c1c) {}}] (TTNLS), }; % FDs (TSno) -- [funcdep left,arrows={-{crossbar}>},bend left,output] (TSSL); (TSno) -- [funcdep right,arrows={-{crossbar}>},output] (TSSNL); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTLS) -- [input keep] (TSno), (TTLSPlusNLS) -- [output] (TSno), }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTNLS) -- [output] (TSno), }; { [edges=projection left] (TTLS) -- [input keep,edge node={node (c2a) {}}] (TSSL), (TSSL) -- { (TStatus), (LSCity) }, }; { [edges=projection right] (TSSC) -- { (TSname), (TStatus) }, (TSSL) -- (TSname), (TSSNL) -- { (NLSCity), (TStatus) }, (TSSNL) -- [bend left=10] (TSname), (TTNLS) -- [arrows={:->},output,edge node={node (c2c) {}}] (TSSNL), }; % selection edges { [edges=selection left] (TSSC) -- { (TSSL), (TSSNL) }, (TCity) -- (NLSCity), }; { [edges=selection right] (TCity) -- (LSCity), }; { [edges=bijection,edge label'={\scriptsize\(\RelRestrict\)}] { (TLS), (TTLS) } -- { (TLSPlusNLS), (TTLSPlusNLS) }, { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) }, }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TSno) -- (TSSC), (TSSC) -- (TCity), (TTLSPlusNLS) -- [bend right] (TSSC), }; (TSno) -- [funcdep right,input keep] (TSSC); (TSSC) -- [projection left] (TCity); (TTLSPlusNLS) -- [projection right,arrows={:{crossbar}->},bend right,output,edge node={node (c2b) {}}] (TSSC), % transformation indicators { [edges={->,gray,dashed}] (c1a) -- (c1b), (c1a) -- [bend right=12,edge node={node[above,sloped,pos=0.175]{\small\emph{copy \& move}}}] (c1c), (c2a) -- [bend right=12] (c2b), (c2a) -- [bend right=12,edge node={node[above,sloped,pos=0.175]{\small\emph{copy \& move}}}] (c2c) }; }; \end{tikzpicture} \caption{SIG for transformed schema \(\schema{3}'\) after copying and moving various edges. [\(\preceq\)]} \label{fig-transform-edge-moves} \end{figure*} %%%%%%%%%%%%%%%%%%%% If we compare Figure~\ref{fig-sig-s-i-restrict} with Figure~\ref{fig-transform-edge-moves}, we can see that the transformed SIG for \(\schema{3}'\) now has the same topology as the SIG for \(\schema{1}\). We still need to remove the totality annotations from the newly created trivial selection edges between the relation and tuple types on the left, but even when this is done, we do not achieve the desired isomorphism, as the edges \(\TT{S} \LabelledEdge{\sigedge{12}}{\RelProject} \TSSC\), \(\TT{\NLS} \LabelledEdge{\sigedge{02}}{\RelProject} \TSSNL\), \(\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSL\), and \(\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSNL\) all have different annotations from the corresponding edges in the SIG for \(\schema{1}\). This clearly shows that the information capacities of \(\schema{1}\) and \(\schema{3}\) are incomparable, and that view updates based on this pair of schemas will be problematic. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Additional notes} If we were to create a fourth sub-schema \(\schema{4} = \{\NLS\}\) and compare this with \(\schema{1}\), the result would be similar to that for \(\schema{3}\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{Nonloss projection views (Date, Chapter 5)} \label{sec-nonloss-projection} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Relvars} \begin{itemize} \item base relvar \(S\{\Sno, \Status, \City\}\) \item view \(\ST = \RelProject_{\{\Sno, \Status\}}S\) [supplier status] \eqnnumber\label{eqn-st} \item view \(\SC = \RelProject_{\{\Sno, \City\}}S\) [supplier city] \eqnnumber\label{eqn-sc} \end{itemize} (Incidentally, the views form a 6NF nonloss decomposition of \(S\).) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Tuple types} \begin{align} \TT{S} &= \Type{\Sno} \times \Type{\Status} \times \Type{\City} \nonumber \\ \TT{\ST} &= \Type{\Sno} \times \Type{\Status} \nonumber \\ \TT{\SC} &= \Type{\Sno} \times \Type{\City} \nonumber \end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Base schema \(\bm{\schema{0} = \{S, \ST, \SC\}}\)} \begin{ConstraintList} \item\label{constraint-project-key} \(\Sno\) is a key for each of \(S\), \(\ST\), and \(\SC\), so the functional dependency \(\{\Sno\} \rightarrow \{\Status\}\) holds in each of (a) \(S\) and (b) \(\ST\); and the functional dependency \(\{\Sno\} \rightarrow \{\City\}\) holds in each of (c) \(S\) and (d) \(\SC\). [(a) and (c) could in principle be merged into a single functional dependency \(\{\Sno\} \rightarrow \{\Status, \City\}\) in \(S\).] % not sure how relevant the FK constraints are? % how to represent? FK is a subset of the original key, so selection edge between S.Sno and each of LS.Sno and NLS.Sno % or perhaps a selection edge from Sno to itself? (labelled FK) % Notation: \Type{\Sno}, \Type{\LS.\Sno}, \Type{\NLS.\Sno}? \item\label{constraint-project-stfk} \(\RelProject_{\{\Sno\}}\ST \subseteq \RelProject_{\{\Sno\}}S\) [foreign key from \(\ST\) to \(S\)] \item\label{constraint-project-scfk} \(\RelProject_{\{\Sno\}}\SC \subseteq \RelProject_{\{\Sno\}}S\) [foreign key from \(\SC\) to \(S\)] \item\label{constraint-project-join} \(S = \SC \RelJoin \ST\) [from (\ref{eqn-st}), (\ref{eqn-sc})] \item\label{constraint-project-s-st-identical} \(\RelProject_{\{\Sno\}}S = \RelProject_{\{\Sno\}}\ST\) [from \Constraint{\ref{constraint-project-join}}, (\ref{eqn-st})] \item\label{constraint-project-s-sc-identical} \(\RelProject_{\{\Sno\}}S = \RelProject_{\{\Sno\}}\SC\) [from \Constraint{\ref{constraint-project-join}}, (\ref{eqn-sc})] \item\label{constraint-project-st-sc-identical} \(\RelProject_{\{\Sno\}}\ST = \RelProject_{\{\Sno\}}\SC\) [from \Constraint{\ref{constraint-project-join}}, (\ref{eqn-st}), (\ref{eqn-sc})] \item\label{constraint-project-relation-type-join} (a) \(\Type{S} = \Type{\ST} \RelJoin \Type{\SC}\); (b) \(\TT{S} = \TT{\ST} \RelJoin \TT{\SC}\) [from \Constraint{\ref{constraint-project-join}}] \item\label{constraint-project-types-st} (a) \(\Type{\ST} = \RelProject_{\{\Sno, \Status\}}\Type{S}\); (b) \(\TT{\ST} = \RelProject_{\{\Sno, \Status\}}\TT{S}\) [from (\ref{eqn-st})] \item\label{constraint-project-types-sc} (a) \(\Type{\SC} = \RelProject_{\{\Sno, \City\}}\Type{S}\); (b) \(\TT{\SC} = \RelProject_{\{\Sno, \City\}}\TT{S}\) [from (\ref{eqn-sc})] \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-schema \(\bm{\schema{1} = \{S\}}\)} \noindent A user granted access only to \(\schema{1}\) can see only relvar \(S\). Constraints \Constraint{\ref{constraint-project-key}}(a) and \Constraint{\ref{constraint-project-key}}(c) can be propagated directly from \(\schema{0}\) into \(\schema{1}\) as they are already expressed in terms of \(S\) only. Constraints \Constraint{\ref{constraint-project-join}}--\Constraint{\ref{constraint-restrict-tuple-types}} cannot be propagated directly from \(\schema{0}\) into \(\schema{1}\), but can all be rewritten in terms of \(S\) only, by substituting the definitions of \(\ST\) (equation~\ref{eqn-st}) and \(\SC\) (equation~\ref{eqn-sc}), i.e., \(\STsub = \RelProject_{\{\Sno, \Status\}}S\) and \(\SCsub = \RelProject_{\{\Sno, \City\}}S\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-i-project} \begin{ConstraintList}[\schema{1}] \item \Sno\ is a key for each of \(\STsub\) and \(\SCsub\), therefore the functional dependency \(\{\Sno\} \rightarrow \{\Status\}\) holds in (b) \(\STsub\), and the functional dependency \(\{\Sno\} \rightarrow \{\City\}\) holds in (d) \(\SCsub\). \setcounter{constraint}{3} \item \(S = \STsub \RelJoin \SCsub\) [or even just \(S = S\)] \item \(\RelProject_{\{\Sno\}}S = \RelProject_{\{\Sno\}}\STsub\) \item \(\RelProject_{\{\Sno\}}S = \RelProject_{\{\Sno\}}\SCsub\) \item \(\RelProject_{\{\Sno\}}\STsub = \RelProject_{\{\Sno\}}\SCsub\) \item (a) \(\Type{S} = \Type{\STsub} \RelJoin \Type{\SCsub}\); (b) \(\TT{S} = \TT{\STsub} \RelJoin \TT{\SCsub}\) \newline {[or even just \(\Type{S} = \Type{S}\) and \(\TT{S} = \TT{S}\)]} \item (a) \(\Type{\STsub} = \RelProject_{\{\Sno, \Status\}}\Type{S}\); (b) \(\TT{\STsub} = \RelProject_{\{\Sno, \Status\}}\TT{S}\) \item (a) \(\Type{\SCsub} = \RelProject_{\{\Sno, \City\}}\Type{S}\); (b) \(\TT{\SCsub} = \RelProject_{\{\Sno, \City\}}\TT{S}\) \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-i-project} \noindent The constraints appearing in \(\schema{1}\) are \Constraint{\ref{constraint-project-key}}(a), \Constraint[\schema{1}]{\ref{constraint-project-key}}(b), \Constraint{\ref{constraint-project-key}}(c), \Constraint[\schema{1}]{\ref{constraint-project-key}}(d), and \Constraint[\schema{1}]{\ref{constraint-project-join}}--\Constraint[\schema{1}]{\ref{constraint-project-types-sc}}. \Constraint[\schema{1}]{\ref{constraint-project-join}}, \Constraint[\schema{1}]{\ref{constraint-project-relation-type-join}}(a), and \Constraint[\schema{1}]{\ref{constraint-project-relation-type-join}}(b) are effectively tautologies that add no new information, and can thus be safely ignored. The completed SIG for \(\schema{1}\) is shown in Figure~\ref{fig-sig-s-i-project}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=1.5cm] { \node (TST) {\(\Type{\STsub}\)}; &[7mm] \node (TTST) {\(\TT{\STsub}\)}; &[10mm] &[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TSTPlusSC) {\(\Type{S}\)}; & \node (TTSTPlusSC) {\(\TT{S}\)}; & \node (TSno) {\(\Type{\Sno}\)}; & \node (TSSC) {\(\Type{\Status} \times \Type{\City}\)}; \\ \node (TSC) {\(\Type{\SCsub}\)}; & \node (TTSC) {\(\TT{\SCsub}\)}; & & \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TST), (TSC), (TSTPlusSC) } -- { (TTST), (TTSC), (TTSTPlusSC) }, }; % FDs (TSno) -- [funcdep left] (TStatus); (TSno) -- [funcdep right] (TCity); (TSno) -- [funcdep right] (TSSC); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTSTPlusSC), (TTST) } -- (TSno), }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTSC) -- (TSno), }; { [edges=projection left] { (TSTPlusSC), (TTSTPlusSC) } -- { (TST), (TTST) }, (TSSC) -- (TCity), (TTST) -- (TStatus), }; { [edges=projection right] { (TSTPlusSC), (TTSTPlusSC) } -- { (TSC), (TTSC) }, (TSSC) -- (TStatus), (TTSC) -- (TCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TTSTPlusSC) -- [bend right] (TSSC), }; (TTSTPlusSC) -- [projection right, bend right] (TSSC), }; \end{tikzpicture} \caption{SIG for schema \(\schema{1} = \{S\}\).} \label{fig-sig-s-i-project} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-schema \(\bm{\schema{2} = \{\ST, \SC\}}\)} \noindent A user granted access only to \(\schema{2}\) can see only relvars \(\ST\) and \(\SC\). Constraints \Constraint{\ref{constraint-project-key}}(b), \Constraint{\ref{constraint-project-key}}(d), and \Constraint{\ref{constraint-project-st-sc-identical}} can be propagated directly from \(\schema{0}\) into \(\schema{2}\) as they are already expressed in terms of \(\ST\) and \(\SC\) only. The remaining constraints cannot be propagated directly from \(\schema{0}\) into \(\schema{2}\), but can all be rewritten in terms of \(\ST\) and \(\SC\) only, by substituting \(J = \ST \RelJoin \SC\) for \(S\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-ii-project} \begin{ConstraintList}[\schema{2}] \item \Sno\ is a key for \(J\), therefore (a) the functional dependency \(\{\Sno\} \rightarrow \{\Status\}\) holds in \(J\), and (c) the functional dependency \(\{\Sno\} \rightarrow \{\City\}\) holds in \(J\). \setcounter{constraint}{3} \item \(J = \ST \RelJoin \SC\) [or even just \(\ST \RelJoin \SC = \ST \RelJoin \SC\)] \item \(\RelProject_{\{\Sno\}}J = \RelProject_{\{\Sno\}}\ST\) \item \(\RelProject_{\{\Sno\}}J = \RelProject_{\{\Sno\}}\SC\) \setcounter{constraint}{7} \item (a) \(\Type{J} = \Type{\ST} \RelJoin \Type{\SC}\); (b) \(\TT{J} = \TT{\ST} \RelJoin \TT{\SC}\) \item (a) \(\Type{\ST} = \RelProject_{\{\Sno, \Status\}}\Type{J}\); (b) \(\TT{\ST} = \RelProject_{\{\Sno, \Status\}}\TT{J}\) \item (a) \(\Type{\SC} = \RelProject_{\{\Sno, \City\}}\Type{J}\); (b) \(\TT{\SC} = \RelProject_{\{\Sno, \City\}}\TT{J}\) \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-ii-project} \noindent The constraints appearing in \(\schema{2}\) are \Constraint[\schema{2}]{\ref{constraint-project-key}}(a), \Constraint{\ref{constraint-project-key}}(b), \Constraint[\schema{2}]{\ref{constraint-project-key}}(c), \Constraint{\ref{constraint-project-key}}(d), \Constraint[\schema{2}]{\ref{constraint-project-join}}--\Constraint[\schema{2}]{\ref{constraint-project-s-sc-identical}}, \Constraint{\ref{constraint-project-st-sc-identical}}, and \Constraint[\schema{2}]{\ref{constraint-project-relation-type-join}}--\Constraint[\schema{2}]{\ref{constraint-project-types-sc}}. \Constraint[\schema{2}]{\ref{constraint-project-join}} is effectively a tautology that adds no new information, and can be ignored. The completed SIG for \(\schema{2}\) is shown in Figure~\ref{fig-sig-s-ii-project}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=1.5cm] { \node (TST) {\(\Type{\ST}\)}; &[7mm] \node (TTST) {\(\TT{\ST}\)}; &[10mm] &[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TSTPlusSC) {\(\Type{J}\)}; & \node (TTSTPlusSC) {\(\TT{J}\)}; & \node (TSno) {\(\Type{\Sno}\)}; & \node (TSSC) {\(\Type{\Status} \times \Type{\City}\)}; \\ \node (TSC) {\(\Type{\SC}\)}; & \node (TTSC) {\(\TT{\SC}\)}; & & \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TST), (TSC), (TSTPlusSC) } -- { (TTST), (TTSC), (TTSTPlusSC) }, }; % FDs (TSno) -- [funcdep left] (TStatus); (TSno) -- [funcdep right] (TCity); (TSno) -- [funcdep right] (TSSC); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTSTPlusSC), (TTST) } -- (TSno), }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTSC) -- (TSno), }; { [edges=projection left] { (TSTPlusSC), (TTSTPlusSC) } -- { (TST), (TTST) }, (TSSC) -- (TCity), (TTST) -- (TStatus), }; { [edges=projection right] { (TSTPlusSC), (TTSTPlusSC) } -- { (TSC), (TTSC) }, (TSSC) -- (TStatus), (TTSC) -- (TCity), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TTSTPlusSC) -- [bend right] (TSSC), }; (TTSTPlusSC) -- [projection right, bend right] (TSSC), }; \end{tikzpicture} \caption{SIG for schema \(\schema{2} = \{\ST, \SC\}\).} \label{fig-sig-s-ii-project} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-schema \(\bm{\schema{3} = \{\ST\}}\)} \noindent A user granted access only to \(\schema{3}\) can see only relvar \(\ST\). Only constraint \Constraint{\ref{constraint-project-key}}(b) can be propagated directly from \(\schema{0}\) into \(\schema{3}\) as it is the only one expressed in terms of \(\ST\) only. None of the remaining constraints can be rewritten in terms of \(\ST\) only, and so cannot be propagated into \(\schema{3}\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-iii-project} N/A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-iii-project} \noindent The only constraint appearing in \(\schema{3}\) is \Constraint{\ref{constraint-project-key}}(b). The SIG will only include edges and nodes that relate either directly to \(\ST\) or are independent of any relvar. The completed SIG for \(\schema{3}\) is shown in Figure~\ref{fig-sig-s-iii-project}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=1.5cm] { \node (TST) {\(\Type{\ST}\)}; &[7mm] \node (TTST) {\(\TT{\ST}\)}; &[10mm] &[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ & & \node (TSno) {\(\Type{\Sno}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TST) -- (TTST), }; % FDs (TSno) -- [funcdep left] (TStatus); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTST) -- (TSno), }; { [edges=projection left] (TTST) -- (TStatus), }; }; \end{tikzpicture} \caption{SIG for schema \(\schema{3} = \{\ST\}\).} \label{fig-sig-s-iii-project} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Transformations} \label{sec-transforming-project} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{\(\bm{\schema{1}, \schema{2}}\)} \noindent The SIGs for \(\schema{1}\) and \(\schema{2}\) are immediately isomorphic, so \(\Equivalent{\schema{1}}{\schema{2}}\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{\(\bm{\schema{1}, \schema{3}}\)} Despite their differences, it's clear that the comparison between \(\schema{1}\) and \(\schema{3}\) for projection views will be similar to that for disjoint restriction views (Section~\ref{sec-transforming-restrict}), as the SIG structures are topologically similar. It's therefore reasonable to conclude that the information capacities of \(\schema{1}\) and \(\schema{3}\) are incomparable, and that view updates based on this pair of schemas will be problematic. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Additional notes} If we were to create a fourth sub-schema \(\schema{4} = \{\SC\}\) and compare this with \(\schema{1}\), the result would be similar to that for \(\schema{3}\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{Lossy projection views (Date, Chapter 5)} \label{sec-lossy-projection} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Relvars} \begin{itemize} \item base relvar \(S\{\Sno, \Status, \City\}\) \item view \(\ST = \RelProject_{\{\Sno, \Status\}}S\) [supplier status] \eqnnumber\label{eqn-st-lossy} \item view \(\TC = \RelProject_{\{\Status, \City\}}S\) [status \& city] \eqnnumber\label{eqn-tc-lossy} \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Tuple types} \begin{align} \TT{S} &= \Type{\Sno} \times \Type{\Status} \times \Type{\City} \nonumber \\ \TT{\ST} &= \Type{\Sno} \times \Type{\Status} \nonumber \\ \TT{\TC} &= \Type{\Status} \times \Type{\City} \nonumber \end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Base schema \(\bm{\schema{0} = \{S, \ST, \TC\}}\)} \begin{ConstraintList} \item\label{constraint-project-lossy-key} \(\Sno\) is a key for each of \(S\) and \(\ST\), so the functional dependency \(\{\Sno\} \rightarrow \{\Status\}\) holds in each of (a) \(S\) and (b) \(\ST\); and the functional dependency \(\{\Sno\} \rightarrow \{\City\}\) holds in (c) \(S\). [(a) and (c) could in principle be merged into a single functional dependency \(\{\Sno\} \rightarrow \{\Status, \City\}\) in \(S\).] \item\label{constraint-project-lossy-key-ii} \(\{\Status, \City\}\) is a key for \(\TC\), so the functional dependencies (a) \(\{\Status, \City\} \rightarrow \{\Status\}\) and (b) \(\{\Status, \City\} \rightarrow \{\City\}\) hold in \(\TC\). % not sure how relevant the FK constraints are? % how to represent? FK is a subset of the original key, so selection edge between S.Sno and each of LS.Sno and NLS.Sno % or perhaps a selection edge from Sno to itself? (labelled FK) % Notation: \Type{\Sno}, \Type{\LS.\Sno}, \Type{\NLS.\Sno}? \item\label{constraint-project-lossy-stfk} \(\RelProject_{\{\Sno\}}\ST \subseteq \RelProject_{\{\Sno\}}S\) [foreign key from \(\ST\) to \(S\)] \item\label{constraint-project-lossy-s-st-identical} \(\RelProject_{\{\Sno\}}S = \RelProject_{\{\Sno\}}\ST\) [from (\ref{eqn-st-lossy})] \item\label{constraint-project-lossy-st-tc-status} \(\RelProject_{\{\Status\}}\ST = \RelProject_{\{\Status\}}\TC\) [from (\ref{eqn-tc-lossy})] \item\label{constraint-project-lossy-types-st} (a) \(\Type{\ST} = \RelProject_{\{\Sno, \Status\}}\Type{S}\); (b) \(\TT{\ST} = \RelProject_{\{\Sno, \Status\}}\TT{S}\) [from (\ref{eqn-st-lossy})] \item\label{constraint-project-lossy-types-tc} (a) \(\Type{\TC} = \RelProject_{\{\Status, \City\}}\Type{S}\); (b) \(\TT{\TC} = \RelProject_{\{\Status, \City\}}\TT{S}\) [from (\ref{eqn-tc-lossy})] \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-schema \(\bm{\schema{1} = \{S\}}\)} \noindent A user granted access only to \(\schema{1}\) can see only relvar \(S\). Constraints \Constraint{\ref{constraint-project-lossy-key}}(a) and \Constraint{\ref{constraint-project-lossy-key}}(c) can be propagated directly from \(\schema{0}\) into \(\schema{1}\) as they are already expressed in terms of \(S\) only. Constraints \Constraint{\ref{constraint-project-lossy-key-ii}} and \Constraint{\ref{constraint-project-lossy-s-st-identical}}--\Constraint{\ref{constraint-project-lossy-types-tc}} cannot be propagated directly from \(\schema{0}\) into \(\schema{1}\), but can all be rewritten in terms of \(S\) only, by substituting the definitions of \(\ST\) (equation~\ref{eqn-st-lossy}) and \(\TC\) (equation~\ref{eqn-tc-lossy}), i.e., \(\STsub = \RelProject_{\{\Sno, \Status\}}S\) and \(\TCsub = \RelProject_{\{\Status, \City\}}S\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-i-project-lossy} \begin{ConstraintList}[\schema{1}] \item \(\Sno\) is a key for \(\STsub\), therefore the functional dependency \(\{\Sno\} \rightarrow \{\Status\}\) holds in (b) \(\STsub\). \item \(\{\Status, \City\}\) is a key for \(\TCsub\), so the functional dependencies (a) \(\{\Status, \City\} \rightarrow \{\Status\}\) and (b) \(\{\Status, \City\} \rightarrow \{\City\}\) hold in \(\TCsub\). [Ignoring the trivial case of \(\{\Status, \City\} \rightarrow \{\Status, \City\}\).] \setcounter{constraint}{3} \item \(\RelProject_{\{\Sno\}}S = \RelProject_{\{\Sno\}}\STsub\) \item \(\RelProject_{\{\Sno\}}\STsub = \RelProject_{\{\Sno\}}\TCsub\) \item (a) \(\Type{\STsub} = \RelProject_{\{\Sno, \Status\}}\Type{S}\); (b) \(\TT{\STsub} = \RelProject_{\{\Sno, \Status\}}\TT{S}\) \item (a) \(\Type{\TCsub} = \RelProject_{\{\Status, \City\}}\Type{S}\); (b) \(\TT{\TCsub} = \RelProject_{\{\Status, \City\}}\TT{S}\) \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-i-project-lossy} \noindent The constraints appearing in \(\schema{1}\) are \Constraint{\ref{constraint-project-lossy-key}}(a), \Constraint[\schema{1}]{\ref{constraint-project-lossy-key}}(b), \Constraint{\ref{constraint-project-lossy-key}}(c), \Constraint[\schema{1}]{\ref{constraint-project-lossy-key-ii}}, and \Constraint[\schema{1}]{\ref{constraint-project-lossy-s-st-identical}}--\Constraint[\schema{1}]{\ref{constraint-project-lossy-types-tc}}. The completed SIG for \(\schema{1}\) is shown in Figure~\ref{fig-sig-s-i-project-lossy}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=1.5cm] { \node (TST) {\(\Type{\STsub}\)}; &[7mm] \node (TTST) {\(\TT{\STsub}\)}; &[10mm] &[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TSTPlusTC) {\(\Type{S}\)}; & \node (TTSTPlusTC) {\(\TT{S}\)}; & \node (TSno) {\(\Type{\Sno}\)}; & \node (TSSC) {\(\Type{\Status} \times \Type{\City}\)}; \\ \node (TTC) {\(\Type{\TCsub}\)}; & \node (TTTC) {\(\TT{\TCsub}\)}; & & \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TST), (TTC), (TSTPlusTC) } -- { (TTST), (TTTC), (TTSTPlusTC) }, }; % FDs { [edges=funcdep right] (TSno) -- { (TCity), (TSSC) }, (TSSC) -- [bend right] (TStatus), }; { [edges=funcdep left] (TSno) -- (TStatus), (TSSC) -- [bend left] (TCity), }; % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTSTPlusTC), (TTST) } -- (TSno), }; { [edges=projection left] { (TSTPlusTC), (TTSTPlusTC) } -- { (TST), (TTST) }, (TTST) -- (TStatus), (TSSC) -- [bend right] (TCity), }; { [edges=projection right] { (TSTPlusTC), (TTSTPlusTC) } -- { (TTC), (TTTC) }, (TSSC) -- [bend left] (TStatus), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TTSTPlusTC) -- [bend right] (TSSC), (TTTC) -- [bend right=10] (TSSC), }; (TTSTPlusTC) -- [projection right, bend right] (TSSC), (TTTC) -- [edges=bijection,edge label'={\scriptsize\(\RelProject\)}, bend right=10] (TSSC), }; \end{tikzpicture} \caption{SIG for schema \(\schema{1} = \{S\}\).} \label{fig-sig-s-i-project-lossy} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-schema \(\bm{\schema{2} = \{\ST, \TC\}}\)} \noindent A user granted access only to \(\schema{2}\) can see only relvars \(\ST\) and \(\TC\). Constraints \Constraint{\ref{constraint-project-lossy-key}}(b), \Constraint{\ref{constraint-project-lossy-key-ii}}, and \Constraint{\ref{constraint-project-lossy-st-tc-status}} can be propagated directly from \(\schema{0}\) into \(\schema{2}\) as they are already expressed in terms of \(\ST\) and \(\TC\) only. The remaining constraints cannot be propagated from \(\schema{0}\) into \(\schema{2}\) at all, as none of them can be rewritten in terms of \(\ST\) and \(\TC\) only (due to the lossy decomposition of \(S\) into \(\ST\) and \(\TC\)). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-ii-project-lossy} N/A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-ii-project-lossy} \noindent The constraints appearing in \(\schema{2}\) are \Constraint[\schema{2}]{\ref{constraint-project-lossy-key}}(b), \Constraint[\schema{2}]{\ref{constraint-project-lossy-key-ii}}, and \Constraint[\schema{2}]{\ref{constraint-project-lossy-st-tc-status}}. The completed SIG for \(\schema{2}\) is shown in Figure~\ref{fig-sig-s-ii-project-lossy}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=1.5cm] { \node (TST) {\(\Type{\ST}\)}; &[7mm] \node (TTST) {\(\TT{\ST}\)}; &[10mm] &[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ & & \node (TSno) {\(\Type{\Sno}\)}; & \node (TSSC) {\(\Type{\Status} \times \Type{\City}\)}; \\ \node (TTC) {\(\Type{\TC}\)}; & \node (TTTC) {\(\TT{\TC}\)}; & & \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TST), (TTC) } -- { (TTST), (TTTC) }, }; % FDs { [edges=funcdep right] (TSSC) -- [bend right] (TStatus), }; { [edges=funcdep left] (TSno) -- (TStatus), (TSSC) -- [bend left] (TCity), }; % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTST) } -- (TSno), }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTTC) -- (TSSC), }; { [edges=projection left] (TTST) -- (TStatus), (TSSC) -- [bend right] (TCity), }; { [edges=projection right] (TSSC) -- [bend left] (TStatus), }; }; \end{tikzpicture} \caption{SIG for schema \(\schema{2} = \{\ST, \TC\}\).} \label{fig-sig-s-ii-project-lossy} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-schema \(\bm{\schema{3} = \{\ST\}}\)} \noindent A user granted access only to \(\schema{3}\) can see only relvar \(\ST\). Only constraint \Constraint{\ref{constraint-project-lossy-key}}(b) can be propagated directly from \(\schema{0}\) into \(\schema{3}\) as it is the only one expressed in terms of \(\ST\) only. None of the remaining constraints can be rewritten in terms of \(\ST\) only, and so cannot be propagated into \(\schema{3}\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-iii-project-lossy} N/A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-iii-project-lossy} \noindent The only constraint appearing in \(\schema{3}\) is \Constraint{\ref{constraint-project-key}}(b). The SIG will only include edges and nodes that relate either directly to \(\ST\) or are independent of any relvar. The completed SIG for \(\schema{3}\) is shown in Figure~\ref{fig-sig-s-iii-project-lossy}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=1.5cm] { \node (TST) {\(\Type{\ST}\)}; &[7mm] \node (TTST) {\(\TT{\ST}\)}; &[10mm] &[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ & & \node (TSno) {\(\Type{\Sno}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TST) -- (TTST), }; % FDs (TSno) -- [funcdep left] (TStatus); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTST) -- (TSno), }; { [edges=projection left] (TTST) -- (TStatus), }; }; \end{tikzpicture} \caption{SIG for schema \(\schema{3} = \{\ST\}\).} \label{fig-sig-s-iii-project-lossy} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-schema \(\bm{\schema{4} = \{\TC\}}\)} \noindent A user granted access only to \(\schema{4}\) can see only relvar \(\TC\). Only constraint \Constraint{\ref{constraint-project-lossy-key-ii}} can be propagated directly from \(\schema{0}\) into \(\schema{4}\) as it the only one expressed in terms of \(\TC\) only. None of the remaining constraints can be rewritten in terms of \(\TC\) only, and so cannot be propagated into \(\schema{4}\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-iv-project-lossy} N/A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-iv-project-lossy} \noindent The only constraint appearing in \(\schema{4}\) is \Constraint[\schema{4}]{\ref{constraint-project-lossy-key-ii}}. The completed SIG for \(\schema{4}\) is shown in Figure~\ref{fig-sig-s-iv-project-lossy}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=1.5cm] { &[7mm] &[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TTC) {\(\Type{\TC}\)}; & \node (TTTC) {\(\TT{\TC}\)}; & \node (TSSC) {\(\Type{\Status} \times \Type{\City}\)}; \\ & & \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TTC) } -- { (TTTC) }, }; % FDs { [edges=funcdep right] (TSSC) -- [bend right] (TStatus), }; { [edges=funcdep left] (TSSC) -- [bend left] (TCity), }; % projection edges { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTTC) -- (TSSC), }; { [edges=projection left] (TSSC) -- [bend right] (TCity), }; { [edges=projection right] (TSSC) -- [bend left] (TStatus), }; }; \end{tikzpicture} \caption{SIG for schema \(\schema{4} = \{\TC\}\).} \label{fig-sig-s-iv-project-lossy} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Transformations} \label{sec-transforming-project-lossy} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{\(\bm{\schema{1}, \schema{2}}\)} \noindent The SIGs for \(\schema{1}\) and \(\schema{2}\) can't be made isomorphic. We can duplicate nodes \(\Type{\ST}\) and \(\TT{\ST}\), but there is no way to link these to the existing nodes \(\Type{\TC}\) and \(\TT{\TC}\) (and vice versa if we duplicate \(\Type{\TC}\) and \(\TT{\TC}\) instead). There are insufficient bijective edges across which to copy other edges. \(\schema{1}\) and \(\schema{2}\) are therefore incomparable. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{\(\bm{\schema{1}, \schema{3}}\)} Essentially identical to the nonloss projection case (compare Figure~\ref{fig-sig-s-iv-project-lossy} with Figure~\ref{fig-sig-s-iii-project}), so \(\schema{1}\) and \(\schema{3}\) are incomparable. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{\(\bm{\schema{1}, \schema{4}}\)} The SIG for \(\schema{4}\) is a subset of that for \(\schema{2}\), so \(\schema{1}\) and \(\schema{4}\) must also be incomparable. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{Lossy one to one join views (Date, Chapter 6)} \label{sec-one-to-one-join} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Relvars} \begin{itemize} \item base relvar \(\ST\{\Sno, \Status\}\) [supplier status] \item base relvar \(\SC\{\Sno, \City\}\) [supplier city] \item view \(S = \ST \RelJoin \SC\) \eqnnumber\label{eqn-s-join-one} \end{itemize} There are two possible interpretations consistent with these relvars: \begin{description} \item[nonloss join:] All suppliers must have a value for both \(\Status\) and \(\City\). This is essentially just the inverse of the nonloss projection view case (see Section~\ref{sec-nonloss-projection}), and thus has the same constraints and characteristics. We therefore don't need to consider this interpretation any further. \item[lossy join:] Suppliers must have a value for at least one of \(\Status\) or \(\City\). This implies that there may be tuples in \(\ST\) that don't have a corresponding tuple in \(\SC\) (and vice versa), and thus also don't appear in \(S\). In other words, \(S\) is the join of \(\ST\) and \(\SC\), but \(\ST\) and \(\SC\) aren't necessarily projections of \(S\). This interpretation also implies that \(\Sno\) in \(\ST\) and \(\SC\) is not a foreign key to \(S\). This is the interpretation that we will focus on. \end{description} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Tuple types} \begin{align} \TT{S} &= \Type{\Sno} \times \Type{\Status} \times \Type{\City} \nonumber \\ \TT{\ST} &= \Type{\Sno} \times \Type{\Status} \nonumber \\ \TT{\SC} &= \Type{\Sno} \times \Type{\City} \nonumber \end{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Base schema \(\bm{\schema{0} = \{S, \ST, \SC\}}\)} \begin{ConstraintList} \item\label{constraint-join-one-key} \(\Sno\) is a key for each of \(S\), \(\ST\), and \(\SC\), so the functional dependency \(\{\Sno\} \rightarrow \{\Status\}\) holds in each of (a) \(S\) and (b) \(\ST\); and the functional dependency \(\{\Sno\} \rightarrow \{\City\}\) holds in each of (c) \(S\) and (d) \(\SC\). [(a) and (c) could in principle be merged into a single functional dependency \(\{\Sno\} \rightarrow \{\Status, \City\}\) in \(S\).] \item\label{constraint-join-one-join} \(S = \SC \RelJoin \ST\) [from (\ref{eqn-s-join-one})] \item\label{constraint-join-one-relation-type-join} (a) \(\Type{S} = \Type{\ST} \RelJoin \Type{\SC}\); (b) \(\TT{S} = \TT{\ST} \RelJoin \TT{\SC}\) [from \Constraint{\ref{constraint-join-one-join}}] \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-schema \(\bm{\schema{1} = \{S\}}\)} \noindent A user granted access only to \(\schema{1}\) can see only relvar \(S\). Constraints \Constraint{\ref{constraint-project-key}}(a) and \Constraint{\ref{constraint-project-key}}(c) can be propagated directly from \(\schema{0}\) into \(\schema{1}\) as they are already expressed in terms of \(S\) only. Constraints \Constraint{\ref{constraint-join-one-join}} and \Constraint{\ref{constraint-join-one-relation-type-join}} cannot be propagated directly from \(\schema{0}\) into \(\schema{1}\), but can be rewritten in terms of \(S\) only, by substituting \(S\) for the expression \(\ST \RelJoin \SC\) (equation~\ref{eqn-s-join-one}). Constraints \Constraint{\ref{constraint-project-key}}(b) and \Constraint{\ref{constraint-project-key}}(d) cannot be propagated into \(\schema{1}\), as they cannot be rewritten in terms of \(S\) only. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-i-join-one} \begin{ConstraintList}[\schema{1}] \setcounter{constraint}{1} \item \(S = S\) \item (a) \(\Type{S} = \Type{S}\); (b) \(\TT{S} = \TT{S}\) \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-i-join-one} \noindent The constraints appearing in \(\schema{1}\) are \Constraint{\ref{constraint-join-one-key}}(a), \Constraint{\ref{constraint-join-one-key}}(c), \Constraint[\schema{1}]{\ref{constraint-join-one-join}}, and \Constraint[\schema{1}]{\ref{constraint-join-one-relation-type-join}}. \Constraint[\schema{1}]{\ref{constraint-join-one-join}} and \Constraint[\schema{1}]{\ref{constraint-join-one-relation-type-join}} are effectively tautologies that add no new information, and can thus be safely ignored. The completed SIG for \(\schema{1}\) is shown in Figure~\ref{fig-sig-s-i-join-one}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=1.5cm] { &[7mm] &[10mm] &[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ \node (TSTPlusSC) {\(\Type{S}\)}; & \node (TTSTPlusSC) {\(\TT{S}\)}; & \node (TSno) {\(\Type{\Sno}\)}; & \node (TSSC) {\(\Type{\Status} \times \Type{\City}\)}; \\ & & & \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TSTPlusSC) -- (TTSTPlusSC), }; % FDs (TSno) -- [funcdep left] (TStatus); (TSno) -- [funcdep right] (TCity); (TSno) -- [funcdep right] (TSSC); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTSTPlusSC) } -- (TSno), }; { [edges=projection left] (TSSC) -- (TCity), }; { [edges=projection right] (TSSC) -- (TStatus), }; % insert "gaps" into crossed edges { [edges={white,line width=1.5mm}] (TTSTPlusSC) -- [bend right] (TSSC), }; (TTSTPlusSC) -- [projection right, bend right] (TSSC), }; \end{tikzpicture} \caption{SIG for schema \(\schema{1} = \{S\}\).} \label{fig-sig-s-i-join-one} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-schema \(\bm{\schema{2} = \{\ST, \SC\}}\)} \noindent A user granted access only to \(\schema{2}\) can see only relvars \(\ST\) and \(\SC\). Constraints \Constraint{\ref{constraint-join-one-key}}(b) and \Constraint{\ref{constraint-join-one-key}}(d) can be propagated directly from \(\schema{0}\) into \(\schema{2}\) as they are already expressed in terms of \(\ST\) and \(\SC\) only. Constraints \Constraint{\ref{constraint-join-one-join}} and \Constraint{\ref{constraint-join-one-relation-type-join}} cannot be propagated directly from \(\schema{0}\) into \(\schema{2}\), but can be rewritten in terms of \(\ST\) and \(\SC\) only, by substituting \(\ST \RelJoin \SC\) for the expression \(S\) (equation~\ref{eqn-s-join-one}). Constraints \Constraint{\ref{constraint-project-key}}(a) and \Constraint{\ref{constraint-project-key}}(c) cannot be propagated into \(\schema{1}\), as they cannot be rewritten in terms of \(\ST\) and \(\SC\) only. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-ii-join-one} \begin{ConstraintList}[\schema{2}] \setcounter{constraint}{1} \item \(\ST \RelJoin \SC = \ST \RelJoin \SC\) \item (a) \(\Type{\ST} \RelJoin \Type{\SC} = \Type{\ST} \RelJoin \Type{\SC}\); (b) \(\TT{\ST} \RelJoin \TT{\SC} = \TT{\ST} \RelJoin \TT{\SC}\) \end{ConstraintList} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-ii-join-one} \noindent The constraints appearing in \(\schema{2}\) are \Constraint{\ref{constraint-join-one-key}}(b), \Constraint{\ref{constraint-join-one-key}}(d), \Constraint[\schema{1}]{\ref{constraint-join-one-join}}, and \Constraint[\schema{1}]{\ref{constraint-join-one-relation-type-join}}. \Constraint[\schema{1}]{\ref{constraint-join-one-join}} and \Constraint[\schema{1}]{\ref{constraint-join-one-relation-type-join}} are effectively tautologies that add no new information, and can thus be safely ignored. The completed SIG for \(\schema{2}\) is shown in Figure~\ref{fig-sig-s-ii-join-one}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=1.5cm] { \node (TST) {\(\Type{\ST}\)}; &[7mm] \node (TTST) {\(\TT{\ST}\)}; &[10mm] &[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ & & \node (TSno) {\(\Type{\Sno}\)}; & \\ \node (TSC) {\(\Type{\SC}\)}; & \node (TTSC) {\(\TT{\SC}\)}; & & \node (TCity) {\(\Type{\City}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] { (TST), (TSC) } -- { (TTST), (TTSC) }, }; % FDs (TSno) -- [funcdep left] (TStatus); (TSno) -- [funcdep right] (TCity); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] { (TTST) } -- (TSno), }; { [edges=bijection,edge label'={\scriptsize\(\RelProject\)}] (TTSC) -- (TSno), }; { [edges=projection left] (TTST) -- (TStatus), }; { [edges=projection right] (TTSC) -- (TCity), }; }; \end{tikzpicture} \caption{SIG for schema \(\schema{2} = \{\ST, \SC\}\).} \label{fig-sig-s-ii-join-one} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Sub-schema \(\bm{\schema{3} = \{\ST\}}\)} \noindent A user granted access only to \(\schema{3}\) can see only relvar \(\ST\). Only constraint \Constraint{\ref{constraint-join-one-key}}(b) can be propagated directly from \(\schema{0}\) into \(\schema{3}\) as it is the only one expressed in terms of \(\ST\) only. None of the remaining constraints can be rewritten in terms of \(\ST\) only, and so cannot be propagated into \(\schema{3}\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{Rewritten constraints} \label{sec-constraints-s-iii-join-one} N/A %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{SIG} \label{sec-sigs-s-iii-join-one} \noindent The only constraint appearing in \(\schema{3}\) is \Constraint{\ref{constraint-join-one-key}}(b). The SIG will only include edges and nodes that relate either directly to \(\ST\) or are independent of any relvar. The completed SIG for \(\schema{3}\) is shown in Figure~\ref{fig-sig-s-iii-join-one}. %%%%%%%%%%%%%%%%%%%% \begin{figure} \centering \begin{tikzpicture} \matrix[row sep=1.5cm] { \node (TST) {\(\Type{\ST}\)}; &[7mm] \node (TTST) {\(\TT{\ST}\)}; &[10mm] &[10mm] \node (TStatus) {\(\Type{\Status}\)}; \\ & & \node (TSno) {\(\Type{\Sno}\)}; \\ }; \graph{ % relation types to tuple types { [edges=surtotal] (TST) -- (TTST), }; % FDs (TSno) -- [funcdep left] (TStatus); % projection edges { [edges=bijection,edge label={\scriptsize\(\RelProject\)}] (TTST) -- (TSno), }; { [edges=projection left] (TTST) -- (TStatus), }; }; \end{tikzpicture} \caption{SIG for schema \(\schema{3} = \{\ST\}\).} \label{fig-sig-s-iii-join-one} \end{figure} %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Transformations} \label{sec-transforming-join-one} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{\(\bm{\schema{1}, \schema{2}}\)} \noindent The SIGs for \(\schema{1}\) and \(\schema{2}\) can't be made isomorphic, as there's no way using SIG transformations to create in \(\schema{3}\) a node equivalent to \(\Type{\Status} \times \Type{\City}\) in \(\schema{1}\). \(\schema{1}\) and \(\schema{2}\) are therefore incomparable. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsubsection{\(\bm{\schema{1}, \schema{3}}\)} This is essentially identical to the lossy projection views example (see Section~\ref{sec-sigs-s-iii-project-lossy}). \(\schema{1}\) and \(\schema{3}\) are therefore incomparable. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Additional notes} If we were to create a fourth sub-schema \(\schema{4} = \{\SC\}\) and compare this with \(\schema{1}\), the result would be similar to that for \(\schema{3}\). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}

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\documentclass{article}

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    \path (#2) to node[above,#1] {\scriptsize$\RelProject$} (#3);%
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    \path (#2) to node[above,#1] {\scriptsize$\RelRestrict$} (#3);%
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% SIG notation (in text)
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\begin{document}

% OUTLINE (20 mins max)
% Quick overview of view update problem and Date's effort
%   - problem: Date's approach appears little different from earlier metadata-heavy approaches
%   - problem: effectively punts on the hardest and most interesting cases
%   - problem: operational definition of information equivalence
%   - not focusing on view update problem, but rather how it can be characterised
% Quick overview of information capacity
%   - relative measure, not quantitative
%   - designed for relational schemas, so ideal
%   - basic concepts of SIG transformation and isomorphism
%   - my notation extensions
% Examples:
%   - quickly go through restriction view example in more detail
%   - other examples (summary): projection views, join views?
% Future work?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Disjoint restriction views (Date, Chapter 4)}
\label{sec-disjoint-restriction}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Relvars}
\begin{itemize}
    \item base relvar $S\{\Sno, \Sname, \Status, \City\}$
    \item view $\LS = \RelRestrict_{\City = \mathit{'London'}}S$ [London suppliers] \eqnnumber\label{eqn-ls}
    \item view $\NLS = \RelRestrict_{\City \ne \mathit{'London'}}S$ [non-London suppliers] \eqnnumber\label{eqn-nls}
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Tuple types}
\begin{align}
    \TT{S}    &= \Type{\Sno} \times \Type{\Sname} \times \Type{\Status} \times \Type{\City} \nonumber   \\
    \TT{\LS}  &= \Type{\Sno} \times \Type{\Sname} \times \Type{\Status} \times \CityLondon \nonumber    \\
    \TT{\NLS} &= \Type{\Sno} \times \Type{\Sname} \times \Type{\Status} \times (\TCityMinusLondon) \nonumber
\end{align}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Base schema $\bm{\schema{0} = \{S, \LS, \NLS\}}$}
\begin{ConstraintList}

\item\label{constraint-restrict-key} \Sno\ is a key for each of $S$, $\LS$, and $\NLS$, so the functional dependency $\{\Sno\} \rightarrow \{\Sname, \Status, \City\}$ also holds in each of (a) $S$; (b) $\LS$; and (c) $\NLS$.
    
    % not sure how relevant the FK constraints are?
    % how to represent? FK is a subset of the original key, so selection edge between S.Sno and each of LS.Sno and NLS.Sno
    % or perhaps a selection edge from Sno to itself? (labelled FK)
    % Notation: \Type{\Sno}, \Type{\LS.\Sno}, \Type{\NLS.\Sno}?
    \item\label{constraint-restrict-lsfk} $\RelProject_{\{\Sno\}}\LS \subseteq \RelProject_{\{\Sno\}}S$ [foreign key from $\LS$ to $S$]
    
    \item\label{constraint-restrict-nlsfk} $\RelProject_{\{\Sno\}}\NLS \subseteq \RelProject_{\{\Sno\}}S$ [foreign key from $\NLS$ to $S$]
    
    \item\label{constraint-restrict-union} $S = \LS \RelUnion \NLS$ [from (\ref{eqn-ls}) and (\ref{eqn-nls}) above]
    
    \item\label{constraint-restrict-notlondon} $\RelRestrict_{\City \ne \mathit{'London'}}\LS = \emptyset$ [from (\ref{eqn-ls}) above]
    
    \item\label{constraint-restrict-london} $\RelRestrict_{\City = \mathit{'London'}}\NLS = \emptyset$ [from (\ref{eqn-nls}) above]
    
    \item\label{constraint-restrict-disjoint} $\LS \RelIntersect \NLS = \emptyset$ [from \Constraint{\ref{constraint-restrict-notlondon}} and \Constraint{\ref{constraint-restrict-london}}]
    
    \item\label{constraint-restrict-relation-type-union} (a) $\Type{S} = \Type{\LS} \RelUnion \Type{\NLS}$; (b) $\TT{S} = \TT{\LS} \RelUnion \TT{\NLS}$ [from \Constraint{\ref{constraint-restrict-union}}]
    
    \item\label{constraint-restrict-relation-types} (a) $\Type{\LS} \subset \Type{S}$; (b) $\TT{\LS} \subset \TT{S}$ [from (\ref{eqn-ls}) above]
    
    \item\label{constraint-restrict-tuple-types} (a) $\Type{\NLS} \subset \Type{S}$; (b) $\TT{\NLS} \subset \TT{S}$ [from (\ref{eqn-nls}) above]
    
    \item\label{constraint-restrict-implied-types-london} $\CityLondon \subset \Type{\City}$
    
    \item\label{constraint-restrict-implied-types-nonlondon} $(\TCityMinusLondon) \subset \Type{\City}$
    
    \item\label{constraint-restrict-implied-types-TSSL} $\TSSL \subset \TSSC$
    
    \item\label{constraint-restrict-implied-types-TSSNL} $\TSSNL \subset \TSSC$
    
\end{ConstraintList}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-schema $\bm{\schema{1} = \{S\}}$}

\noindent A user granted access only to $\schema{1}$ can see only relvar $S$. Constraint \Constraint{\ref{constraint-restrict-key}}(a) can be propagated directly from $\schema{0}$ into $\schema{1}$ as it is already expressed in terms of $S$ only. Constraints \Constraint{\ref{constraint-restrict-implied-types-london}}--\Constraint{\ref{constraint-restrict-implied-types-TSSNL}} do not include any named relvars in $\schema{0}$, and thus can also be propagated directly from $\schema{0}$ into $\schema{1}$. Constraints \Constraint{\ref{constraint-restrict-key}}(b), \Constraint{\ref{constraint-restrict-key}}(c), and \Constraint{\ref{constraint-restrict-union}}--\Constraint{\ref{constraint-restrict-tuple-types}} cannot be propagated directly from $\schema{0}$ into $\schema{1}$, but can all be rewritten in terms of $S$ only, by substituting the definitions of $\LS$ (equation~\ref{eqn-ls}) and $\NLS$ (equation~\ref{eqn-nls}). To simplify expressions, we shall use $\LSsub$ to denote $\RelRestrict_{\City = \mathit{'London'}}S$ (\ref{eqn-ls}) and $\NLSsub$ to denote $\RelRestrict_{\City \ne \mathit{'London'}}S$ (\ref{eqn-nls}). (This also clearly distinguishes the substitutions from the $\LS$ and $\NLS$ relvars in $\schema{0}$, $\schema{2}$, and $\schema{3}$.)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-i-restrict}

\begin{ConstraintList}[\schema{1}]
    
    \item \Sno\ is a key for each of $\LSsub$ and $\NLSsub$, therefore the functional dependency $\{\Sno\} \rightarrow \{\Sname, \Status, \City\}$ also holds in each of (b) $\LSsub$; and (c) $\NLSsub$. [The fact that we are dealing with two disjoint subsets of $\City$ does not change this.]
    
    \setcounter{constraint}{3}
    \item $S = \LSsub \RelUnion \NLSsub$ [or even just $S = S$]
    
    \item $\RelRestrict_{\City \ne \mathit{'London'}}\LSsub = \emptyset$
    
    \item $\RelRestrict_{\City = \mathit{'London'}}\NLSsub = \emptyset$
    
    \item $\LSsub \RelIntersect \NLSsub = \emptyset$

\item (a) $\Type{S} = \Type{\LSsub} \RelUnion \Type{\NLSsub}$; (b) $\TT{S} = \TT{\LSsub} \RelUnion \TT{\NLSsub}$ \newline
    {[or even just $\Type{S} = \Type{S}$ and $\TT{S} = \TT{S}$]}
    
    \item (a) $\Type{\LSsub} \subset \Type{S}$; (b) $\TT{\LSsub} \subset \TT{S}$
    
    \item (a) $\Type{\NLSsub} \subset \Type{S}$; (b) $\TT{\NLSsub} \subset \TT{S}$
    
\end{ConstraintList}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-i-restrict}

\noindent The constraints appearing in $\schema{1}$ are \Constraint{\ref{constraint-restrict-key}}(a), \Constraint[\schema{1}]{\ref{constraint-restrict-key}}(b), \Constraint[\schema{1}]{\ref{constraint-restrict-key}}(c), \Constraint[\schema{1}]{\ref{constraint-restrict-union}}--\Constraint[\schema{1}]{\ref{constraint-restrict-tuple-types}}, and \Constraint{\ref{constraint-restrict-implied-types-london}}--\Constraint{\ref{constraint-restrict-implied-types-TSSNL}}. The constraints within the schema also imply the existence of several subtype nodes (e.g., the two subtypes of $\Type{City}$ implied by \Constraint{\ref{constraint-restrict-implied-types-london}} and \Constraint{\ref{constraint-restrict-implied-types-nonlondon}}), along with corresponding selection and projection edges. \Constraint[\schema{1}]{\ref{constraint-restrict-union}}, \Constraint[\schema{1}]{\ref{constraint-restrict-relation-type-union}}(a), and \Constraint[\schema{1}]{\ref{constraint-restrict-relation-type-union}}(b) are effectively tautologies that add no new information, and can thus be safely ignored. The completed SIG for $\schema{1}$ is shown in Figure~\ref{fig-sig-s-i-restrict}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\Type{\LSsub}$};   &[7mm]  \node (TTLS) {$\TT{\LSsub}$};     &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                            &                                           &                                   &   \node (TSname) {$\Type{\Sname}$};    \\
            \node (TLSPlusNLS) {$\Type{S}$}; &       \node (TTLSPlusNLS) {$\TT{S}$};   &   \node (TSno) {$\Type{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\Type{City}$};   \\
                                            &                                           &                                   &                                       &                                           &   \node (TStatus) {$\Type{\Status}$};  \\
            \node (TNLS) {$\Type{\NLSsub}$}; &       \node (TTNLS) {$\TT{\NLSsub}$};   &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) },
            };
            
            % FDs
            (TSno) -- [funcdep left,bend left] (TSSL);
            (TSno) -- [funcdep right] (TSSNL);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                { (TTLSPlusNLS), (TTLS) } -- (TSno)
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTNLS) -- (TSno)
            };
            
            { [edges=projection left]
                (TTLS)  -- (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TTNLS) -- (TSSNL),
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (NLSCity), (TStatus) },
                (TSSNL) -- [bend left=10] (TSname),
            };
            
            % selection edges
            { [edges=selection left]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) },
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) },
                (TCity) -- (LSCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
                (TTLSPlusNLS) -- [bend right] (TSSC),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
            (TTLSPlusNLS) -- [projection right,bend right] (TSSC),
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{1} = \{S\}$.}
    \label{fig-sig-s-i-restrict}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-schema $\bm{\schema{2} = \{\LS, \NLS\}}$}

\noindent A user granted access only to $\schema{2}$ can see only relvars $\LS$ and $\NLS$. Constraints \Constraint{\ref{constraint-restrict-key}}(b),  \Constraint{\ref{constraint-restrict-key}}(c), and \Constraint{\ref{constraint-restrict-notlondon}}--\Constraint{\ref{constraint-restrict-disjoint}} can be propagated directly from $\schema{0}$ into $\schema{2}$ as they are defined only in terms of $\LS$ and $\NLS$. \Constraint{\ref{constraint-restrict-implied-types-london}}--\Constraint{\ref{constraint-restrict-implied-types-TSSNL}} can be propagated directly into $\schema{2}$ as they are not expressed in terms of any named relvars in $\schema{0}$. \Constraint{\ref{constraint-restrict-key}}(a), \Constraint{\ref{constraint-restrict-union}}, \Constraint{\ref{constraint-restrict-relation-type-union}}, \Constraint{\ref{constraint-restrict-relation-types}}, and \Constraint{\ref{constraint-restrict-tuple-types}} can be rewritten in terms of $\LS$ and $\NLS$ only.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-ii-restrict}

\begin{ConstraintList}[\schema{2}]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-ii-restrict}

\noindent The constraints appearing in $\schema{2}$ are \Constraint[\schema{2}]{\ref{constraint-restrict-key}}(a), \Constraint{\ref{constraint-restrict-key}}(b), \Constraint{\ref{constraint-restrict-key}}(c), \Constraint[\schema{2}]{\ref{constraint-restrict-union}}, \Constraint{\ref{constraint-restrict-notlondon}}--\Constraint{\ref{constraint-restrict-disjoint}}, \Constraint[\schema{2}]{\ref{constraint-restrict-relation-type-union}}--\Constraint[\schema{2}]{\ref{constraint-restrict-tuple-types}}, and \Constraint{\ref{constraint-restrict-implied-types-london}}--\Constraint{\ref{constraint-restrict-implied-types-TSSNL}}. This is essentially the same set of constraints as $\schema{1}$, but with different rewritings. This means that the SIG for $\schema{2}$ can be built in the same manner as for $\schema{1}$, with some nodes named differently. In particular, \Constraint[\schema{2}]{\ref{constraint-restrict-relation-type-union}} implies that there will be nodes $\TLSPlusNLS$ and $\TTLSPlusNLS$ instead of $\Type{S}$ and $\TT{S}$, respectively. There will also be $\Type{\LS}$ instead of $\Type{\LSsub}$, etc. The complete SIG for $\schema{2}$ is shown in Figure~\ref{fig-sig-s-ii-restrict}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\Type{\LS}$};                  &[7mm]  \node (TTLS) {$\TT{\LS}$};                    &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                                        &                                                       &                                   &   \node (TSname) {$\Type{\Sname}$};    \\
            \node (TLSPlusNLS) {$\StackTLSPlusNLS$};  &       \node (TTLSPlusNLS) {$\StackTTLSPlusNLS$};    &   \node (TSno) {$\Type{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\Type{City}$};   \\
                                                        &                                                       &                                   &                                       &                                           &   \node (TStatus) {$\Type{\Status}$};  \\
            \node (TNLS) {$\Type{\NLS}$};                &       \node (TTNLS) {$\TT{\NLS}$};                  &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TLS), (TNLS), (TLSPlusNLS) } -- { (TTLS), (TTNLS), (TTLSPlusNLS) },
            };
            
            % FDs
            (TSno) -- [funcdep left,bend left] (TSSL);
            (TSno) -- [funcdep right] (TSSNL);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                { (TTLSPlusNLS), (TTLS) } -- (TSno)
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTNLS) -- (TSno)
            };
            
            { [edges=projection left]
                (TTLS)  -- (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TTNLS) -- (TSSNL),
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (NLSCity), (TStatus) },
                (TSSNL) -- [bend left=10] (TSname),
            };
            
            % selection edges
            { [edges=selection left]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TLS), (TTLS) },
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) },
                (TCity) -- (LSCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
                (TTLSPlusNLS) -- [bend right] (TSSC),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
            (TTLSPlusNLS) -- [projection right,bend right] (TSSC),
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{2} = \{\LS, \NLS\}$.}
    \label{fig-sig-s-ii-restrict}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-chema $\bm{\schema{3} = \{LS\}}$}

\noindent A user granted access only to $\schema{3}$ can see only relvar $\LS$. Constraints \Constraint{\ref{constraint-restrict-key}}(b) and \Constraint{\ref{constraint-restrict-notlondon}} can be propagated directly into $\schema{3}$, as they are defined in terms of $\LS$ only. \Constraint{\ref{constraint-restrict-implied-types-london}}--\Constraint{\ref{constraint-restrict-implied-types-TSSNL}} can also be propagated directly into $\schema{3}$ as previously discussed. The remaining constraints cannot be rewritten in terms of $\LS$ only, and so cannot be propagated into $\schema{3}$. As we were able to propagate all constraints into $\schema{1}$, this indicates a qualitative difference between $\schema{1}$ and $\schema{3}$. It therefore seems likely that the relative information capacities of $\schema{1}$ and $\schema{3}$ will also differ.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-iii-restrict}

\noindent N/A

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-iii-restrict}

\noindent The constraints appearing in $\schema{3}$ are \Constraint{\ref{constraint-restrict-key}}(b), \Constraint{\ref{constraint-restrict-notlondon}}, and \Constraint{\ref{constraint-restrict-implied-types-london}}--\Constraint{\ref{constraint-restrict-implied-types-TSSNL}}. The SIG will only include edges and nodes that relate either directly to $\LS$ or are independent of any relvar, such as the subtype node $\TSSNL$. The complete SIG for $\schema{3}$ is shown in Figure~\ref{fig-sig-s-iii-restrict} on the previous page.

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\Type{\LS}$};  &[7.5mm]  \node (TTLS) {$\TT{\LS}$};  &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                        &                                       &                                   &   \node (TSname) {$\Type{\Sname}$};    \\
                                        &                                       &   \node (TSno) {$\Type{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\Type{City}$};   \\
                                        &                                       &                                   &                                       &                                           &   \node (TStatus) {$\Type{\Status}$};  \\
                                        &                                       &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TLS) -- (TTLS),
            };
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                (TTLS) -- (TSno)
            };
            
            { [edges=projection left]
                (TTLS)  -- (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (NLSCity), (TStatus) },
                (TSSNL) -- [bend left=10] (TSname),
            };
            
            % selection edges
            { [edges=selection left]
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                (TCity) -- (LSCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{3} = \{\LS\}$.}
    \label{fig-sig-s-iii-restrict}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Transformations}
\label{sec-transforming-restrict}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{$\bm{\schema{1}, \schema{2}}$}

\noindent Looking at the SIGs for $\schema{1}$ and $\schema{2}$, we can see almost immediately that both SIGs have the same node and edge structure; indeed the only difference is in the names of some of the nodes. However, we already know from the discussion in Sections~\ref{sec-constraints-s-i-restrict} and~\ref{sec-constraints-s-ii-restrict} that these differently named nodes are just rewritings that represent the same domains. We therefore have an immediate isomorphism between the two SIGs and can conclude that the information capacities of $\schema{1}$ and $\schema{2}$ are equivalent, i.e., $\Equivalent{\schema{1}}{\schema{2}}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{$\bm{\schema{1}, \schema{3}}$}

There are some shared structures, but the two SIGs are clearly different. Since SIG transformations only ever preserve or enhance information capacity, intuitively we should attempt to transform the ``smaller'' SIG for $\schema{3}$ into the ``larger'' SIG for $\schema{1}$.

First, duplicate node $\Type{\LS}$ to $\Type{S}$ and node $\TT{\LS}$ to $\TT{S}$. To obtain a node structure similar to that of $\schema{1}$, we also need to further duplicate $\Type{S}$ to $\Type{\NLS}$ and $\TT{S}$ to $\TT{\NLS}$. The result of these duplications is shown in Figure~\ref{fig-transform-all-duplicates}. The transformed SIG now has the same node structure as that for $\schema{1}$.

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}[t]
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node [input keep] (TLS) {$\Type{\LS}$};     &[7mm]  \node [input keep] (TTLS) {$\TT{\LS}$};   &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                                        &                                                   &                                   &   \node (TSname) {$\Type{\Sname}$};    \\
            \node [output] (TLSPlusNLS) {$\Type{S}$};    &       \node [output] (TTLSPlusNLS) {$\TT{S}$};  &   \node (TSno) {$\Type{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\Type{City}$};   \\
                                                        &                                                   &                                   &                                       &                                           &   \node (TStatus) {$\Type{\Status}$};  \\
            \node [output] (TNLS) {$\Type{\NLS}$};       &       \node [output] (TTNLS) {$\TT{\NLS}$};     &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TLS) -- (TTLS),
            };
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                (TTLS) -- (TSno)
            };
            
            { [edges=projection left]
                (TTLS)  -- (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TSSC)  -- { (TSname),  (TStatus) },
                (TSSL)  -- (TSname),
                (TSSNL) -- { (NLSCity), (TStatus) },
                (TSSNL) -- [bend left=10] (TSname),
            };
            
            % selection edges
            { [edges=selection left]
                (TSSC)  -- { (TSSL), (TSSNL) },
                (TCity) -- (NLSCity),
            };
            { [edges=selection right]
                (TCity) -- (LSCity),
            };
            
            { [edges={bijection,output},edge label'={\scriptsize$\RelRestrict$}]
                { (TLS), (TTLS) } -- { (TLSPlusNLS), (TTLSPlusNLS) },
                { (TLSPlusNLS), (TTLSPlusNLS) } -- { (TNLS), (TTNLS) },
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TSno) -- (TSSC),
                (TSSC) -- (TCity),
            };
            (TSno) -- [funcdep right] (TSSC);
            (TSSC) -- [projection left] (TCity);
            
            % transformation indicators
            { [edges={->,gray,dashed,bend left=45}]
                (TLS) -- [edge node={node[sloped,above]{\small\emph{copy}}}] (TLSPlusNLS),
                (TLSPlusNLS) -- [edge node={node[sloped,above,rotate=180]{\small\emph{copy}}}] (TNLS)
            };
        };
    \end{tikzpicture}
    \caption{SIG for transformed schema $\schema{3}'$ after duplicating \Type{\LS} and \TT{\LS} to produce analogues of the additional nodes in $\schema{1}$. Inputs to the transformations are coloured \textcolor{blue}{blue}, outputs are coloured \textcolor{red}{red}. [$\equiv$]}
    \label{fig-transform-all-duplicates}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%

\begin{figure*}[t]
    \centering
    \begin{tikzpicture}
        \matrix[row sep=0.5cm]
        { 
            \node (TLS) {$\Type{\LS}$};      &[7mm]  \node (TTLS) {$\TT{\LS}$};        &[7mm]                              &[4mm]                                  &[-4mm] \node (TSSL) {$\StackTSSL$};      &                                       &   \node (LSCity) {$\CityLondon$};   \\
                                            &                                           &                                   &   \node (TSname) {$\Type{\Sname}$};    \\
            \node (TLSPlusNLS) {$\Type{S}$}; &       \node (TTLSPlusNLS) {$\TT{S}$};   &   \node (TSno) {$\Type{\Sno}$};    &                                       &       \node (TSSC) {$\StackTSSC$};      &                                       &   \node (TCity) {$\Type{City}$};   \\
                                            &                                           &                                   &                                       &                                           &   \node (TStatus) {$\Type{\Status}$};  \\
            \node (TNLS) {$\Type{\NLS}$};    &       \node (TTNLS) {$\TT{\NLS}$};      &                                   &                                       &       \node (TSSNL) {$\StackTSSNL$};    &                                       &   \node (NLSCity) {$\StackTCityMinusLondon$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TLS)        -- [input keep,edge node={node (c1a) {}}] (TTLS),
                (TLSPlusNLS) -- [output,edge node={node (c1b) {}}]     (TTLSPlusNLS),
                (TNLS)       -- [output,edge node={node (c1c) {}}]     (TTNLS),
            };
            
            % FDs
            (TSno) -- [funcdep left,arrows={-{crossbar}>},bend left,output] (TSSL);
            (TSno) -- [funcdep right,arrows={-{crossbar}>},output] (TSSNL);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                (TTLS)        -- [input keep] (TSno),
                (TTLSPlusNLS) -- [output] (TSno),
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTNLS)       -- [output] (TSno),
            };
            
            { [edges=projection left]
                (TTLS)  -- [input keep,edge node={node (c2a) {}}] (TSSL),
                (TSSL)  -- { (TStatus), (LSCity) },
            };
            
            { [edges=projection right]
                (TSSC)   -- { (TSname),  (TStatus) },
                (TSSL)   -- (TSname),
                (TSSNL) -- { (NLSCity), (TStatus) },
                (TSSNL) -- [bend left=10] (TSname),
                (TTNLS)  -- [arrows={:->},output,edge node={node (c2c) {}}] (TSSNL),
            };

%%%%%%%%%%%%%%%%%%%%

If we compare Figure~\ref{fig-sig-s-i-restrict} with Figure~\ref{fig-transform-edge-moves}, we can see that the transformed SIG for $\schema{3}'$ now has the same topology as the SIG for $\schema{1}$. We still need to remove the totality annotations from the newly created trivial selection edges between the relation and tuple types on the left, but even when this is done, we do not achieve the desired isomorphism, as the edges $\TT{S} \LabelledEdge{\sigedge{12}}{\RelProject} \TSSC$, $\TT{\NLS} \LabelledEdge{\sigedge{02}}{\RelProject} \TSSNL$, $\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSL$, and $\Type{\Sno} \LabelledEdge{\sigedge{03}}{\mathit{key}} \TSSNL$ all have different annotations from the corresponding edges in the SIG for $\schema{1}$. This clearly shows that the information capacities of $\schema{1}$ and $\schema{3}$ are incomparable, and that view updates based on this pair of schemas will be problematic.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Additional notes}

If we were to create a fourth sub-schema $\schema{4} = \{\NLS\}$ and compare this with $\schema{1}$, the result would be similar to that for $\schema{3}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

\section{Nonloss projection views (Date, Chapter 5)}
\label{sec-nonloss-projection}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Relvars}
\begin{itemize}
    \item base relvar $S\{\Sno, \Status, \City\}$
    \item view $\ST = \RelProject_{\{\Sno, \Status\}}S$ [supplier status] \eqnnumber\label{eqn-st}
    \item view $\SC = \RelProject_{\{\Sno, \City\}}S$ [supplier city] \eqnnumber\label{eqn-sc}
\end{itemize}
(Incidentally, the views form a 6NF nonloss decomposition of $S$.)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Tuple types}
\begin{align}
    \TT{S}   &= \Type{\Sno} \times \Type{\Status} \times \Type{\City} \nonumber   \\
    \TT{\ST} &= \Type{\Sno} \times \Type{\Status} \nonumber    \\
    \TT{\SC} &= \Type{\Sno} \times \Type{\City} \nonumber
\end{align}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Base schema $\bm{\schema{0} = \{S, \ST, \SC\}}$}
\begin{ConstraintList}

\item\label{constraint-project-key} $\Sno$ is a key for each of $S$, $\ST$, and $\SC$, so the functional dependency $\{\Sno\} \rightarrow \{\Status\}$ holds in each of (a) $S$ and (b) $\ST$; and the functional dependency $\{\Sno\} \rightarrow \{\City\}$ holds in each of (c) $S$ and (d) $\SC$. [(a) and (c) could in principle be merged into a single functional dependency $\{\Sno\} \rightarrow \{\Status, \City\}$ in $S$.]
    
    % not sure how relevant the FK constraints are?
    % how to represent? FK is a subset of the original key, so selection edge between S.Sno and each of LS.Sno and NLS.Sno
    % or perhaps a selection edge from Sno to itself? (labelled FK)
    % Notation: \Type{\Sno}, \Type{\LS.\Sno}, \Type{\NLS.\Sno}?
    \item\label{constraint-project-stfk} $\RelProject_{\{\Sno\}}\ST \subseteq \RelProject_{\{\Sno\}}S$ [foreign key from $\ST$ to $S$]
    
    \item\label{constraint-project-scfk} $\RelProject_{\{\Sno\}}\SC \subseteq \RelProject_{\{\Sno\}}S$ [foreign key from $\SC$ to $S$]
    
    \item\label{constraint-project-join} $S = \SC \RelJoin \ST$ [from (\ref{eqn-st}), (\ref{eqn-sc})]
    
    \item\label{constraint-project-s-st-identical} $\RelProject_{\{\Sno\}}S = \RelProject_{\{\Sno\}}\ST$ [from \Constraint{\ref{constraint-project-join}}, (\ref{eqn-st})]
    
    \item\label{constraint-project-s-sc-identical} $\RelProject_{\{\Sno\}}S = \RelProject_{\{\Sno\}}\SC$ [from \Constraint{\ref{constraint-project-join}}, (\ref{eqn-sc})]
    
    \item\label{constraint-project-st-sc-identical} $\RelProject_{\{\Sno\}}\ST = \RelProject_{\{\Sno\}}\SC$ [from \Constraint{\ref{constraint-project-join}}, (\ref{eqn-st}), (\ref{eqn-sc})]
    
    \item\label{constraint-project-relation-type-join} (a) $\Type{S} = \Type{\ST} \RelJoin \Type{\SC}$; (b) $\TT{S} = \TT{\ST} \RelJoin \TT{\SC}$ [from \Constraint{\ref{constraint-project-join}}]
    
    \item\label{constraint-project-types-st} (a) $\Type{\ST} = \RelProject_{\{\Sno, \Status\}}\Type{S}$; (b) $\TT{\ST} = \RelProject_{\{\Sno, \Status\}}\TT{S}$ [from (\ref{eqn-st})]
    
    \item\label{constraint-project-types-sc} (a) $\Type{\SC} = \RelProject_{\{\Sno, \City\}}\Type{S}$; (b) $\TT{\SC} = \RelProject_{\{\Sno, \City\}}\TT{S}$ [from (\ref{eqn-sc})]
    
\end{ConstraintList}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-schema $\bm{\schema{1} = \{S\}}$}

\noindent A user granted access only to $\schema{1}$ can see only relvar $S$. Constraints \Constraint{\ref{constraint-project-key}}(a) and \Constraint{\ref{constraint-project-key}}(c) can be propagated directly from $\schema{0}$ into $\schema{1}$ as they are already expressed in terms of $S$ only. Constraints \Constraint{\ref{constraint-project-join}}--\Constraint{\ref{constraint-restrict-tuple-types}} cannot be propagated directly from $\schema{0}$ into $\schema{1}$, but can all be rewritten in terms of $S$ only, by substituting the definitions of $\ST$ (equation~\ref{eqn-st}) and $\SC$ (equation~\ref{eqn-sc}), i.e., $\STsub = \RelProject_{\{\Sno, \Status\}}S$ and $\SCsub = \RelProject_{\{\Sno, \City\}}S$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-i-project}

\begin{ConstraintList}[\schema{1}]
    
    \item \Sno\ is a key for each of $\STsub$ and $\SCsub$, therefore the functional dependency $\{\Sno\} \rightarrow \{\Status\}$ holds in (b) $\STsub$, and the functional dependency $\{\Sno\} \rightarrow \{\City\}$ holds in (d) $\SCsub$.
    
    \setcounter{constraint}{3}
    \item $S = \STsub \RelJoin \SCsub$ [or even just $S = S$]
    
    \item $\RelProject_{\{\Sno\}}S = \RelProject_{\{\Sno\}}\STsub$
    
    \item $\RelProject_{\{\Sno\}}S = \RelProject_{\{\Sno\}}\SCsub$
    
    \item $\RelProject_{\{\Sno\}}\STsub = \RelProject_{\{\Sno\}}\SCsub$
    
    \item (a) $\Type{S} = \Type{\STsub} \RelJoin \Type{\SCsub}$; (b) $\TT{S} = \TT{\STsub} \RelJoin \TT{\SCsub}$ \newline
    {[or even just $\Type{S} = \Type{S}$ and $\TT{S} = \TT{S}$]}
    
    \item (a) $\Type{\STsub} = \RelProject_{\{\Sno, \Status\}}\Type{S}$; (b) $\TT{\STsub} = \RelProject_{\{\Sno, \Status\}}\TT{S}$
    
    \item (a) $\Type{\SCsub} = \RelProject_{\{\Sno, \City\}}\Type{S}$; (b) $\TT{\SCsub} = \RelProject_{\{\Sno, \City\}}\TT{S}$
    
\end{ConstraintList}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-i-project}

\noindent The constraints appearing in $\schema{1}$ are \Constraint{\ref{constraint-project-key}}(a), \Constraint[\schema{1}]{\ref{constraint-project-key}}(b), \Constraint{\ref{constraint-project-key}}(c), \Constraint[\schema{1}]{\ref{constraint-project-key}}(d), and \Constraint[\schema{1}]{\ref{constraint-project-join}}--\Constraint[\schema{1}]{\ref{constraint-project-types-sc}}. \Constraint[\schema{1}]{\ref{constraint-project-join}}, \Constraint[\schema{1}]{\ref{constraint-project-relation-type-join}}(a), and \Constraint[\schema{1}]{\ref{constraint-project-relation-type-join}}(b) are effectively tautologies that add no new information, and can thus be safely ignored. The completed SIG for $\schema{1}$ is shown in Figure~\ref{fig-sig-s-i-project}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=1.5cm]
        { 
            \node (TST) {$\Type{\STsub}$};    &[7mm]  \node (TTST) {$\TT{\STsub}$};     &[10mm]                             &[10mm] \node (TStatus) {$\Type{\Status}$};              \\
            \node (TSTPlusSC) {$\Type{S}$};   &       \node (TTSTPlusSC) {$\TT{S}$};    &   \node (TSno) {$\Type{\Sno}$}; & \node (TSSC) {$\Type{\Status} \times \Type{\City}$};  \\
            \node (TSC) {$\Type{\SCsub}$};    &       \node (TTSC) {$\TT{\SCsub}$};     &                                   & \node (TCity) {$\Type{\City}$};                       \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TST), (TSC), (TSTPlusSC) } -- { (TTST), (TTSC), (TTSTPlusSC) },
            };
            
            % FDs
            (TSno) -- [funcdep left] (TStatus);
            (TSno) -- [funcdep right] (TCity);
            (TSno) -- [funcdep right] (TSSC);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                { (TTSTPlusSC), (TTST) } -- (TSno),
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTSC) -- (TSno),
            };
            
            { [edges=projection left]
                { (TSTPlusSC), (TTSTPlusSC) } -- { (TST), (TTST) },
                (TSSC)  -- (TCity),
                (TTST)  -- (TStatus),
            };
            
            { [edges=projection right]
                { (TSTPlusSC), (TTSTPlusSC) } -- { (TSC), (TTSC) },
                (TSSC)       -- (TStatus),
                (TTSC) -- (TCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TTSTPlusSC) -- [bend right] (TSSC),
            };
            (TTSTPlusSC) -- [projection right, bend right] (TSSC),
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{1} = \{S\}$.}
    \label{fig-sig-s-i-project}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-schema $\bm{\schema{2} = \{\ST, \SC\}}$}

\noindent A user granted access only to $\schema{2}$ can see only relvars $\ST$ and $\SC$. Constraints \Constraint{\ref{constraint-project-key}}(b), \Constraint{\ref{constraint-project-key}}(d), and  \Constraint{\ref{constraint-project-st-sc-identical}} can be propagated directly from $\schema{0}$ into $\schema{2}$ as they are already expressed in terms of $\ST$ and $\SC$ only. The remaining constraints cannot be propagated directly from $\schema{0}$ into $\schema{2}$, but can all be rewritten in terms of $\ST$ and $\SC$ only, by substituting $J = \ST \RelJoin \SC$ for $S$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-ii-project}

\begin{ConstraintList}[\schema{2}]
    
    \item \Sno\ is a key for $J$, therefore (a) the functional dependency $\{\Sno\} \rightarrow \{\Status\}$ holds in $J$, and (c) the functional dependency $\{\Sno\} \rightarrow \{\City\}$ holds in $J$.
    
    \setcounter{constraint}{3}
    \item $J = \ST \RelJoin \SC$ [or even just $\ST \RelJoin \SC = \ST \RelJoin \SC$]
    
    \item $\RelProject_{\{\Sno\}}J = \RelProject_{\{\Sno\}}\ST$
    
    \item $\RelProject_{\{\Sno\}}J = \RelProject_{\{\Sno\}}\SC$
    
    \setcounter{constraint}{7}
    \item (a) $\Type{J} = \Type{\ST} \RelJoin \Type{\SC}$; (b) $\TT{J} = \TT{\ST} \RelJoin \TT{\SC}$
    
    \item (a) $\Type{\ST} = \RelProject_{\{\Sno, \Status\}}\Type{J}$; (b) $\TT{\ST} = \RelProject_{\{\Sno, \Status\}}\TT{J}$
    
    \item (a) $\Type{\SC} = \RelProject_{\{\Sno, \City\}}\Type{J}$; (b) $\TT{\SC} = \RelProject_{\{\Sno, \City\}}\TT{J}$
    
\end{ConstraintList}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-ii-project}

\noindent The constraints appearing in $\schema{2}$ are \Constraint[\schema{2}]{\ref{constraint-project-key}}(a), \Constraint{\ref{constraint-project-key}}(b), \Constraint[\schema{2}]{\ref{constraint-project-key}}(c), \Constraint{\ref{constraint-project-key}}(d), \Constraint[\schema{2}]{\ref{constraint-project-join}}--\Constraint[\schema{2}]{\ref{constraint-project-s-sc-identical}}, \Constraint{\ref{constraint-project-st-sc-identical}}, and \Constraint[\schema{2}]{\ref{constraint-project-relation-type-join}}--\Constraint[\schema{2}]{\ref{constraint-project-types-sc}}. \Constraint[\schema{2}]{\ref{constraint-project-join}} is effectively a tautology that adds no new information, and can be ignored. The completed SIG for $\schema{2}$ is shown in Figure~\ref{fig-sig-s-ii-project}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=1.5cm]
        { 
            \node (TST) {$\Type{\ST}$};       &[7mm]  \node (TTST) {$\TT{\ST}$};        &[10mm]                             &[10mm] \node (TStatus) {$\Type{\Status}$};              \\
            \node (TSTPlusSC) {$\Type{J}$};   &       \node (TTSTPlusSC) {$\TT{J}$};    &   \node (TSno) {$\Type{\Sno}$}; & \node (TSSC) {$\Type{\Status} \times \Type{\City}$};  \\
            \node (TSC) {$\Type{\SC}$};       &       \node (TTSC) {$\TT{\SC}$};        &                                   & \node (TCity) {$\Type{\City}$};                       \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TST), (TSC), (TSTPlusSC) } -- { (TTST), (TTSC), (TTSTPlusSC) },
            };
            
            % FDs
            (TSno) -- [funcdep left] (TStatus);
            (TSno) -- [funcdep right] (TCity);
            (TSno) -- [funcdep right] (TSSC);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                { (TTSTPlusSC), (TTST) } -- (TSno),
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTSC) -- (TSno),
            };
            
            { [edges=projection left]
                { (TSTPlusSC), (TTSTPlusSC) } -- { (TST), (TTST) },
                (TSSC)  -- (TCity),
                (TTST)  -- (TStatus),
            };
            
            { [edges=projection right]
                { (TSTPlusSC), (TTSTPlusSC) } -- { (TSC), (TTSC) },
                (TSSC)       -- (TStatus),
                (TTSC) -- (TCity),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TTSTPlusSC) -- [bend right] (TSSC),
            };
            (TTSTPlusSC) -- [projection right, bend right] (TSSC),
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{2} = \{\ST, \SC\}$.}
    \label{fig-sig-s-ii-project}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-schema $\bm{\schema{3} = \{\ST\}}$}

\noindent A user granted access only to $\schema{3}$ can see only relvar $\ST$. Only constraint \Constraint{\ref{constraint-project-key}}(b) can be propagated directly from $\schema{0}$ into $\schema{3}$ as it is the only one expressed in terms of $\ST$ only. None of the remaining constraints can be rewritten in terms of $\ST$ only, and so cannot be propagated into $\schema{3}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-iii-project}

N/A

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-iii-project}

\noindent The only constraint appearing in $\schema{3}$ is \Constraint{\ref{constraint-project-key}}(b). The SIG will only include edges and nodes that relate either directly to $\ST$ or are independent of any relvar. The completed SIG for $\schema{3}$ is shown in Figure~\ref{fig-sig-s-iii-project}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=1.5cm]
        { 
            \node (TST) {$\Type{\ST}$};   &[7mm]  \node (TTST) {$\TT{\ST}$};    &[10mm]                             &[10mm] \node (TStatus) {$\Type{\Status}$};    \\
                                            &                                       &   \node (TSno) {$\Type{\Sno}$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TST) -- (TTST),
            };
            
            % FDs
            (TSno) -- [funcdep left] (TStatus);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                (TTST) -- (TSno),
            };
            
            { [edges=projection left]
                (TTST)  -- (TStatus),
            };
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{3} = \{\ST\}$.}
    \label{fig-sig-s-iii-project}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Transformations}
\label{sec-transforming-project}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{$\bm{\schema{1}, \schema{2}}$}

\noindent The SIGs for $\schema{1}$ and $\schema{2}$ are immediately isomorphic, so $\Equivalent{\schema{1}}{\schema{2}}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{$\bm{\schema{1}, \schema{3}}$}

Despite their differences, it's clear that the comparison between $\schema{1}$ and $\schema{3}$ for projection views will be similar to that for disjoint restriction views (Section~\ref{sec-transforming-restrict}), as the SIG structures are topologically similar. It's therefore reasonable to conclude that the information capacities of $\schema{1}$ and $\schema{3}$ are incomparable, and that view updates based on this pair of schemas will be problematic.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Additional notes}

If we were to create a fourth sub-schema $\schema{4} = \{\SC\}$ and compare this with $\schema{1}$, the result would be similar to that for $\schema{3}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

\section{Lossy projection views (Date, Chapter 5)}
\label{sec-lossy-projection}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Relvars}
\begin{itemize}
    \item base relvar $S\{\Sno, \Status, \City\}$
    \item view $\ST = \RelProject_{\{\Sno, \Status\}}S$ [supplier status] \eqnnumber\label{eqn-st-lossy}
    \item view $\TC = \RelProject_{\{\Status, \City\}}S$ [status \& city] \eqnnumber\label{eqn-tc-lossy}
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Tuple types}
\begin{align}
    \TT{S}   &= \Type{\Sno} \times \Type{\Status} \times \Type{\City} \nonumber   \\
    \TT{\ST} &= \Type{\Sno} \times \Type{\Status} \nonumber    \\
    \TT{\TC} &= \Type{\Status} \times \Type{\City} \nonumber
\end{align}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Base schema $\bm{\schema{0} = \{S, \ST, \TC\}}$}
\begin{ConstraintList}

\item\label{constraint-project-lossy-key} $\Sno$ is a key for each of $S$ and $\ST$, so the functional dependency $\{\Sno\} \rightarrow \{\Status\}$ holds in each of (a) $S$ and (b) $\ST$; and the functional dependency $\{\Sno\} \rightarrow \{\City\}$ holds in (c) $S$. [(a) and (c) could in principle be merged into a single functional dependency $\{\Sno\} \rightarrow \{\Status, \City\}$ in $S$.]
    
    \item\label{constraint-project-lossy-key-ii} $\{\Status, \City\}$ is a key for $\TC$, so the functional dependencies (a) $\{\Status, \City\} \rightarrow \{\Status\}$ and (b) $\{\Status, \City\} \rightarrow \{\City\}$ hold in $\TC$.
    
    % not sure how relevant the FK constraints are?
    % how to represent? FK is a subset of the original key, so selection edge between S.Sno and each of LS.Sno and NLS.Sno
    % or perhaps a selection edge from Sno to itself? (labelled FK)
    % Notation: \Type{\Sno}, \Type{\LS.\Sno}, \Type{\NLS.\Sno}?
    \item\label{constraint-project-lossy-stfk} $\RelProject_{\{\Sno\}}\ST \subseteq \RelProject_{\{\Sno\}}S$ [foreign key from $\ST$ to $S$]
    
    \item\label{constraint-project-lossy-s-st-identical} $\RelProject_{\{\Sno\}}S = \RelProject_{\{\Sno\}}\ST$ [from (\ref{eqn-st-lossy})]
    
    \item\label{constraint-project-lossy-st-tc-status} $\RelProject_{\{\Status\}}\ST = \RelProject_{\{\Status\}}\TC$ [from (\ref{eqn-tc-lossy})]
    
    \item\label{constraint-project-lossy-types-st} (a) $\Type{\ST} = \RelProject_{\{\Sno, \Status\}}\Type{S}$; (b) $\TT{\ST} = \RelProject_{\{\Sno, \Status\}}\TT{S}$ [from (\ref{eqn-st-lossy})]
    
    \item\label{constraint-project-lossy-types-tc} (a) $\Type{\TC} = \RelProject_{\{\Status, \City\}}\Type{S}$; (b) $\TT{\TC} = \RelProject_{\{\Status, \City\}}\TT{S}$ [from (\ref{eqn-tc-lossy})]
    
\end{ConstraintList}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-schema $\bm{\schema{1} = \{S\}}$}

\noindent A user granted access only to $\schema{1}$ can see only relvar $S$. Constraints \Constraint{\ref{constraint-project-lossy-key}}(a) and \Constraint{\ref{constraint-project-lossy-key}}(c) can be propagated directly from $\schema{0}$ into $\schema{1}$ as they are already expressed in terms of $S$ only. Constraints \Constraint{\ref{constraint-project-lossy-key-ii}} and \Constraint{\ref{constraint-project-lossy-s-st-identical}}--\Constraint{\ref{constraint-project-lossy-types-tc}} cannot be propagated directly from $\schema{0}$ into $\schema{1}$, but can all be rewritten in terms of $S$ only, by substituting the definitions of $\ST$ (equation~\ref{eqn-st-lossy}) and $\TC$ (equation~\ref{eqn-tc-lossy}), i.e., $\STsub = \RelProject_{\{\Sno, \Status\}}S$ and $\TCsub = \RelProject_{\{\Status, \City\}}S$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-i-project-lossy}

\begin{ConstraintList}[\schema{1}]
    
    \item $\Sno$ is a key for $\STsub$, therefore the functional dependency $\{\Sno\} \rightarrow \{\Status\}$ holds in (b) $\STsub$.
    
    \item $\{\Status, \City\}$ is a key for $\TCsub$, so the functional dependencies (a) $\{\Status, \City\} \rightarrow \{\Status\}$ and (b) $\{\Status, \City\} \rightarrow \{\City\}$ hold in $\TCsub$. [Ignoring the trivial case of $\{\Status, \City\} \rightarrow \{\Status, \City\}$.]
    
    \setcounter{constraint}{3}
    \item $\RelProject_{\{\Sno\}}S = \RelProject_{\{\Sno\}}\STsub$
    
    \item $\RelProject_{\{\Sno\}}\STsub = \RelProject_{\{\Sno\}}\TCsub$
    
    \item (a) $\Type{\STsub} = \RelProject_{\{\Sno, \Status\}}\Type{S}$; (b) $\TT{\STsub} = \RelProject_{\{\Sno, \Status\}}\TT{S}$
    
    \item (a) $\Type{\TCsub} = \RelProject_{\{\Status, \City\}}\Type{S}$; (b) $\TT{\TCsub} = \RelProject_{\{\Status, \City\}}\TT{S}$
    
\end{ConstraintList}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-i-project-lossy}

\noindent The constraints appearing in $\schema{1}$ are \Constraint{\ref{constraint-project-lossy-key}}(a), \Constraint[\schema{1}]{\ref{constraint-project-lossy-key}}(b), \Constraint{\ref{constraint-project-lossy-key}}(c), \Constraint[\schema{1}]{\ref{constraint-project-lossy-key-ii}}, and \Constraint[\schema{1}]{\ref{constraint-project-lossy-s-st-identical}}--\Constraint[\schema{1}]{\ref{constraint-project-lossy-types-tc}}. The completed SIG for $\schema{1}$ is shown in Figure~\ref{fig-sig-s-i-project-lossy}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=1.5cm]
        { 
            \node (TST) {$\Type{\STsub}$};    &[7mm]  \node (TTST) {$\TT{\STsub}$};     &[10mm]                             &[10mm] \node (TStatus) {$\Type{\Status}$};              \\
            \node (TSTPlusTC) {$\Type{S}$};   &       \node (TTSTPlusTC) {$\TT{S}$};    &   \node (TSno) {$\Type{\Sno}$}; & \node (TSSC) {$\Type{\Status} \times \Type{\City}$};  \\
            \node (TTC) {$\Type{\TCsub}$};    &       \node (TTTC) {$\TT{\TCsub}$};     &                                   & \node (TCity) {$\Type{\City}$};                       \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TST), (TTC), (TSTPlusTC) } -- { (TTST), (TTTC), (TTSTPlusTC) },
            };
            
            % FDs
            { [edges=funcdep right]
                (TSno) -- { (TCity),  (TSSC) },
                (TSSC) -- [bend right] (TStatus),
            };
            { [edges=funcdep left]
                (TSno) -- (TStatus),
                (TSSC) -- [bend left] (TCity),
            };
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                { (TTSTPlusTC), (TTST) } -- (TSno),
            };
            
            { [edges=projection left]
                { (TSTPlusTC), (TTSTPlusTC) } -- { (TST), (TTST) },
                (TTST)  -- (TStatus),
                (TSSC) -- [bend right] (TCity),
            };
            
            { [edges=projection right]
                { (TSTPlusTC), (TTSTPlusTC) } -- { (TTC), (TTTC) },
                (TSSC) -- [bend left] (TStatus),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TTSTPlusTC) -- [bend right] (TSSC),
                (TTTC) -- [bend right=10] (TSSC),
            };
            (TTSTPlusTC) -- [projection right, bend right] (TSSC),
            (TTTC) -- [edges=bijection,edge label'={\scriptsize$\RelProject$}, bend right=10] (TSSC),
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{1} = \{S\}$.}
    \label{fig-sig-s-i-project-lossy}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-schema $\bm{\schema{2} = \{\ST, \TC\}}$}

\noindent A user granted access only to $\schema{2}$ can see only relvars $\ST$ and $\TC$. Constraints \Constraint{\ref{constraint-project-lossy-key}}(b), \Constraint{\ref{constraint-project-lossy-key-ii}}, and  \Constraint{\ref{constraint-project-lossy-st-tc-status}} can be propagated directly from $\schema{0}$ into $\schema{2}$ as they are already expressed in terms of $\ST$ and $\TC$ only. The remaining constraints cannot be propagated from $\schema{0}$ into $\schema{2}$ at all, as none of them can be rewritten in terms of $\ST$ and $\TC$ only (due to the lossy decomposition of $S$ into $\ST$ and $\TC$).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-ii-project-lossy}

N/A

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-ii-project-lossy}

\noindent The constraints appearing in $\schema{2}$ are \Constraint[\schema{2}]{\ref{constraint-project-lossy-key}}(b), \Constraint[\schema{2}]{\ref{constraint-project-lossy-key-ii}}, and \Constraint[\schema{2}]{\ref{constraint-project-lossy-st-tc-status}}. The completed SIG for $\schema{2}$ is shown in Figure~\ref{fig-sig-s-ii-project-lossy}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=1.5cm]
        { 
            \node (TST) {$\Type{\ST}$};    &[7mm]  \node (TTST) {$\TT{\ST}$};     &[10mm]                             &[10mm] \node (TStatus) {$\Type{\Status}$};              \\
                                             &                                        &   \node (TSno) {$\Type{\Sno}$}; & \node (TSSC) {$\Type{\Status} \times \Type{\City}$};  \\
            \node (TTC) {$\Type{\TC}$};    &       \node (TTTC) {$\TT{\TC}$};     &                                   & \node (TCity) {$\Type{\City}$};                       \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TST), (TTC) } -- { (TTST), (TTTC) },
            };
            
            % FDs
            { [edges=funcdep right]
                (TSSC) -- [bend right] (TStatus),
            };
            { [edges=funcdep left]
                (TSno) -- (TStatus),
                (TSSC) -- [bend left] (TCity),
            };
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                { (TTST) } -- (TSno),
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTTC) -- (TSSC),
            };
            
            { [edges=projection left]
                (TTST)  -- (TStatus),
                (TSSC) -- [bend right] (TCity),
            };
            
            { [edges=projection right]
                (TSSC) -- [bend left] (TStatus),
            };
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{2} = \{\ST, \TC\}$.}
    \label{fig-sig-s-ii-project-lossy}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-schema $\bm{\schema{3} = \{\ST\}}$}

\noindent A user granted access only to $\schema{3}$ can see only relvar $\ST$. Only constraint \Constraint{\ref{constraint-project-lossy-key}}(b) can be propagated directly from $\schema{0}$ into $\schema{3}$ as it is the only one expressed in terms of $\ST$ only. None of the remaining constraints can be rewritten in terms of $\ST$ only, and so cannot be propagated into $\schema{3}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-iii-project-lossy}

N/A

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-iii-project-lossy}

%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=1.5cm]
        { 
            \node (TST) {$\Type{\ST}$};   &[7mm]  \node (TTST) {$\TT{\ST}$};    &[10mm]                             &[10mm] \node (TStatus) {$\Type{\Status}$};    \\
                                            &                                       &   \node (TSno) {$\Type{\Sno}$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TST) -- (TTST),
            };
            
            % FDs
            (TSno) -- [funcdep left] (TStatus);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                (TTST) -- (TSno),
            };
            
            { [edges=projection left]
                (TTST)  -- (TStatus),
            };
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{3} = \{\ST\}$.}
    \label{fig-sig-s-iii-project-lossy}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-schema $\bm{\schema{4} = \{\TC\}}$}

\noindent A user granted access only to $\schema{4}$ can see only relvar $\TC$. Only constraint \Constraint{\ref{constraint-project-lossy-key-ii}} can be propagated directly from $\schema{0}$ into $\schema{4}$ as it the only one expressed in terms of $\TC$ only. None of the remaining constraints can be rewritten in terms of $\TC$ only, and so cannot be propagated into $\schema{4}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-iv-project-lossy}

N/A

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-iv-project-lossy}

\noindent The only constraint appearing in $\schema{4}$ is \Constraint[\schema{4}]{\ref{constraint-project-lossy-key-ii}}. The completed SIG for $\schema{4}$ is shown in Figure~\ref{fig-sig-s-iv-project-lossy}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=1.5cm]
        { 
                                                &[7mm]                                      &[10mm] \node (TStatus) {$\Type{\Status}$};              \\
            \node (TTC) {$\Type{\TC}$};       &       \node (TTTC) {$\TT{\TC}$};        & \node (TSSC) {$\Type{\Status} \times \Type{\City}$};  \\
                                                &                                           & \node (TCity) {$\Type{\City}$};                       \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TTC) } -- { (TTTC) },
            };
            
            % FDs
            { [edges=funcdep right]
                (TSSC) -- [bend right] (TStatus),
            };
            { [edges=funcdep left]
                (TSSC) -- [bend left] (TCity),
            };
            
            % projection edges
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTTC) -- (TSSC),
            };
            
            { [edges=projection left]
                (TSSC) -- [bend right] (TCity),
            };
            
            { [edges=projection right]
                (TSSC) -- [bend left] (TStatus),
            };
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{4} = \{\TC\}$.}
    \label{fig-sig-s-iv-project-lossy}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Transformations}
\label{sec-transforming-project-lossy}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{$\bm{\schema{1}, \schema{2}}$}

\noindent The SIGs for $\schema{1}$ and $\schema{2}$ can't be made isomorphic. We can duplicate nodes $\Type{\ST}$ and $\TT{\ST}$, but there is no way to link these to the existing nodes $\Type{\TC}$ and $\TT{\TC}$ (and vice versa if we duplicate $\Type{\TC}$ and $\TT{\TC}$ instead). There are insufficient bijective edges across which to copy other edges. $\schema{1}$ and $\schema{2}$ are therefore incomparable.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{$\bm{\schema{1}, \schema{3}}$}

Essentially identical to the nonloss projection case (compare Figure~\ref{fig-sig-s-iv-project-lossy} with Figure~\ref{fig-sig-s-iii-project}), so $\schema{1}$ and $\schema{3}$ are incomparable.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{$\bm{\schema{1}, \schema{4}}$}

The SIG for $\schema{4}$ is a subset of that for $\schema{2}$, so $\schema{1}$ and $\schema{4}$ must also be incomparable.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

\section{Lossy one to one join views (Date, Chapter 6)}
\label{sec-one-to-one-join}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Relvars}
\begin{itemize}
    \item base relvar $\ST\{\Sno, \Status\}$ [supplier status]
    \item base relvar $\SC\{\Sno, \City\}$ [supplier city]
    \item view $S = \ST \RelJoin \SC$ \eqnnumber\label{eqn-s-join-one}
\end{itemize}
There are two possible interpretations consistent with these relvars:
\begin{description}
    \item[nonloss join:] All suppliers must have a value for both $\Status$ and $\City$. This is essentially just the inverse of the nonloss projection view case (see Section~\ref{sec-nonloss-projection}), and thus has the same constraints and characteristics. We therefore don't need to consider this interpretation any further.
    
    \item[lossy join:] Suppliers must have a value for at least one of $\Status$ or $\City$. This implies that there may be tuples in $\ST$ that don't have a corresponding tuple in $\SC$ (and vice versa), and thus also don't appear in $S$. In other words, $S$ is the join of $\ST$ and $\SC$, but $\ST$ and $\SC$ aren't necessarily projections of $S$. This interpretation also implies that $\Sno$ in $\ST$ and $\SC$ is not a foreign key to $S$. This is the interpretation that we will focus on.
\end{description}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Base schema $\bm{\schema{0} = \{S, \ST, \SC\}}$}
\begin{ConstraintList}

\item\label{constraint-join-one-key} $\Sno$ is a key for each of $S$, $\ST$, and $\SC$, so the functional dependency $\{\Sno\} \rightarrow \{\Status\}$ holds in each of (a) $S$ and (b) $\ST$; and the functional dependency $\{\Sno\} \rightarrow \{\City\}$ holds in each of (c) $S$ and (d) $\SC$. [(a) and (c) could in principle be merged into a single functional dependency $\{\Sno\} \rightarrow \{\Status, \City\}$ in $S$.]
    
    \item\label{constraint-join-one-join} $S = \SC \RelJoin \ST$ [from (\ref{eqn-s-join-one})]
    
    \item\label{constraint-join-one-relation-type-join} (a) $\Type{S} = \Type{\ST} \RelJoin \Type{\SC}$; (b) $\TT{S} = \TT{\ST} \RelJoin \TT{\SC}$ [from \Constraint{\ref{constraint-join-one-join}}]
    
\end{ConstraintList}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-schema $\bm{\schema{1} = \{S\}}$}

\noindent A user granted access only to $\schema{1}$ can see only relvar $S$. Constraints \Constraint{\ref{constraint-project-key}}(a) and \Constraint{\ref{constraint-project-key}}(c) can be propagated directly from $\schema{0}$ into $\schema{1}$ as they are already expressed in terms of $S$ only. Constraints \Constraint{\ref{constraint-join-one-join}} and \Constraint{\ref{constraint-join-one-relation-type-join}} cannot be propagated directly from $\schema{0}$ into $\schema{1}$, but can be rewritten in terms of $S$ only, by substituting $S$ for the expression $\ST \RelJoin \SC$ (equation~\ref{eqn-s-join-one}). Constraints \Constraint{\ref{constraint-project-key}}(b) and \Constraint{\ref{constraint-project-key}}(d) cannot be propagated into $\schema{1}$, as they cannot be rewritten in terms of $S$ only.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-i-join-one}

\begin{ConstraintList}[\schema{1}]
    
    \setcounter{constraint}{1}
    \item $S = S$
    
    \item (a) $\Type{S} = \Type{S}$; (b) $\TT{S} = \TT{S}$
    
\end{ConstraintList}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-i-join-one}

\noindent The constraints appearing in $\schema{1}$ are \Constraint{\ref{constraint-join-one-key}}(a), \Constraint{\ref{constraint-join-one-key}}(c), \Constraint[\schema{1}]{\ref{constraint-join-one-join}}, and \Constraint[\schema{1}]{\ref{constraint-join-one-relation-type-join}}. \Constraint[\schema{1}]{\ref{constraint-join-one-join}} and \Constraint[\schema{1}]{\ref{constraint-join-one-relation-type-join}} are effectively tautologies that add no new information, and can thus be safely ignored. The completed SIG for $\schema{1}$ is shown in Figure~\ref{fig-sig-s-i-join-one}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=1.5cm]
        { 
                                                &[7mm]                                      &[10mm]                             &[10mm] \node (TStatus) {$\Type{\Status}$};              \\
            \node (TSTPlusSC) {$\Type{S}$};   &       \node (TTSTPlusSC) {$\TT{S}$};    &   \node (TSno) {$\Type{\Sno}$}; & \node (TSSC) {$\Type{\Status} \times \Type{\City}$};  \\
                                                &                                           &                                   & \node (TCity) {$\Type{\City}$};                       \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TSTPlusSC) -- (TTSTPlusSC),
            };
            
            % FDs
            (TSno) -- [funcdep left] (TStatus);
            (TSno) -- [funcdep right] (TCity);
            (TSno) -- [funcdep right] (TSSC);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                { (TTSTPlusSC) } -- (TSno),
            };
            
            { [edges=projection left]
                (TSSC) -- (TCity),
            };
            
            { [edges=projection right]
                (TSSC) -- (TStatus),
            };
            
            % insert "gaps" into crossed edges
            { [edges={white,line width=1.5mm}]
                (TTSTPlusSC) -- [bend right] (TSSC),
            };
            (TTSTPlusSC) -- [projection right, bend right] (TSSC),
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{1} = \{S\}$.}
    \label{fig-sig-s-i-join-one}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-schema $\bm{\schema{2} = \{\ST, \SC\}}$}

\noindent A user granted access only to $\schema{2}$ can see only relvars $\ST$ and $\SC$. Constraints \Constraint{\ref{constraint-join-one-key}}(b) and \Constraint{\ref{constraint-join-one-key}}(d) can be propagated directly from $\schema{0}$ into $\schema{2}$ as they are already expressed in terms of $\ST$ and $\SC$ only. Constraints \Constraint{\ref{constraint-join-one-join}} and \Constraint{\ref{constraint-join-one-relation-type-join}} cannot be propagated directly from $\schema{0}$ into $\schema{2}$, but can be rewritten in terms of $\ST$ and $\SC$ only, by substituting $\ST \RelJoin \SC$ for the expression $S$ (equation~\ref{eqn-s-join-one}). Constraints \Constraint{\ref{constraint-project-key}}(a) and \Constraint{\ref{constraint-project-key}}(c) cannot be propagated into $\schema{1}$, as they cannot be rewritten in terms of $\ST$ and $\SC$ only.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-ii-join-one}

\begin{ConstraintList}[\schema{2}]
    
    \setcounter{constraint}{1}
    \item $\ST \RelJoin \SC = \ST \RelJoin \SC$
    
    \item (a) $\Type{\ST} \RelJoin \Type{\SC} = \Type{\ST} \RelJoin \Type{\SC}$; (b) $\TT{\ST} \RelJoin \TT{\SC} = \TT{\ST} \RelJoin \TT{\SC}$
    
\end{ConstraintList}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-ii-join-one}

\noindent The constraints appearing in $\schema{2}$ are \Constraint{\ref{constraint-join-one-key}}(b), \Constraint{\ref{constraint-join-one-key}}(d), \Constraint[\schema{1}]{\ref{constraint-join-one-join}}, and \Constraint[\schema{1}]{\ref{constraint-join-one-relation-type-join}}. \Constraint[\schema{1}]{\ref{constraint-join-one-join}} and \Constraint[\schema{1}]{\ref{constraint-join-one-relation-type-join}} are effectively tautologies that add no new information, and can thus be safely ignored. The completed SIG for $\schema{2}$ is shown in Figure~\ref{fig-sig-s-ii-join-one}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=1.5cm]
        { 
            \node (TST) {$\Type{\ST}$};       &[7mm]  \node (TTST) {$\TT{\ST}$};        &[10mm]                             &[10mm] \node (TStatus) {$\Type{\Status}$};              \\
                                                &                                           &   \node (TSno) {$\Type{\Sno}$}; &   \\
            \node (TSC) {$\Type{\SC}$};       &       \node (TTSC) {$\TT{\SC}$};        &                                   & \node (TCity) {$\Type{\City}$};                       \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                { (TST), (TSC) } -- { (TTST), (TTSC) },
            };
            
            % FDs
            (TSno) -- [funcdep left] (TStatus);
            (TSno) -- [funcdep right] (TCity);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                { (TTST) } -- (TSno),
            };
            
            { [edges=bijection,edge label'={\scriptsize$\RelProject$}]
                (TTSC) -- (TSno),
            };
            
            { [edges=projection left]
                (TTST)  -- (TStatus),
            };
            
            { [edges=projection right]
                (TTSC) -- (TCity),
            };
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{2} = \{\ST, \SC\}$.}
    \label{fig-sig-s-ii-join-one}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Sub-schema $\bm{\schema{3} = \{\ST\}}$}

\noindent A user granted access only to $\schema{3}$ can see only relvar $\ST$. Only constraint \Constraint{\ref{constraint-join-one-key}}(b) can be propagated directly from $\schema{0}$ into $\schema{3}$ as it is the only one expressed in terms of $\ST$ only. None of the remaining constraints can be rewritten in terms of $\ST$ only, and so cannot be propagated into $\schema{3}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Rewritten constraints}
\label{sec-constraints-s-iii-join-one}

N/A

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{SIG}
\label{sec-sigs-s-iii-join-one}

\noindent The only constraint appearing in $\schema{3}$ is \Constraint{\ref{constraint-join-one-key}}(b). The SIG will only include edges and nodes that relate either directly to $\ST$ or are independent of any relvar. The completed SIG for $\schema{3}$ is shown in Figure~\ref{fig-sig-s-iii-join-one}.

%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \begin{tikzpicture}
        \matrix[row sep=1.5cm]
        { 
            \node (TST) {$\Type{\ST}$};   &[7mm]  \node (TTST) {$\TT{\ST}$};    &[10mm]                             &[10mm] \node (TStatus) {$\Type{\Status}$};    \\
                                            &                                       &   \node (TSno) {$\Type{\Sno}$}; \\
        };
        
        \graph{
            % relation types to tuple types
            { [edges=surtotal]
                (TST) -- (TTST),
            };
            
            % FDs
            (TSno) -- [funcdep left] (TStatus);
            
            % projection edges
            { [edges=bijection,edge label={\scriptsize$\RelProject$}]
                (TTST) -- (TSno),
            };
            
            { [edges=projection left]
                (TTST)  -- (TStatus),
            };
        };
    \end{tikzpicture}
    \caption{SIG for schema $\schema{3} = \{\ST\}$.}
    \label{fig-sig-s-iii-join-one}
\end{figure}

%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Transformations}
\label{sec-transforming-join-one}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{$\bm{\schema{1}, \schema{2}}$}

\noindent The SIGs for $\schema{1}$ and $\schema{2}$ can't be made isomorphic, as there's no way using SIG transformations to create in $\schema{3}$ a node equivalent to $\Type{\Status} \times \Type{\City}$ in $\schema{1}$. $\schema{1}$ and $\schema{2}$ are therefore incomparable.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{$\bm{\schema{1}, \schema{3}}$}

This is essentially identical to the lossy projection views example (see Section~\ref{sec-sigs-s-iii-project-lossy}). $\schema{1}$ and $\schema{3}$ are therefore incomparable.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Additional notes}

If we were to create a fourth sub-schema $\schema{4} = \{\SC\}$ and compare this with $\schema{1}$, the result would be similar to that for $\schema{3}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

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Request permissions from permissions@acm.org. \or % rightsretained Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner\hspace{.5pt}/author(s). \or % usgov This paper is authored by an employee(s) of the United States Government and is in the public domain. Non-exclusive copying or redistribution is allowed, provided that the article citation is given and the authors and agency are clearly identified as its source. \or % usgovmixed ACM acknowledges that this contribution was authored or co-authored by an employee, or contractor of the national government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only. Permission to make digital or hard copies for personal or classroom use is granted. Copies must bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. To copy otherwise, distribute, republish, or post, requires prior specific permission and\hspace{.5pt}/or a fee. Request permissions from permissions@acm.org. \or % cagov This article was authored by employees of the Government of Canada. As such, the Canadian government retains all interest in the copyright to this work and grants to ACM a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, provided that clear attribution is given both to the authors and the Canadian government agency employing them. Permission to make digital or hard copies for personal or classroom use is granted. Copies must bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the Canadain Government must be honored. To copy otherwise, distribute, republish, or post, requires prior specific permission and\hspace{.5pt}/or a fee. Request permissions from permissions@acm.org. \or % cagovmixed ACM acknowledges that this contribution was co-authored by an affiliate of the national government of Canada. As such, the Crown in Right of Canada retains an equal interest in the copyright. Reprints must include clear attribution to ACM and the author's government agency affiliation. Permission to make digital or hard copies for personal or classroom use is granted. Copies must bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. To copy otherwise, distribute, republish, or post, requires prior specific permission and\hspace*{.5pt}/or a fee. Request permissions from permissions@acm.org. \or % licensedusgovmixed Publication rights licensed to ACM. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of the United States government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only. \or % licensedcagovmixed Publication rights licensed to ACM. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of the national government of Canada. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only. \or % othergov ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of a national government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only. \or % licensedothergov Publication rights licensed to ACM. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of a national government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only. \fi} \endinput %% %% End of file `acmcopyright.sty'.

Ignore Space Show notes View

APCCM_2017/acmcopyright.sty 0 → 100644

%%
%% This is file `acmcopyright.sty',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% acmcopyright.dtx  (with options: `style')
%% 
%% IMPORTANT NOTICE:
%% 
%% For the copyright see the source file.
%% 
%% Any modified versions of this file must be renamed
%% with new filenames distinct from acmcopyright.sty.
%% 
%% For distribution of the original source see the terms
%% for copying and modification in the file acmcopyright.dtx.
%% 
%% This generated file may be distributed as long as the
%% original source files, as listed above, are part of the
%% same distribution. (The sources need not necessarily be
%% in the same archive or directory.)
%% \CharacterTable
%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
%%   Digits        \0\1\2\3\4\5\6\7\8\9
%%   Exclamation   \!     Double quote  \"     Hash (number) \#
%%   Dollar        \$     Percent       \%     Ampersand     \&
%%   Acute accent  \'     Left paren    $     Right paren   $
%%   Asterisk      \*     Plus          \+     Comma         \,
%%   Minus         \-     Point         \.     Solidus       \/
%%   Colon         \:     Semicolon     \;     Less than     \<
%%   Equals        \=     Greater than  \>     Question mark \?
%%   Commercial at \@     Left bracket  \[     Backslash     \\
%%   Right bracket \]     Circumflex    \^     Underscore    \_
%%   Grave accent  \`     Left brace    \{     Vertical bar  \|
%%   Right brace   \}     Tilde         \~}
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{acmcopyright}
[2014/06/29 v1.2 Copyright statemens for ACM classes]
\newif\if@printcopyright
\@printcopyrighttrue
\newif\if@printpermission
\@printpermissiontrue
\newif\if@acmowned
\@acmownedtrue
\RequirePackage{xkeyval}
\define@choicekey*{ACM@}{acmcopyrightmode}[%
  \acm@copyrightinput\acm@copyrightmode]{none,acmcopyright,acmlicensed,%
  rightsretained,usgov,usgovmixed,cagov,cagovmixed,%
  licensedusgovmixed,licensedcagovmixed,othergov,licensedothergov}{%
  \@printpermissiontrue
  \@printcopyrighttrue
  \@acmownedtrue
  \ifnum\acm@copyrightmode=0\relax % none
   \@printpermissionfalse
   \@printcopyrightfalse
   \@acmownedfalse
  \fi
  \ifnum\acm@copyrightmode=2\relax % acmlicensed
   \@acmownedfalse
  \fi
  \ifnum\acm@copyrightmode=3\relax % rightsretained
   \@acmownedfalse
  \fi
  \ifnum\acm@copyrightmode=4\relax % usgov
   \@printpermissiontrue
   \@printcopyrightfalse
   \@acmownedfalse
  \fi
  \ifnum\acm@copyrightmode=6\relax % cagov
   \@acmownedfalse
  \fi
  \ifnum\acm@copyrightmode=8\relax % licensedusgovmixed
   \@acmownedfalse
  \fi
  \ifnum\acm@copyrightmode=9\relax % licensedcagovmixed
   \@acmownedfalse
  \fi
  \ifnum\acm@copyrightmode=10\relax % othergov
   \@acmownedtrue
  \fi
  \ifnum\acm@copyrightmode=11\relax % licensedothergov
   \@acmownedfalse
   \@printcopyrightfalse
  \fi}
\def\setcopyright#1{\setkeys{ACM@}{acmcopyrightmode=#1}}
\setcopyright{acmcopyright}
\def\@copyrightowner{%
  \ifcase\acm@copyrightmode\relax % none
  \or % acmcopyright
  ACM.
  \or % acmlicensed
  Copyright held by the owner/author(s). Publication rights licensed to
  ACM.
  \or % rightsretained
  Copyright held by the owner/author(s).
  \or % usgov
  \or % usgovmixed
  ACM.
  \or % cagov
  Crown in Right of Canada.
  \or %cagovmixed
  ACM.
  \or %licensedusgovmixed
  Copyright held by the owner/author(s). Publication rights licensed to
  ACM.
  \or %licensedcagovmixed
  Copyright held by the owner/author(s). Publication rights licensed to
  ACM.
  \or % othergov
  ACM.
  \or % licensedothergov
  \fi}
\def\@copyrightpermission{%
  \ifcase\acm@copyrightmode\relax % none
  \or % acmcopyright
   Permission to make digital or hard copies of all or part of this
   work for personal or classroom use is granted without fee provided
   that copies are not made or distributed for profit or commercial
   advantage and that copies bear this notice and the full citation on
   the first page. Copyrights for components of this work owned by
   others than ACM must be honored. Abstracting with credit is
   permitted. To copy otherwise, or republish, to post on servers or to
   redistribute to lists, requires prior specific permission
   and\hspace*{.5pt}/or  a fee. Request permissions from
   permissions@acm.org.
  \or % acmlicensed
   Permission to make digital or hard copies of all or part of this
   work for personal or classroom use is granted without fee provided
   that copies are not made or distributed for profit or commercial
   advantage and that copies bear this notice and the full citation on
   the first page. Copyrights for components of this work owned by
   others than the author(s) must be honored. Abstracting with credit
   is permitted.  To copy otherwise, or republish, to post on servers
   or to  redistribute to lists, requires prior specific permission
   and\hspace*{.5pt}/or  a fee. Request permissions from
   permissions@acm.org.
  \or % rightsretained
   Permission to make digital or hard copies of part or all of this work
   for personal or classroom use is granted without fee provided that
   copies are not made or distributed for profit or commercial advantage
   and that copies bear this notice and the full citation on the first
   page. Copyrights for third-party components of this work must be
   honored. For all other uses, contact the
   owner\hspace*{.5pt}/author(s).
  \or % usgov
   This paper is authored by an employee(s) of the United States
   Government and is in the public domain. Non-exclusive copying or
   redistribution is allowed, provided that the article citation is
   given and the authors and agency are clearly identified as its
   source.
  \or % usgovmixed
   ACM acknowledges that this contribution was authored or co-authored
   by an employee, or contractor of the national government. As such,
   the Government retains a nonexclusive, royalty-free right to
   publish or reproduce this article, or to allow others to do so, for
   Government purposes only. Permission to make digital or hard copies
   for personal or classroom use is granted. Copies must bear this
   notice and the full citation on the first page. Copyrights for
   components of this work owned by others than ACM must be
   honored. To copy otherwise, distribute, republish, or post,
   requires prior specific permission and\hspace*{.5pt}/or a
   fee. Request permissions from permissions@acm.org.
  \or % cagov
   This article was authored by employees of the Government of Canada.
   As such, the Canadian government retains all interest in the
   copyright to this work and grants to ACM a nonexclusive,
   royalty-free right to publish or reproduce this article, or to allow
   others to do so, provided that clear attribution is given both to
   the authors and the Canadian government agency employing them.
   Permission to make digital or hard copies for personal or classroom
   use is granted. Copies must bear this notice and the full citation
   on the first page.  Copyrights for components of this work owned by
   others than the Canadain Government must be honored. To copy
   otherwise, distribute, republish, or post, requires prior specific
   permission and\hspace*{.5pt}/or a fee. Request permissions from
   permissions@acm.org.
  \or % cagovmixed
   ACM acknowledges that this contribution was co-authored by an
   affiliate of the national government of Canada. As such, the Crown
   in Right of Canada retains an equal interest in the copyright.
   Reprints must include clear attribution to ACM and the author's
   government agency affiliation.  Permission to make digital or hard
   copies for personal or classroom use is granted.  Copies must bear
   this notice and the full citation on the first page. Copyrights for
   components of this work owned by others than ACM must be honored.
   To copy otherwise, distribute, republish, or post, requires prior
   specific permission and\hspace*{.5pt}/or a fee. Request permissions
   from permissions@acm.org.
  \or % licensedusgovmixed
   Publication rights licensed to ACM. ACM acknowledges that this
   contribution was authored or co-authored by an employee, contractor
   or affiliate of the United States government. As such, the
   Government retains a nonexclusive, royalty-free right to publish or
   reproduce this article, or to allow others to do so, for Government
   purposes only.
  \or % licensedcagovmixed
   Publication rights licensed to ACM. ACM acknowledges that this
   contribution was authored or co-authored by an employee, contractor
   or affiliate of the national government of Canada. As such, the
   Government retains a nonexclusive, royalty-free right to publish or
   reproduce this article, or to allow others to do so, for Government
   purposes only.
  \or % othergov
   ACM acknowledges that this contribution was authored or co-authored
   by an employee, contractor or affiliate of a national government. As
   such, the Government retains a nonexclusive, royalty-free right to
   publish or reproduce this article, or to allow others to do so, for
   Government purposes only.
  \or % licensedothergov
   Publication rights licensed to ACM. ACM acknowledges that this
   contribution was authored or co-authored by an employee, contractor
   or affiliate of a national government. As such, the Government
   retains a nonexclusive, royalty-free right to publish or reproduce
   this article, or to allow others to do so, for Government purposes
   only.
  \fi}
\endinput
%%
%% End of file `acmcopyright.sty'.

Ignore Space Show notes View APCCM_2017/sig-alternate-05-2015.cls 0 → 100644
% SIG-ALTERNATE.CLS - VERSION 2.8 % "COMPATIBLE" WITH THE "ACM_PROC_ARTICLE-SP.CLS" V3.2SP % Gerald Murray - May 23rd 2012 % Boris Veytsman - April 23 2013 % Boris Veytsman - May 12 2013 % Boris Veytsman - June 09 2013 % Boris Veytsman - August 12 2013 % % ---- Start of 'updates' ---- % Added new permission/copyright statement - BV % Changed $10 fee to $15 -- May 2012 -- Gerry % Changed $5 fee to $10 -- April 2009 -- Gerry % April 22nd. 2009 - Fixed 'Natbib' incompatibility problem - Gerry % April 22nd. 2009 - Fixed 'Babel' incompatibility problem - Gerry % April 22nd. 2009 - Inserted various bug-fixes and improvements - Gerry % % To produce Type 1 fonts in the document plus allow for 'normal LaTeX accenting' in the critical areas; % title, author block, section-heads, confname, etc. etc. % i.e. the whole purpose of this version update is to NOT resort to 'inelegant accent patches'. % After much research, three extra .sty packages were added to the the tail (ae, aecompl, aeguill) to solve, % in particular, the accenting problem(s). We _could_ ask authors (via instructions/sample file) to 'include' these in % the source .tex file - in the preamble - but if everything is already provided ('behind the scenes' - embedded IN the .cls) % then this is less work for authors and also makes everything appear 'vanilla'. % NOTE: all 'patchwork accenting" has been commented out (here) and is no longer 'used' in the sample .tex file (either). % Gerry June 2007 % % Patch for accenting in conference name/location. Gerry May 3rd. 2007 % Rule widths changed to .5, author count (>6) fixed, roll-back for Type 3 problem. Gerry March 20th. 2007 % Changes made to 'modernize' the fontnames but esp. for MikTeX users V2.4/2.5 - Nov. 30th. 2006 % Updated the \email definition to allow for its use inside of 'shared affiliations' - Nov. 30th. 2006 % Fixed the 'section number depth value' - Nov. 30th. 2006 % % Footnotes inside table cells using \minipage (Oct. 2002) % Georgia fixed bug in sub-sub-section numbering in paragraphs (July 29th. 2002) % JS/GM fix to vertical spacing before Proofs (July 30th. 2002) % % Made the Permission Statement / Conference Info / Copyright Info % 'user definable' in the source .tex file OR automatic if % not specified. % % Allowance made to switch default fonts between those systems using % normal/modern font names and those using 'Type 1' or 'Truetype' fonts. % See LINE NUMBER 255 for details. % Also provided for enumerated/annotated Corollaries 'surrounded' by % enumerated Theorems (line 848). % Gerry November 11th. 1999 % % ---- End of 'updates' ---- % \def\fileversion{v2.9} % for ACM's tracking purposes \def\filedate{August 12, 2013} % Gerry Murray's tracking data \def\docdate {\filedate} \usepackage{epsfig} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % Need this for accents in Arial/Helvetica %\usepackage[T1]{fontenc} % Gerry March 12, 2007 - causes Type 3 problems (body text) %\usepackage{textcomp} % % SIG-ALTERNATE DOCUMENT STYLE % G.K.M. Tobin August-October 1999 % adapted from ARTICLE document style by Ken Traub, Olin Shivers % also using elements of esub2acm.cls % HEAVILY MODIFIED, SUBSEQUENTLY, BY GERRY MURRAY 2000 % ARTICLE DOCUMENT STYLE -- Released 16 March 1988 % for LaTeX version 2.09 % Copyright (C) 1988 by Leslie Lamport % % %%% sig-alternate.cls is an 'ALTERNATE' document style for producing %%% two-column camera-ready pages for ACM conferences. %%% THIS FILE DOES NOT STRICTLY ADHERE TO THE SIGS (BOARD-ENDORSED) %%% PROCEEDINGS STYLE. It has been designed to produce a 'tighter' %%% paper in response to concerns over page budgets. %%% The main features of this style are: %%% %%% 1) Two columns. %%% 2) Side and top margins of 4.5pc, bottom margin of 6pc, column gutter of %%% 2pc, hence columns are 20pc wide and 55.5pc tall. (6pc =3D 1in, approx) %%% 3) First page has title information, and an extra 6pc of space at the %%% bottom of the first column for the ACM copyright notice. %%% 4) Text is 9pt on 10pt baselines; titles (except main) are 9pt bold. %%% %%% %%% There are a few restrictions you must observe: %%% %%% 1) You cannot change the font size; ACM wants you to use 9pt. %%% 3) You must start your paper with the \maketitle command. Prior to the %%% \maketitle you must have \title and \author commands. If you have a %%% \date command it will be ignored; no date appears on the paper, since %%% the proceedings will have a date on the front cover. %%% 4) Marginal paragraphs, tables of contents, lists of figures and tables, %%% and page headings are all forbidden. %%% 5) The `figure' environment will produce a figure one column wide; if you %%% want one that is two columns wide, use `figure'. %%% % %%% Copyright Space: %%% This style automatically reserves 1" blank space at the bottom of page 1/ %%% column 1. This space can optionally be filled with some text using the %%% \toappear{...} command. If used, this command must be BEFORE the \maketitle %%% command. If this command is defined AND [preprint] is on, then the %%% space is filled with the {...} text (at the bottom); otherwise, it is %%% blank. If you use \toappearbox{...} instead of \toappear{...} then a %%% box will be drawn around the text (if [preprint] is on). %%% %%% A typical usage looks like this: %%% \toappear{To appear in the Ninth AES Conference on Medievil Lithuanian %%% Embalming Technique, June 1991, Alfaretta, Georgia.} %%% This will be included in the preprint, and left out of the conference %%% version. %%% %%% WARNING: %%% Some dvi-ps converters heuristically allow chars to drift from their %%% true positions a few pixels. This may be noticeable with the 9pt sans-serif %%% bold font used for section headers. %%% You may turn this hackery off via the -e option: %%% dvips -e 0 foo.dvi >foo.ps %%% \typeout{Document Class 'sig-alternate' <9th June '13>. Modified by G.K.M. Tobin/Gerry Murray/Boris Veytsman} \typeout{Based in part upon document Style `acmconf' <22 May 89>. Hacked 4/91 by} \typeout{shivers@cs.cmu.edu, 4/93 by theobald@cs.mcgill.ca} \typeout{Excerpts were taken from (Journal Style) 'esub2acm.cls'.} \typeout{*** Bugs/comments/suggestions/technicalities to Gerry Murray -- murray@hq.acm.org ***} \typeout{Questions on the style, SIGS policies, etc. to Adrienne Griscti griscti@acm.org} % New option code by BV \newcount\ACM@basesize \ACM@basesize=9\relax \DeclareOption{9pt}{\ACM@basesize=9\relax} \DeclareOption{10pt}{\ACM@basesize=10\relax} \DeclareOption{11pt}{\ClassError{sig-alternate}{The `11pt' option is not allowed}{sig-alternate now exists in 9pt and 10pt versions only}} \DeclareOption{12pt}{\ClassError{sig-alternate}{The `12pt' option is not allowed}{sig-alternate now exists in 9pt and 10pt versions only}} \ExecuteOptions{9pt} \ProcessOptions \let\@concepts\@empty % Support for CCSXML file \RequirePackage{comment} \excludecomment{CCSXML} % New concepts scheme % % The first argument is the significance, the % second is the concept(s) % \newcommand\ccsdesc[2][100]{% \ccsdesc@parse#1~#2~} % % The parser of the expression Significance~General~Specific % \def\ccsdesc@parse#1~#2~#3~{% \expandafter\ifx\csname CCS@#2\endcsname\relax \expandafter\gdef\csname CCS@#2\endcsname{\textbullet\textbf{#2} $\to$ }% \g@addto@macro{\@concepts}{\csname CCS@#2\endcsname}\fi \expandafter\g@addto@macro\expandafter{\csname CCS@#2\endcsname}{% \ifnum#1>499\textbf{#3; }\else \ifnum#1>299\textit{#3; }\else #3; \fi\fi}} \newcommand\printccsdesc{% \ifx\@concepts\@empty\else \if@twocolumn \section{CCS Concepts} \@concepts \else \small \quotation{\@concepts}% \fi \fi} \def\doi#1{\def\@doi{#1}} \doi{http://dx.doi.org/10.1145/0000000.0000000} \oddsidemargin 4.5pc \evensidemargin 4.5pc \advance\oddsidemargin by -1in % Correct for LaTeX gratuitousness \advance\evensidemargin by -1in % Correct for LaTeX gratuitousness \marginparwidth 0pt % Margin pars are not allowed. \marginparsep 11pt % Horizontal space between outer margin and % marginal note % Top of page: \topmargin 4.5pc % Nominal distance from top of page to top of % box containing running head. \advance\topmargin by -1in % Correct for LaTeX gratuitousness \headheight 0pt % Height of box containing running head. \headsep 0pt % Space between running head and text. % Bottom of page: \footskip 30pt % Distance from baseline of box containing foot % to baseline of last line of text. \@ifundefined{footheight}{\newdimen\footheight}{}% this is for LaTeX2e \footheight 12pt % Height of box containing running foot. %% Must redefine the top margin so there's room for headers and %% page numbers if you are using the preprint option. Footers %% are OK as is. Olin. \advance\topmargin by -37pt % Leave 37pt above text for headers \headheight 12pt % Height of box containing running head. \headsep 25pt % Space between running head and text. \textheight 666pt % 9 1/4 column height \textwidth 42pc % Width of text line. % For two-column mode: \columnsep 2pc % Space between columns \columnseprule 0pt % Width of rule between columns. \hfuzz 1pt % Allow some variation in column width, otherwise it's % too hard to typeset in narrow columns. \ifnum\ACM@basesize=9\relax \footnotesep 5.6pt % Height of strut placed at the beginning of every % footnote =3D height of normal \footnotesize strut, % so no extra space between footnotes. \fi \ifnum\ACM@basesize=10\relax \footnotesep 6.22pt % Height of strut placed at the beginning of every % footnote =3D height of normal \footnotesize strut, % so no extra space between footnotes. \fi \skip\footins 8.1pt plus 4pt minus 2pt % Space between last line of text and % top of first footnote. \floatsep 11pt plus 2pt minus 2pt % Space between adjacent floats moved % to top or bottom of text page. \textfloatsep 18pt plus 2pt minus 4pt % Space between main text and floats % at top or bottom of page. \intextsep 11pt plus 2pt minus 2pt % Space between in-text figures and % text. \@ifundefined{@maxsep}{\newdimen\@maxsep}{}% this is for LaTeX2e \@maxsep 18pt % The maximum of \floatsep, % \textfloatsep and \intextsep (minus % the stretch and shrink). \dblfloatsep 11pt plus 2pt minus 2pt % Same as \floatsep for double-column % figures in two-column mode. \dbltextfloatsep 18pt plus 2pt minus 4pt% \textfloatsep for double-column % floats. \@ifundefined{@dblmaxsep}{\newdimen\@dblmaxsep}{}% this is for LaTeX2e \@dblmaxsep 18pt % The maximum of \dblfloatsep and % \dbltexfloatsep. \@fptop 0pt plus 1fil % Stretch at top of float page/column. (Must be % 0pt plus ...) \@fpsep 8pt plus 2fil % Space between floats on float page/column. \@fpbot 0pt plus 1fil % Stretch at bottom of float page/column. (Must be % 0pt plus ... ) \@dblfptop 0pt plus 1fil % Stretch at top of float page. (Must be 0pt plus ...) \@dblfpsep 8pt plus 2fil % Space between floats on float page. \@dblfpbot 0pt plus 1fil % Stretch at bottom of float page. (Must be % 0pt plus ... ) \marginparpush 5pt % Minimum vertical separation between two marginal % notes. \parskip 0pt plus 1pt % Extra vertical space between % paragraphs. \ifnum\ACM@basesize=9\relax \parindent 9pt % GM July 2000 / was 0pt - width of paragraph % indentation. \fi \ifnum\ACM@basesize=10\relax \parindent 10pt % GM July 2000 / was 0pt - width of paragraph % indentation. \fi \partopsep 2pt plus 1pt minus 1pt% Extra vertical space, in addition to % \parskip and \topsep, added when user % leaves blank line before environment. \@lowpenalty 51 % Produced by \nopagebreak[1] or \nolinebreak[1] \@medpenalty 151 % Produced by \nopagebreak[2] or \nolinebreak[2] \@highpenalty 301 % Produced by \nopagebreak[3] or \nolinebreak[3] \@beginparpenalty -\@lowpenalty % Before a list or paragraph environment. \@endparpenalty -\@lowpenalty % After a list or paragraph environment. \@itempenalty -\@lowpenalty % Between list items. \RequirePackage{ifpdf}% \ifpdf \pdfpagewidth=8.5in \pdfpageheight=11in \fi \lineskip 2pt % \lineskip is 1pt for all font sizes. \normallineskip 2pt \def\baselinestretch{1} \ifnum\ACM@basesize=9\relax \abovedisplayskip 9pt plus2pt minus4.5pt% \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus3pt% \belowdisplayshortskip 5.4pt plus3pt minus3pt% \let\@listi\@listI % Setting of \@listi added 9 Jun 87 \def\small{\@setsize\small{9pt}\viiipt\@viiipt \abovedisplayskip 7.6pt plus 3pt minus 4pt% \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus2pt% \belowdisplayshortskip 3.6pt plus2pt minus 2pt \def\@listi{\leftmargin\leftmargini %% Added 22 Dec 87 \topsep 4pt plus 2pt minus 2pt\parsep 2pt plus 1pt minus 1pt \itemsep \parsep}} \def\footnotesize{\@setsize\footnotesize{9pt}\ixpt\@ixpt \abovedisplayskip 6.4pt plus 2pt minus 4pt% \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus 1pt% \belowdisplayshortskip 2.7pt plus 1pt minus 2pt \def\@listi{\leftmargin\leftmargini %% Added 22 Dec 87 \topsep 3pt plus 1pt minus 1pt\parsep 2pt plus 1pt minus 1pt \itemsep \parsep}} \fi \ifnum\ACM@basesize=10\relax \abovedisplayskip 10pt plus2pt minus4.5pt% \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus3pt% \belowdisplayshortskip 6pt plus3pt minus3pt% \let\@listi\@listI % Setting of \@listi added 9 Jun 87 \def\small{\@setsize\small{10pt}\ixpt\@ixpt \abovedisplayskip 8.5pt plus 3pt minus 4pt% \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus2pt% \belowdisplayshortskip 4pt plus2pt minus 2pt \def\@listi{\leftmargin\leftmargini %% Added 22 Dec 87 \topsep 4.5pt plus 2pt minus 2pt\parsep 2pt plus 1pt minus 1pt \itemsep \parsep}} \def\footnotesize{\@setsize\footnotesize{10pt}\xpt\@xpt \abovedisplayskip 7.6pt plus 2pt minus 4pt% \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus 1pt% \belowdisplayshortskip 3.0pt plus 1pt minus 2pt \def\@listi{\leftmargin\leftmargini %% Added 22 Dec 87 \topsep 3.2pt plus 1pt minus 1pt\parsep 2pt plus 1pt minus 1pt \itemsep \parsep}} \fi \newcount\aucount \newcount\originalaucount \newdimen\auwidth \auwidth=\textwidth \newdimen\auskip \newcount\auskipcount \newdimen\auskip \global\auskip=1pc \newdimen\allauboxes \allauboxes=\auwidth \newtoks\addauthors \newcount\addauflag \global\addauflag=0 %Haven't shown additional authors yet \newtoks\subtitletext \gdef\subtitle#1{\subtitletext={#1}} \gdef\additionalauthors#1{\addauthors={#1}} \gdef\numberofauthors#1{\global\aucount=#1 \ifnum\aucount>3\global\originalaucount=\aucount \global\aucount=3\fi %g} % 3 OK - Gerry March 2007 \global\auskipcount=\aucount\global\advance\auskipcount by 1 \global\multiply\auskipcount by 2 \global\multiply\auskip by \auskipcount \global\advance\auwidth by -\auskip \global\divide\auwidth by \aucount} % \and was modified to count the number of authors. GKMT 12 Aug 1999 \def\alignauthor{% % \begin{tabular} \end{tabular}% \begin{tabular}[t]{p{\auwidth}}\centering}% % * NOTE * NOTE * NOTE * NOTE * % If you have 'font problems' then you may need % to change these, e.g. 'arialb' instead of "arialbd". % Gerry Murray 11/11/1999 % * OR comment out block A and activate block B or vice versa. % ****************************************** % % -- Start of block A -- (Type 1 or Truetype fonts) %\newfont{\secfnt}{timesbd at 12pt} % was timenrb originally - now is timesbd %\newfont{\secit}{timesbi at 12pt} %13 Jan 00 gkmt %\newfont{\subsecfnt}{timesi at 11pt} % was timenrri originally - now is timesi %\newfont{\subsecit}{timesbi at 11pt} % 13 Jan 00 gkmt -- was times changed to timesbi gm 2/4/2000 % % because "normal" is italic, "italic" is Roman %\newfont{\ttlfnt}{arialbd at 18pt} % was arialb originally - now is arialbd %\newfont{\ttlit}{arialbi at 18pt} % 13 Jan 00 gkmt %\newfont{\subttlfnt}{arial at 14pt} % was arialr originally - now is arial %\newfont{\subttlit}{ariali at 14pt} % 13 Jan 00 gkmt %\newfont{\subttlbf}{arialbd at 14pt} % 13 Jan 00 gkmt %\newfont{\aufnt}{arial at 12pt} % was arialr originally - now is arial %\newfont{\auit}{ariali at 12pt} % 13 Jan 00 gkmt %\newfont{\affaddr}{arial at 10pt} % was arialr originally - now is arial %\newfont{\affaddrit}{ariali at 10pt} %13 Jan 00 gkmt %\newfont{\eaddfnt}{arial at 12pt} % was arialr originally - now is arial %\newfont{\ixpt}{times at 9pt} % was timenrr originally - now is times %\newfont{\confname}{timesi at 8pt} % was timenrri - now is timesi %\newfont{\crnotice}{times at 8pt} % was timenrr originally - now is times %\newfont{\ninept}{times at 9pt} % was timenrr originally - now is times % ***************************************** % -- End of block A -- % % % -- Start of block B -- UPDATED FONT NAMES % ***************************************** % Gerry Murray 11/30/2006 % ******************************************* \ifnum\ACM@basesize=9\relax \newfont{\secfnt}{ptmb8t at 12pt} \newfont{\secit}{ptmbi8t at 12pt} %13 Jan 00 gkmt \newfont{\subsecfnt}{ptmri8t at 11pt} \newfont{\subsecit}{ptmbi8t at 11pt} % \newfont{\ttlfnt}{phvb8t at 18pt} \newfont{\ttlit}{phvbo8t at 18pt} % GM 2/4/2000 \newfont{\subttlfnt}{phvr8t at 14pt} \newfont{\subttlit}{phvro8t at 14pt} % GM 2/4/2000 \newfont{\subttlbf}{phvb8t at 14pt} % 13 Jan 00 gkmt \newfont{\aufnt}{phvr8t at 12pt} \newfont{\auit}{phvro8t at 12pt} % GM 2/4/2000 \newfont{\affaddr}{phvr8t at 10pt} \newfont{\affaddrit}{phvro8t at 10pt} % GM 2/4/2000 \newfont{\eaddfnt}{phvr8t at 12pt} \newfont{\ixpt}{ptmr8t at 9pt} \newfont{\confname}{ptmri8t at 8pt} \newfont{\crnotice}{ptmr8t at 8pt} \newfont{\ninept}{ptmr8t at 9pt} \fi \ifnum\ACM@basesize=10\relax \newfont{\secfnt}{ptmb8t at 13pt} \newfont{\secit}{ptmbi8t at 13pt} %13 Jan 00 gkmt \newfont{\subsecfnt}{ptmri8t at 12pt} \newfont{\subsecit}{ptmbi8t at 12pt} % \newfont{\ttlfnt}{phvb8t at 20pt} \newfont{\ttlit}{phvbo8t at 20pt} % GM 2/4/2000 \newfont{\subttlfnt}{phvr8t at 15pt} \newfont{\subttlit}{phvro8t at 15pt} % GM 2/4/2000 \newfont{\subttlbf}{phvb8t at 15pt} % 13 Jan 00 gkmt \newfont{\aufnt}{phvr8t at 12pt} \newfont{\auit}{phvro8t at 12pt} % GM 2/4/2000 \newfont{\affaddr}{phvr8t at 11pt} \newfont{\affaddrit}{phvro8t at 11pt} % GM 2/4/2000 \newfont{\eaddfnt}{phvr8t at 12pt} \newfont{\ixpt}{ptmr8t at 10pt} \newfont{\confname}{ptmri8t at 9pt} \newfont{\crnotice}{ptmr8t at 9pt} \newfont{\ninept}{ptmr8t at 10pt} \fi % +++++++++++++++++++++++++++++++++++++++++++++ % -- End of block B -- %\def\email#1{{{\eaddfnt{\vskip 4pt#1}}}} % If we have an email, inside a "shared affiliation" then we need the following instead \def\email#1{{{\eaddfnt{\par #1}}}} % revised - GM - 11/30/2006 \def\addauthorsection{\ifnum\originalaucount>6 % was 3 - Gerry March 2007 \section{Additional Authors}\the\addauthors \fi} \newcount\savesection \newcount\sectioncntr \global\sectioncntr=1 \setcounter{secnumdepth}{3} \def\appendix{\par \section{APPENDIX} \setcounter{section}{0} \setcounter{subsection}{0} \def\thesection{\Alph{section}} } \leftmargini 22.5pt \leftmarginii 19.8pt % > \labelsep + width of '(m)' \leftmarginiii 16.8pt % > \labelsep + width of 'vii.' \leftmarginiv 15.3pt % > \labelsep + width of 'M.' \leftmarginv 9pt \leftmarginvi 9pt \leftmargin\leftmargini \labelsep 4.5pt \labelwidth\leftmargini\advance\labelwidth-\labelsep \def\@listI{\leftmargin\leftmargini \parsep 3.6pt plus 2pt minus 1pt% \topsep 7.2pt plus 2pt minus 4pt% \itemsep 3.6pt plus 2pt minus 1pt} \let\@listi\@listI \@listi \def\@listii{\leftmargin\leftmarginii \labelwidth\leftmarginii\advance\labelwidth-\labelsep \topsep 3.6pt plus 2pt minus 1pt \parsep 1.8pt plus 0.9pt minus 0.9pt \itemsep \parsep} \def\@listiii{\leftmargin\leftmarginiii \labelwidth\leftmarginiii\advance\labelwidth-\labelsep \topsep 1.8pt plus 0.9pt minus 0.9pt \parsep \z@ \partopsep 1pt plus 0pt minus 1pt \itemsep \topsep} \def\@listiv{\leftmargin\leftmarginiv \labelwidth\leftmarginiv\advance\labelwidth-\labelsep} \def\@listv{\leftmargin\leftmarginv \labelwidth\leftmarginv\advance\labelwidth-\labelsep} \def\@listvi{\leftmargin\leftmarginvi \labelwidth\leftmarginvi\advance\labelwidth-\labelsep} \def\labelenumi{\theenumi.} \def\theenumi{\arabic{enumi}} \def\labelenumii{(\theenumii)} \def\theenumii{\alph{enumii}} \def\p@enumii{\theenumi} \def\labelenumiii{\theenumiii.} \def\theenumiii{\roman{enumiii}} \def\p@enumiii{\theenumi(\theenumii)} \def\labelenumiv{\theenumiv.} \def\theenumiv{\Alph{enumiv}} \def\p@enumiv{\p@enumiii\theenumiii} \def\labelitemi{$\bullet$} \def\labelitemii{\bf --} \def\labelitemiii{$\ast$} \def\labelitemiv{$\cdot$} \def\verse{\let\\=\@centercr \list{}{\itemsep\z@ \itemindent -1.5em\listparindent \itemindent \rightmargin\leftmargin\advance\leftmargin 1.5em}\item[]} \let\endverse\endlist \def\quotation{\list{}{\listparindent 1.5em \itemindent\listparindent \rightmargin\leftmargin \parsep 0pt plus 1pt}\item[]} \let\endquotation=\endlist \def\quote{\list{}{\rightmargin\leftmargin}\item[]} \let\endquote=\endlist \def\descriptionlabel#1{\hspace\labelsep \bf #1} \def\description{\list{}{\labelwidth\z@ \itemindent-\leftmargin \let\makelabel\descriptionlabel}} \let\enddescription\endlist \def\theequation{\arabic{equation}} \ifnum\ACM@basesize=9\relax \arraycolsep 4.5pt % Half the space between columns in an array environment. \tabcolsep 5.4pt % Half the space between columns in a tabular environment. \arrayrulewidth .5pt % Width of rules in array and tabular environment. % (was .4) updated Gerry March 20 2007 \doublerulesep 1.8pt % Space between adjacent rules in array or tabular env. \fi \ifnum\ACM@basesize=10\relax \arraycolsep 5pt % Half the space between columns in an array environment. \tabcolsep 6pt % Half the space between columns in a tabular environment. \arrayrulewidth .5pt % Width of rules in array and tabular environment. % (was .4) updated Gerry March 20 2007 \doublerulesep 1.8pt % Space between adjacent rules in array or tabular env. \fi \tabbingsep \labelsep % Space used by the \' command. (See LaTeX manual.) \skip\@mpfootins =\skip\footins \fboxsep =2.7pt % Space left between box and text by \fbox and \framebox. \fboxrule =.5pt % Width of rules in box made by \fbox and \framebox. % (was .4) updated Gerry March 20 2007 \def\thepart{\Roman{part}} % Roman numeral part numbers. \def\thesection {\arabic{section}} \def\thesubsection {\thesection.\arabic{subsection}} %\def\thesubsubsection {\thesubsection.\arabic{subsubsection}} % GM 7/30/2002 %\def\theparagraph {\thesubsubsection.\arabic{paragraph}} % GM 7/30/2002 \def\thesubparagraph {\theparagraph.\arabic{subparagraph}} \def\@pnumwidth{1.55em} \def\@tocrmarg {2.55em} \def\@dotsep{4.5} \setcounter{tocdepth}{3} %\def\tableofcontents{\@latexerr{\tableofcontents: Tables of contents are not % allowed in the `acmconf' document style.}\@eha} \def\tableofcontents{\ClassError{% \string\tableofcontents\space is not allowed in the `acmconf' document % January 2008 style}\@eha} \def\l@part#1#2{\addpenalty{\@secpenalty} \addvspace{2.25em plus 1pt} % space above part line \begingroup \@tempdima 3em % width of box holding part number, used by \parindent \z@ \rightskip \@pnumwidth %% \numberline \parfillskip -\@pnumwidth {\large \bf % set line in \large boldface \leavevmode % TeX command to enter horizontal mode. #1\hfil \hbox to\@pnumwidth{\hss #2}}\par \nobreak % Never break after part entry \endgroup} \def\l@section#1#2{\addpenalty{\@secpenalty} % good place for page break \addvspace{1.0em plus 1pt} % space above toc entry \@tempdima 1.5em % width of box holding section number \begingroup \parindent \z@ \rightskip \@pnumwidth \parfillskip -\@pnumwidth \bf % Boldface. \leavevmode % TeX command to enter horizontal mode. \advance\leftskip\@tempdima %% added 5 Feb 88 to conform to \hskip -\leftskip %% 25 Jan 88 change to \numberline #1\nobreak\hfil \nobreak\hbox to\@pnumwidth{\hss #2}\par \endgroup} \def\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}} \def\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}} \def\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}} \def\l@subparagraph{\@dottedtocline{5}{10em}{5em}} %\def\listoffigures{\@latexerr{\listoffigures: Lists of figures are not % allowed in the `acmconf' document style.}\@eha} \def\listoffigures{\ClassError{% \string\listoffigures\space is not allowed in the `acmconf' document % January 2008 style}\@eha} \def\l@figure{\@dottedtocline{1}{1.5em}{2.3em}} %\def\listoftables{\@latexerr{\listoftables: Lists of tables are not % allowed in the `acmconf' document style.}\@eha} %\let\l@table\l@figure \def\listoftables{\ClassError{% \string\listoftables\space is not allowed in the `acmconf' document % January 2008 style}\@eha} \let\l@table\l@figure \def\footnoterule{\kern-3\p@ \hrule width .5\columnwidth % (was .4) updated Gerry March 20 2007 \kern 2.6\p@} % The \hrule has default height of .4pt % (was .4) updated Gerry March 20 2007 % ------ \long\def\@makefntext#1{\noindent %\hbox to .5em{\hss$^{\@thefnmark}$}#1} % original \hbox to .5em{\hss\textsuperscript{\@thefnmark}}#1} % C. Clifton / GM Oct. 2nd. 2002 % ------- \long\def\@maketntext#1{\noindent #1} \long\def\@maketitlenotetext#1#2{\noindent \hbox to 1.8em{\hss$^{#1}$}#2} \setcounter{topnumber}{2} \def\topfraction{.7} \setcounter{bottomnumber}{1} \def\bottomfraction{.3} \setcounter{totalnumber}{3} \def\textfraction{.2} \def\floatpagefraction{.5} \setcounter{dbltopnumber}{2} \def\dbltopfraction{.7} \def\dblfloatpagefraction{.5} % \long\def\@makecaption#1#2{ \vskip \baselineskip \setbox\@tempboxa\hbox{\textbf{#1: #2}} \ifdim \wd\@tempboxa >\hsize % IF longer than one line: \textbf{#1: #2}\par % THEN set as ordinary paragraph. \else % ELSE center. \hbox to\hsize{\hfil\box\@tempboxa\hfil}\par \fi} % \long\def\@makecaption#1#2{ \vskip 10pt \setbox\@tempboxa\hbox{\textbf{#1: #2}} \ifdim \wd\@tempboxa >\hsize % IF longer than one line: \textbf{#1: #2}\par % THEN set as ordinary paragraph. \else % ELSE center. \hbox to\hsize{\hfil\box\@tempboxa\hfil} \fi} \@ifundefined{figure}{\newcounter {figure}} % this is for LaTeX2e \def\fps@figure{tbp} \def\ftype@figure{1} \def\ext@figure{lof} \def\fnum@figure{Figure \thefigure} \def\figure{\@float{figure}} %\let\endfigure\end@float \def\endfigure{\end@float} % Gerry January 2008 \@namedef{figure}{\@dblfloat{figure}} \@namedef{endfigure}{\end@dblfloat} \@ifundefined{table}{\newcounter {table}} % this is for LaTeX2e \def\fps@table{tbp} \def\ftype@table{2} \def\ext@table{lot} \def\fnum@table{Table \thetable} \def\table{\@float{table}} %\let\endtable\end@float \def\endtable{\end@float} % Gerry January 2008 \@namedef{table}{\@dblfloat{table}} \@namedef{endtable}{\end@dblfloat} \newtoks\titleboxnotes \newcount\titleboxnoteflag \def\maketitle{\par \begingroup \def\thefootnote{\fnsymbol{footnote}} \def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}} \twocolumn[\@maketitle] \@thanks \endgroup \setcounter{footnote}{0} \let\maketitle\relax \let\@maketitle\relax \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\gdef\@subtitle{}\let\thanks\relax \@copyrightspace} %% CHANGES ON NEXT LINES \newif\if@ll % to record which version of LaTeX is in use \expandafter\ifx\csname LaTeXe\endcsname\relax % LaTeX2.09 is used \else% LaTeX2e is used, so set ll to true \global\@lltrue \fi \if@ll \NeedsTeXFormat{LaTeX2e} \ProvidesClass{sig-alternate} [2013/05/12 v2.7 based on acmproc.cls V1.3 <Nov. 30 '99>] \RequirePackage{latexsym}% QUERY: are these two really needed? \let\dooptions\ProcessOptions \else \let\dooptions\@options \fi %% END CHANGES \def\@height{height} \def\@width{width} \def\@minus{minus} \def\@plus{plus} \def\hb@xt@{\hbox to} \newif\if@faircopy \@faircopyfalse \def\ds@faircopy{\@faircopytrue} \def\ds@preprint{\@faircopyfalse} \@twosidetrue \@mparswitchtrue \def\ds@draft{\overfullrule 5\p@} %% CHANGE ON NEXT LINE \dooptions \lineskip \p@ \normallineskip \p@ \def\baselinestretch{1} \def\@ptsize{0} %needed for amssymbols.sty %% CHANGES ON NEXT LINES \if@ll% allow use of old-style font change commands in LaTeX2e \@maxdepth\maxdepth % \DeclareOldFontCommand{\rm}{\ninept\rmfamily}{\mathrm} \DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf} \DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt} \DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf} \DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit} \DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl} \DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc} \DeclareRobustCommand{\cal}{\@fontswitch{\relax}{\mathcal}} \DeclareRobustCommand{\mit}{\@fontswitch{\relax}{\mathnormal}} \fi % \if@ll \renewcommand{\rmdefault}{cmr} % was 'ttm' % Note! I have also found 'mvr' to work ESPECIALLY well. % Gerry - October 1999 % You may need to change your LV1times.fd file so that sc is % mapped to cmcsc - -for smallcaps -- that is if you decide % to change {cmr} to {times} above. (Not recommended) \ifnum\ACM@basesize=9\relax \renewcommand{\@ptsize}{} \renewcommand{\normalsize}{% \@setfontsize\normalsize\@ixpt{10.5\p@}%\ninept% \abovedisplayskip 6\p@ \@plus2\p@ \@minus\p@ \belowdisplayskip \abovedisplayskip \abovedisplayshortskip 6\p@ \@minus 3\p@ \belowdisplayshortskip 6\p@ \@minus 3\p@ \let\@listi\@listI} \fi \ifnum\ACM@basesize=10\relax \renewcommand{\@ptsize}{} \renewcommand{\normalsize}{% \@setfontsize\normalsize\@xpt{11.5\p@}%\ninept% \abovedisplayskip 6.5\p@ \@plus2\p@ \@minus\p@ \belowdisplayskip \abovedisplayskip \abovedisplayshortskip 6.5\p@ \@minus 3\p@ \belowdisplayshortskip 6.5\p@ \@minus 3\p@ \let\@listi\@listI} \fi \else \def\@normalsize{%changed next to 9 from 10 \@setsize\normalsize{9\p@}\ixpt\@ixpt \abovedisplayskip 6\p@ \@plus2\p@ \@minus\p@ \belowdisplayskip \abovedisplayskip \abovedisplayshortskip 6\p@ \@minus 3\p@ \belowdisplayshortskip 6\p@ \@minus 3\p@ \let\@listi\@listI }% \fi \if@ll \ifnum\ACM@basesize=9\relax \newcommand\scriptsize{\@setfontsize\scriptsize\@viipt{8\p@}} \newcommand\tiny{\@setfontsize\tiny\@vpt{6\p@}} \newcommand\large{\@setfontsize\large\@xiipt{14\p@}} \newcommand\Large{\@setfontsize\Large\@xivpt{18\p@}} \newcommand\LARGE{\@setfontsize\LARGE\@xviipt{20\p@}} \newcommand\huge{\@setfontsize\huge\@xxpt{25\p@}} \newcommand\Huge{\@setfontsize\Huge\@xxvpt{30\p@}} \fi \ifnum\ACM@basesize=10\relax \newcommand\scriptsize{\@setfontsize\scriptsize\@viiipt{9\p@}} \newcommand\tiny{\@setfontsize\tiny\@vipt{7\p@}} \newcommand\large{\@setfontsize\large\@xiiipt{15\p@}} \newcommand\Large{\@setfontsize\Large\@xvpt{20\p@}} \newcommand\LARGE{\@setfontsize\LARGE\@xixpt{22\p@}} \newcommand\huge{\@setfontsize\huge\@xixpt{30\p@}} \newcommand\Huge{\@setfontsize\Huge30pt{36\p@}} \fi \else \def\scriptsize{\@setsize\scriptsize{8\p@}\viipt\@viipt} \def\tiny{\@setsize\tiny{6\p@}\vpt\@vpt} \def\large{\@setsize\large{14\p@}\xiipt\@xiipt} \def\Large{\@setsize\Large{18\p@}\xivpt\@xivpt} \def\LARGE{\@setsize\LARGE{20\p@}\xviipt\@xviipt} \def\huge{\@setsize\huge{25\p@}\xxpt\@xxpt} \def\Huge{\@setsize\Huge{30\p@}\xxvpt\@xxvpt} \fi \normalsize % make aubox hsize/number of authors up to 3, less gutter % then showbox gutter showbox gutter showbox -- GKMT Aug 99 \newbox\@acmtitlebox \ifnum\ACM@basesize=9\relax \def\@maketitle{\newpage \null \setbox\@acmtitlebox\vbox{% \baselineskip 20pt \vskip 2em % Vertical space above title. \begin{center} {\ttlfnt \@title\par} % Title set in 18pt Helvetica (Arial) bold size. \vskip 1.5em % Vertical space after title. %This should be the subtitle. {\subttlfnt \the\subtitletext\par}\vskip 1.25em%\fi {\baselineskip 16pt\aufnt % each author set in \12 pt Arial, in a \lineskip .5em % tabular environment \begin{tabular}[t]{c}\@author \end{tabular}\par} \vskip 1.5em % Vertical space after author. \end{center}} \dimen0=\ht\@acmtitlebox \advance\dimen0 by -12.75pc\relax % Increased space for title box -- KBT \unvbox\@acmtitlebox \ifdim\dimen0<0.0pt\relax\vskip-\dimen0\fi} \fi \ifnum\ACM@basesize=10\relax \def\@maketitle{\newpage \null \setbox\@acmtitlebox\vbox{% \baselineskip 22pt \vskip 2.2em % Vertical space above title. \begin{center} {\ttlfnt \@title\par} % Title set in 18pt Helvetica (Arial) bold size. \vskip 2em % Vertical space after title. %This should be the subtitle. {\subttlfnt \the\subtitletext\par}\vskip 1.25em%\fi {\baselineskip 18pt\aufnt % each author set in \12 pt Arial, in a \lineskip .5em % tabular environment \begin{tabular}[t]{c}\@author \end{tabular}\par} \vskip 2em % Vertical space after author. \end{center}} \dimen0=\ht\@acmtitlebox \advance\dimen0 by -12.75pc\relax % Increased space for title box -- KBT \unvbox\@acmtitlebox \ifdim\dimen0<0.0pt\relax\vskip-\dimen0\fi} \fi \newcount\titlenotecount \global\titlenotecount=0 \newtoks\tntoks \newtoks\tntokstwo \newtoks\tntoksthree \newtoks\tntoksfour \newtoks\tntoksfive \def\abstract{ \ifnum\titlenotecount>0 % was =1 \insert\footins{% \reset@font\footnotesize \interlinepenalty\interfootnotelinepenalty \splittopskip\footnotesep \splitmaxdepth \dp\strutbox \floatingpenalty \@MM \hsize\columnwidth \@parboxrestore \protected@edef\@currentlabel{% }% \color@begingroup \ifnum\titlenotecount=1 \@maketntext{% \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\@finalstrut\strutbox}% \fi \ifnum\titlenotecount=2 \@maketntext{% \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\dagger$}\rule\z@\footnotesep\ignorespaces\the\tntokstwo\@finalstrut\strutbox}% \fi \ifnum\titlenotecount=3 \@maketntext{% \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\dagger$}\rule\z@\footnotesep\ignorespaces\the\tntokstwo\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\ddagger$}\rule\z@\footnotesep\ignorespaces\the\tntoksthree\@finalstrut\strutbox}% \fi \ifnum\titlenotecount=4 \@maketntext{% \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\dagger$}\rule\z@\footnotesep\ignorespaces\the\tntokstwo\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\ddagger$}\rule\z@\footnotesep\ignorespaces\the\tntoksthree\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\S$}\rule\z@\footnotesep\ignorespaces\the\tntoksfour\@finalstrut\strutbox}% \fi \ifnum\titlenotecount=5 \@maketntext{% \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\dagger$}\rule\z@\footnotesep\ignorespaces\the\tntokstwo\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\ddagger$}\rule\z@\footnotesep\ignorespaces\the\tntoksthree\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\S$}\rule\z@\footnotesep\ignorespaces\the\tntoksfour\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\P$}\rule\z@\footnotesep\ignorespaces\the\tntoksfive\@finalstrut\strutbox}% \fi \color@endgroup} %g} \fi \setcounter{footnote}{0} \section{ABSTRACT}\normalsize%\ninept } \def\endabstract{\if@twocolumn\else\endquotation\fi} \def\keywords{\if@twocolumn \section{Keywords} \else \small \quotation \fi} \def\terms#1{% %\if@twocolumn %\section{General Terms} %\else \small %\quotation %\fi } % -- Classification needs to be a bit smart due to optionals - Gerry/Georgia November 2nd. 1999 \newcount\catcount \global\catcount=1 \def\category#1#2#3{% \ifnum\catcount=1 \section{Categories and Subject Descriptors} \advance\catcount by 1\else{\unskip; }\fi \@ifnextchar [{\@category{#1}{#2}{#3}}{\@category{#1}{#2}{#3}[]}% } \def\@category#1#2#3[#4]{% \begingroup \let\and\relax #1 [\textbf{#2}]% \if!#4!% \if!#3!\else : #3\fi \else :\space \if!#3!\else #3\kern\z@---\hskip\z@\fi \textit{#4}% \fi \endgroup } % %%% This section (written by KBT) handles the 1" box in the lower left %%% corner of the left column of the first page by creating a picture, %%% and inserting the predefined string at the bottom (with a negative %%% displacement to offset the space allocated for a non-existent %%% caption). %%% \newtoks\copyrightnotice \def\ftype@copyrightbox{8} \def\@copyrightspace{ \@float{copyrightbox}[b] \begin{center} \setlength{\unitlength}{1pc} \ifnum\ACM@basesize=9 \begin{picture}(20,6) %Space for copyright notice \put(0,-0.95){\crnotice{\@toappear}} \end{picture} \fi \ifnum\ACM@basesize=10 \begin{picture}(20,7) %Space for copyright notice \put(0,-0.95){\crnotice{\@toappear}} \end{picture} \fi \end{center} \end@float} \def\@toappear{} % Default setting blank - commands below change this. \long\def\toappear#1{\def\@toappear{\parbox[b]{20pc}{\baselineskip 9pt#1}}} \def\toappearbox#1{\def\@toappear{\raisebox{5pt}{\framebox[20pc]{\parbox[b]{19pc}{#1}}}}} \newtoks\conf \newtoks\confinfo \def\conferenceinfo#1#2{\global\conf={#1}\global\confinfo{#2}} %\def\marginpar{\@latexerr{The \marginpar command is not allowed in the % `acmconf' document style.}\@eha} \def\marginpar{\ClassError{% \string\marginpar\space is not allowed in the `acmconf' document % January 2008 style}\@eha} \mark{{}{}} % Initializes TeX's marks \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \def\@begintheorem#1#2{% \parskip 0pt % GM July 2000 (for tighter spacing) \trivlist \item[% \hskip 10\p@ \hskip \labelsep {{\sc #1}\hskip 5\p@\relax#2.}% ] \it } \def\@opargbegintheorem#1#2#3{% \parskip 0pt % GM July 2000 (for tighter spacing) \trivlist \item[% \hskip 10\p@ \hskip \labelsep {\sc #1\ #2\ % This mod by Gerry to enumerate corollaries \setbox\@tempboxa\hbox{(#3)} % and bracket the 'corollary title' \ifdim \wd\@tempboxa>\z@ % and retain the correct numbering of e.g. theorems \hskip 5\p@\relax % if they occur 'around' said corollaries. \box\@tempboxa % Gerry - Nov. 1999. \fi.}% ] \it } \newif\if@qeded \global\@qededfalse % -- original %\def\proof{% % \vspace{-\parskip} % GM July 2000 (for tighter spacing) % \global\@qededfalse % \@ifnextchar[{\@xproof}{\@proof}% %} % -- end of original % (JSS) Fix for vertical spacing bug - Gerry Murray July 30th. 2002 \def\proof{% \vspace{-\lastskip}\vspace{-\parsep}\penalty-51% \global\@qededfalse \@ifnextchar[{\@xproof}{\@proof}% } \def\endproof{% \if@qeded\else\qed\fi \endtrivlist } \def\@proof{% \trivlist \item[% \hskip 10\p@ \hskip \labelsep {\sc Proof.}% ] \ignorespaces } \def\@xproof[#1]{% \trivlist \item[\hskip 10\p@\hskip \labelsep{\sc Proof #1.}]% \ignorespaces } \def\qed{% \unskip \kern 10\p@ \begingroup \unitlength\p@ \linethickness{.4\p@}% \framebox(6,6){}% \endgroup \global\@qededtrue } \def\newdef#1#2{% \expandafter\@ifdefinable\csname #1\endcsname {\@definecounter{#1}% \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}% \global\@namedef{#1}{\@defthm{#1}{#2}}% \global\@namedef{end#1}{\@endtheorem}% }% } \def\@defthm#1#2{% \refstepcounter{#1}% \@ifnextchar[{\@ydefthm{#1}{#2}}{\@xdefthm{#1}{#2}}% } \def\@xdefthm#1#2{% \@begindef{#2}{\csname the#1\endcsname}% \ignorespaces } \def\@ydefthm#1#2[#3]{% \trivlist \item[% \hskip 10\p@ \hskip \labelsep {\it #2% % \savebox\@tempboxa{#3}% \saveb@x\@tempboxa{#3}% % January 2008 \ifdim \wd\@tempboxa>\z@ \ \box\@tempboxa \fi.% }]% \ignorespaces } \def\@begindef#1#2{% \trivlist \item[% \hskip 10\p@ \hskip \labelsep {\it #1\ \rm #2.}% ]% } \def\theequation{\arabic{equation}} \newcounter{part} \newcounter{section} \newcounter{subsection}[section] \newcounter{subsubsection}[subsection] \newcounter{paragraph}[subsubsection] \def\thepart{\Roman{part}} \def\thesection{\arabic{section}} \def\thesubsection{\thesection.\arabic{subsection}} \def\thesubsubsection{\thesubsection.\arabic{subsubsection}} %removed \subsecfnt 29 July 2002 gkmt \def\theparagraph{\thesubsubsection.\arabic{paragraph}} %removed \subsecfnt 29 July 2002 gkmt \newif\if@uchead \@ucheadfalse %% CHANGES: NEW NOTE %% NOTE: OK to use old-style font commands below, since they were %% suitably redefined for LaTeX2e %% END CHANGES \setcounter{secnumdepth}{3} \def\part{% \@startsection{part}{9}{\z@}{-10\p@ \@plus -4\p@ \@minus -2\p@} {4\p@}{\normalsize\@ucheadtrue}% } \def\section{% \@startsection{section}{1}{\z@}{-10\p@ \@plus -4\p@ \@minus -2\p@}% GM {4\p@}{\baselineskip 14pt\secfnt\@ucheadtrue}% } \def\subsection{% \@startsection{subsection}{2}{\z@}{-8\p@ \@plus -2\p@ \@minus -\p@} {4\p@}{\secfnt}% } \def\subsubsection{% \@startsection{subsubsection}{3}{\z@}{-8\p@ \@plus -2\p@ \@minus -\p@}% {4\p@}{\subsecfnt}% } %\def\paragraph{% % \vskip 12pt\@startsection{paragraph}{3}{\z@}{6\p@ \@plus \p@}% original % {-5\p@}{\subsecfnt}% %} % If one wants sections, subsections and subsubsections numbered, % but not paragraphs, one usually sets secnumepth to 3. % For that, the "depth" of paragraphs must be given correctly % in the definition (``4'' instead of ``3'' as second argument % of @startsection): \def\paragraph{% \vskip 12pt\@startsection{paragraph}{4}{\z@}{6\p@ \@plus \p@}% % GM and Wolfgang May - 11/30/06 {-5\p@}{\subsecfnt}% } \let\@period=. \def\@startsection#1#2#3#4#5#6{% \if@noskipsec %gkmt, 11 aug 99 \global\let\@period\@empty \leavevmode \global\let\@period.% \fi \par % \@tempskipa #4\relax \@afterindenttrue \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \@afterindentfalse \fi \if@nobreak \everypar{}% \else \addpenalty\@secpenalty \addvspace\@tempskipa \fi \parskip=0pt % GM July 2000 (non numbered) section heads \@ifstar {\@ssect{#3}{#4}{#5}{#6}} {\@dblarg{\@sect{#1}{#2}{#3}{#4}{#5}{#6}}}% } \def\@sect#1#2#3#4#5#6[#7]#8{% \ifnum #2>\c@secnumdepth \let\@svsec\@empty \else \refstepcounter{#1}% \edef\@svsec{% \begingroup %\ifnum#2>2 \noexpand\rm \fi % changed to next 29 July 2002 gkmt \ifnum#2>2 \noexpand#6 \fi \csname the#1\endcsname \endgroup \ifnum #2=1\relax .\fi \hskip 1em }% \fi \@tempskipa #5\relax \ifdim \@tempskipa>\z@ \begingroup #6\relax \@hangfrom{\hskip #3\relax\@svsec}% \begingroup \interlinepenalty \@M \if@uchead \uppercase{#8}% \else #8% \fi \par \endgroup \endgroup \csname #1mark\endcsname{#7}% \vskip -12pt %gkmt, 11 aug 99 and GM July 2000 (was -14) - numbered section head spacing \addcontentsline{toc}{#1}{% \ifnum #2>\c@secnumdepth \else \protect\numberline{\csname the#1\endcsname}% \fi #7% }% \else \def\@svsechd{% #6% \hskip #3\relax \@svsec \if@uchead \uppercase{#8}% \else #8% \fi \csname #1mark\endcsname{#7}% \addcontentsline{toc}{#1}{% \ifnum #2>\c@secnumdepth \else \protect\numberline{\csname the#1\endcsname}% \fi #7% }% }% \fi \@xsect{#5}\hskip 1pt \par } \def\@xsect#1{% \@tempskipa #1\relax \ifdim \@tempskipa>\z@ \par \nobreak \vskip \@tempskipa \@afterheading \else \global\@nobreakfalse \global\@noskipsectrue \everypar{% \if@noskipsec \global\@noskipsecfalse \clubpenalty\@M \hskip -\parindent \begingroup \@svsechd \@period \endgroup \unskip \@tempskipa #1\relax \hskip -\@tempskipa \else \clubpenalty \@clubpenalty \everypar{}% \fi }% \fi \ignorespaces } \def\@trivlist{% \@topsepadd\topsep \if@noskipsec \global\let\@period\@empty \leavevmode \global\let\@period.% \fi \ifvmode \advance\@topsepadd\partopsep \else \unskip \par \fi \if@inlabel \@noparitemtrue \@noparlisttrue \else \@noparlistfalse \@topsep\@topsepadd \fi \advance\@topsep \parskip \leftskip\z@skip \rightskip\@rightskip \parfillskip\@flushglue \@setpar{\if@newlist\else{\@@par}\fi} \global\@newlisttrue \@outerparskip\parskip } %%% Actually, 'abbrev' works just fine as the default %%% Bibliography style. \typeout{Using 'Abbrev' bibliography style} \newcommand\bibyear[2]{% \unskip\quad\ignorespaces#1\unskip \if#2..\quad \else \quad#2 \fi } \newcommand{\bibemph}[1]{{\em#1}} \newcommand{\bibemphic}[1]{{\em#1\/}} \newcommand{\bibsc}[1]{{\sc#1}} \def\@normalcite{% \def\@cite##1##2{[##1\if@tempswa , ##2\fi]}% } \def\@citeNB{% \def\@cite##1##2{##1\if@tempswa , ##2\fi}% } \def\@citeRB{% \def\@cite##1##2{##1\if@tempswa , ##2\fi]}% } \def\start@cite#1#2{% \edef\citeauthoryear##1##2##3{% ###1% \ifnum#2=\z@ \else\ ###2\fi }% \ifnum#1=\thr@@ \let\@@cite\@citeyear \else \let\@@cite\@citenormal \fi \@ifstar{\@citeNB\@@cite}{\@normalcite\@@cite}% } %\def\cite{\start@cite23} \DeclareRobustCommand\cite{\start@cite23} % January 2008 \def\citeNP{\cite} % No Parentheses e.g. 5 %\def\citeA{\start@cite10} \DeclareRobustCommand\citeA{\start@cite10} % January 2008 \def\citeANP{\citeA} %\def\shortcite{\start@cite23} \DeclareRobustCommand\shortcite{\start@cite23} % January 2008 \def\shortciteNP{\shortcite} %\def\shortciteA{\start@cite20} \DeclareRobustCommand\shortciteA{\start@cite20} % January 2008 \def\shortciteANP{\shortciteA} %\def\citeyear{\start@cite30} \DeclareRobustCommand\citeyear{\start@cite30} % January 2008 \def\citeyearNP{\citeyear} %\def\citeN{% \DeclareRobustCommand\citeN{% % January 2008 \@citeRB \def\citeauthoryear##1##2##3{##1\ [##3% \def\reserved@a{##1}% \def\citeauthoryear####1####2####3{% \def\reserved@b{####1}% \ifx\reserved@a\reserved@b ####3% \else \errmessage{Package acmart Error: author mismatch in \string\citeN^^J^^J% See the acmart package documentation for explanation}% \fi }% }% \@ifstar\@citeyear\@citeyear } %\def\shortciteN{% \DeclareRobustCommand\shortciteN{% % January 2008 \@citeRB \def\citeauthoryear##1##2##3{##2\ [##3% \def\reserved@a{##2}% \def\citeauthoryear####1####2####3{% \def\reserved@b{####2}% \ifx\reserved@a\reserved@b ####3% \else \errmessage{Package acmart Error: author mismatch in \string\shortciteN^^J^^J% See the acmart package documentation for explanation}% \fi }% }% \@ifstar\@citeyear\@citeyear % GM July 2000 } \def\@citenormal{% \@ifnextchar [{\@tempswatrue\@citex;}% % original {\@tempswafalse\@citex,[]}% was ; Gerry 2/24/00 {\@tempswafalse\@citex[]}% % GERRY FIX FOR BABEL 3/20/2009 } \def\@citeyear{% \@ifnextchar [{\@tempswatrue\@citex,}% % original {\@tempswafalse\@citex,[]}% {\@tempswafalse\@citex[]}% % GERRY FIX FOR BABEL 3/20/2009 } \def\@citex#1[#2]#3{% \let\@citea\@empty \@cite{% \@for\@citeb:=#3\do{% \@citea % original \def\@citea{#1 }% \def\@citea{#1, }% % GERRY FIX FOR BABEL 3/20/2009 -- SO THAT YOU GET [1, 2] IN THE BODY TEXT \edef\@citeb{\expandafter\@iden\@citeb}% \if@filesw \immediate\write\@auxout{\string\citation{\@citeb}}% \fi \@ifundefined{b@\@citeb}{% {\bf ?}% \@warning{% Citation `\@citeb' on page \thepage\space undefined% }% }% {\csname b@\@citeb\endcsname}% }% }{#2}% } %\let\@biblabel\@gobble % Dec. 2008 - Gerry % ---- \def\@biblabelnum#1{[#1]} % Gerry's solution #1 - for Natbib -- April 2009 \let\@biblabel=\@biblabelnum % Gerry's solution #1 - for Natbib -- April 2009 \def\newblock{\relax} % Gerry Dec. 2008 % --- \newdimen\bibindent \setcounter{enumi}{1} \bibindent=0em \def\thebibliography#1{% \ifnum\addauflag=0\addauthorsection\global\addauflag=1\fi \section[References]{% <=== OPTIONAL ARGUMENT ADDED HERE {References} % was uppercased but this affects pdf bookmarks (SP/GM October 2004) {\vskip -9pt plus 1pt} % GM Nov. 2006 / GM July 2000 (for somewhat tighter spacing) \@mkboth{{\refname}}{{\refname}}% }% \list{[\arabic{enumi}]}{% \settowidth\labelwidth{[#1]}% \leftmargin\labelwidth \advance\leftmargin\labelsep \advance\leftmargin\bibindent \parsep=0pt\itemsep=1pt % GM July 2000 \itemindent -\bibindent \listparindent \itemindent \usecounter{enumi} }% \let\newblock\@empty \raggedright % GM July 2000 \sloppy \sfcode`\.=1000\relax } \gdef\balancecolumns {\vfill\eject \global\@colht=\textheight \global\ht\@cclv=\textheight } \newcount\colcntr \global\colcntr=0 %\newbox\savebox \newbox\saveb@x % January 2008 \gdef \@makecol {% \global\advance\colcntr by 1 \ifnum\colcntr>2 \global\colcntr=1\fi \ifvoid\footins \setbox\@outputbox \box\@cclv \else \setbox\@outputbox \vbox{% \boxmaxdepth \@maxdepth \@tempdima\dp\@cclv \unvbox \@cclv \vskip-\@tempdima \vskip \skip\footins \color@begingroup \normalcolor \footnoterule \unvbox \footins \color@endgroup }% \fi \xdef\@freelist{\@freelist\@midlist}% \global \let \@midlist \@empty \@combinefloats \ifvbox\@kludgeins \@makespecialcolbox \else \setbox\@outputbox \vbox to\@colht {% \@texttop \dimen@ \dp\@outputbox \unvbox \@outputbox \vskip -\dimen@ \@textbottom }% \fi \global \maxdepth \@maxdepth } \def\titlenote{\@ifnextchar[\@xtitlenote{\stepcounter\@mpfn \global\advance\titlenotecount by 1 \ifnum\titlenotecount=1 \raisebox{9pt}{$\ast$} \fi \ifnum\titlenotecount=2 \raisebox{9pt}{$\dagger$} \fi \ifnum\titlenotecount=3 \raisebox{9pt}{$\ddagger$} \fi \ifnum\titlenotecount=4 \raisebox{9pt}{$\S$} \fi \ifnum\titlenotecount=5 \raisebox{9pt}{$\P$} \fi \@titlenotetext }} \long\def\@titlenotetext#1{\insert\footins{% \ifnum\titlenotecount=1\global\tntoks={#1}\fi \ifnum\titlenotecount=2\global\tntokstwo={#1}\fi \ifnum\titlenotecount=3\global\tntoksthree={#1}\fi \ifnum\titlenotecount=4\global\tntoksfour={#1}\fi \ifnum\titlenotecount=5\global\tntoksfive={#1}\fi \reset@font\footnotesize \interlinepenalty\interfootnotelinepenalty \splittopskip\footnotesep \splitmaxdepth \dp\strutbox \floatingpenalty \@MM \hsize\columnwidth \@parboxrestore \protected@edef\@currentlabel{% }% \color@begingroup \color@endgroup}} %%%%%%%%%%%%%%%%%%%%%%%%% \ps@plain \baselineskip=11pt \let\thepage\relax % For NO page numbers - GM Nov. 30th. 1999 and July 2000 \def\setpagenumber#1{\global\setcounter{page}{#1}} %\pagenumbering{arabic} % Arabic page numbers GM July 2000 \twocolumn % Double column. \flushbottom % Even bottom -- alas, does not balance columns at end of document \pagestyle{plain} % Need Copyright Year and Copyright Data to be user definable (in .tex file). % Gerry Nov. 30th. 1999 \newtoks\copyrtyr \newtoks\acmcopyr \newtoks\boilerplate \global\acmcopyr={X-XXXXX-XX-X/XX/XX} % Default - 5/11/2001 * Gerry \global\copyrtyr={\the\year} % Default - 3/3/2003 * Gerry \def\acmPrice#1{\gdef\@acmPrice{#1}} \acmPrice{} %article price % Changed to 15 - June 2012 - Gerry \def\CopyrightYear#1{\global\copyrtyr{#1}} \def\crdata#1{\global\acmcopyr{#1}} \def\permission#1{\global\boilerplate{#1}} % ISBN % \def\isbn#1{\global\acmcopyr={#1}} \isbn{978-1-4503-2138-9} \RequirePackage{url} \urlstyle{rm} \def\doi#1{\def\@doi{#1}} \doi{10.1145/1235} \def\printdoi#1{\url{#1}} % Copyright \RequirePackage{acmcopyright} \setcopyright{none} % \global\boilerplate={\@copyrightpermission} \newtoks\copyrightetc \ifnum\ACM@basesize=9\relax \global\copyrightetc{% {\noindent\confname\ \the\conf\ \the\confinfo}\par\smallskip \if@printcopyright \copyright\ \the\copyrtyr\ \@copyrightowner \fi \if@acmowned ISBN \else\ifnum\acm@copyrightmode=2 ISBN \else \par\smallskip ACM ISBN \fi\fi \the\acmcopyr\ifx\@acmPrice\@empty.\else\dots\@acmPrice\fi\par\smallskip {DOI: \small\expandafter\printdoi\expandafter{\@doi}}} \toappear{\fontsize{7pt}{8pt}\fontfamily{ptm}\selectfont \the\boilerplate\par\smallskip \the\copyrightetc} \fi \ifnum\ACM@basesize=10\relax \global\copyrightetc{% {\noindent\confname\ \the\conf\ \the\confinfo}\par\smallskip \if@printcopyright \copyright\ \the\copyrtyr\ \@copyrightowner \fi \if@acmowned ISBN \else\ifnum\acm@copyrightmode=2 ISBN \else \par\smallskip ACM ISBN \fi\fi \the\acmcopyr\ifx\@acmPrice\@empty.\else\dots\@acmPrice\fi\par\smallskip {DOI: \small\expandafter\printdoi\expandafter{\@doi}}} \toappear{\fontsize{7.5pt}{8.5pt}\fontfamily{ptm}\selectfont \the\boilerplate\par\smallskip \the\copyrightetc} \fi %\DeclareFixedFont{\altcrnotice}{OT1}{tmr}{m}{n}{8} % << patch needed for accenting e.g. Montreal - Gerry, May 2007 %\DeclareFixedFont{\altconfname}{OT1}{tmr}{m}{it}{8} % << patch needed for accenting in italicized confname - Gerry, May 2007 % %{\altconfname{{\the\conf}}} {\altcrnotice\the\confinfo\par} \the\copyrightetc.} % << Gerry, May 2007 % % The following section (i.e. 3 .sty inclusions) was added in May 2007 so as to fix the problems that many % authors were having with accents. Sometimes accents would occur, but the letter-character would be of a different % font. Conversely the letter-character font would be correct but, e.g. a 'bar' would appear superimposed on the % character instead of, say, an unlaut/diaresis. Sometimes the letter-character would NOT appear at all. % Using [T1]{fontenc} outright was not an option as this caused 99% of the authors to 'produce' a Type-3 (bitmapped) % PDF file - useless for production. % % For proper (font) accenting we NEED these packages to be part of the .cls file i.e. 'ae', 'aecompl' and 'aeguil' % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ %% This is file `ae.sty' \def\fileversion{1.3} \def\filedate{2001/02/12} \NeedsTeXFormat{LaTeX2e} %\ProvidesPackage{ae}[\filedate\space\fileversion\space % GM % Almost European Computer Modern] % GM - keeping the log file clean(er) \newif\if@ae@slides \@ae@slidesfalse \DeclareOption{slides}{\@ae@slidestrue} \ProcessOptions \fontfamily{aer} \RequirePackage[T1]{fontenc} \if@ae@slides \renewcommand{\sfdefault}{laess} \renewcommand{\rmdefault}{laess} % no roman \renewcommand{\ttdefault}{laett} \else \renewcommand{\sfdefault}{aess} \renewcommand{\rmdefault}{aer} \renewcommand{\ttdefault}{aett} \fi \endinput %% %% End of file `ae.sty'. % % \def\fileversion{0.9} \def\filedate{1998/07/23} \NeedsTeXFormat{LaTeX2e} %\ProvidesPackage{aecompl}[\filedate\space\fileversion\space % GM %T1 Complements for AE fonts (D. Roegel)] % GM -- keeping the log file clean(er) \def\@ae@compl#1{{\fontencoding{T1}\fontfamily{cmr}\selectfont\symbol{#1}}} \def\guillemotleft{\@ae@compl{19}} \def\guillemotright{\@ae@compl{20}} \def\guilsinglleft{\@ae@compl{14}} \def\guilsinglright{\@ae@compl{15}} \def\TH{\@ae@compl{222}} \def\NG{\@ae@compl{141}} \def\ng{\@ae@compl{173}} \def\th{\@ae@compl{254}} \def\DJ{\@ae@compl{208}} \def\dj{\@ae@compl{158}} \def\DH{\@ae@compl{208}} \def\dh{\@ae@compl{240}} \def\@perthousandzero{\@ae@compl{24}} \def\textperthousand{\%\@perthousandzero} \def\textpertenthousand{\%\@perthousandzero\@perthousandzero} \endinput % % %% This is file `aeguill.sty' % This file gives french guillemets (and not guillemots!) % built with the Polish CMR fonts (default), WNCYR fonts, the LASY fonts % or with the EC fonts. % This is useful in conjunction with the ae package % (this package loads the ae package in case it has not been loaded) % and with or without the french(le) package. % % In order to get the guillemets, it is necessary to either type % \guillemotleft and \guillemotright, or to use an 8 bit encoding % (such as ISO-Latin1) which selects these two commands, % or, if you use the french package (but not the frenchle package), % to type << or >>. % % By default, you get the Polish CMR guillemets; if this package is loaded % with the `cm' option, you get the LASY guillemets; with `ec,' you % get the EC guillemets, and with `cyr,' you get the cyrillic guillemets. % % In verbatim mode, you always get the EC/TT guillemets. % % The default option is interesting in conjunction with PDF, % because there is a Type 1 version of the Polish CMR fonts % and these guillemets are very close in shape to the EC guillemets. % There are no free Type 1 versions of the EC fonts. % % Support for Polish CMR guillemets was kindly provided by % Rolf Niepraschk <niepraschk@ptb.de> in version 0.99 (2000/05/22). % Bernd Raichle provided extensive simplifications to the code % for version 1.00. % % This package is released under the LPPL. % % Changes: % Date version % 2001/04/12 1.01 the frenchle and french package are now distinguished. % \def\fileversion{1.01} \def\filedate{2001/04/12} \NeedsTeXFormat{LaTeX2e} %\ProvidesPackage{aeguill}[2001/04/12 1.01 % % GM %AE fonts with french guillemets (D. Roegel)] % GM - keeping the log file clean(er) %\RequirePackage{ae} % GM May 2007 - already embedded here \newcommand{\@ae@switch}[4]{#4} \DeclareOption{ec}{\renewcommand\@ae@switch[4]{#1}} \DeclareOption{cm}{\renewcommand\@ae@switch[4]{#2}} \DeclareOption{cyr}{\renewcommand\@ae@switch[4]{#3}} \DeclareOption{pl}{\renewcommand\@ae@switch[4]{#4}} % % Load necessary packages % \@ae@switch{% ec % do nothing }{% cm \RequirePackage{latexsym}% GM - May 2007 - already 'mentioned as required' up above }{% cyr \RequirePackage[OT2,T1]{fontenc}% }{% pl \RequirePackage[OT4,T1]{fontenc}% } % The following command will be compared to \frenchname, % as defined in french.sty and frenchle.sty. \def\aeguillfrenchdefault{french}% \let\guill@verbatim@font\verbatim@font \def\verbatim@font{\guill@verbatim@font\ecguills{cmtt}% \let\guillemotleft\@oguills\let\guillemotright\@fguills} \begingroup \catcode`\<=13 \catcode`\>=13 \def\x{\endgroup \def\ae@lfguill{<<}% \def\ae@rfguill{>>}% }\x \newcommand{\ecguills}[1]{% \def\selectguillfont{\fontencoding{T1}\fontfamily{#1}\selectfont}% \def\@oguills{{\selectguillfont\symbol{19}}}% \def\@fguills{{\selectguillfont\symbol{20}}}% } \newcommand{\aeguills}{% \ae@guills % We redefine \guillemotleft and \guillemotright % in order to catch them when they are used % with \DeclareInputText (in latin1.def for instance) % We use \auxWARNINGi as a safe indicator that french.sty is used. \gdef\guillemotleft{\ifx\auxWARNINGi\undefined \@oguills % neither french.sty nor frenchle.sty \else \ifx\aeguillfrenchdefault\frenchname \ae@lfguill % french.sty \else \@oguills % frenchle.sty \fi \fi}% \gdef\guillemotright{\ifx\auxWARNINGi\undefined \@fguills % neither french.sty nor frenchle.sty \else \ifx\aeguillfrenchdefault\frenchname \ae@rfguill % french.sty \else \@fguills % frenchle.sty \fi \fi}% } % % Depending on the class option % define the internal command \ae@guills \@ae@switch{% ec \newcommand{\ae@guills}{% \ecguills{cmr}}% }{% cm \newcommand{\ae@guills}{% \def\selectguillfont{\fontencoding{U}\fontfamily{lasy}% \fontseries{m}\fontshape{n}\selectfont}% \def\@oguills{\leavevmode\nobreak \hbox{\selectguillfont (\kern-.20em(\kern.20em}\nobreak}% \def\@fguills{\leavevmode\nobreak \hbox{\selectguillfont \kern.20em)\kern-.2em)}% \ifdim\fontdimen\@ne\font>\z@\/\fi}}% }{% cyr \newcommand{\ae@guills}{% \def\selectguillfont{\fontencoding{OT2}\fontfamily{wncyr}\selectfont}% \def\@oguills{{\selectguillfont\symbol{60}}}% \def\@fguills{{\selectguillfont\symbol{62}}}} }{% pl \newcommand{\ae@guills}{% \def\selectguillfont{\fontencoding{OT4}\fontfamily{cmr}\selectfont}% \def\@oguills{{\selectguillfont\symbol{174}}}% \def\@fguills{{\selectguillfont\symbol{175}}}} } \AtBeginDocument{% \ifx\GOfrench\undefined \aeguills \else \let\aeguill@GOfrench\GOfrench \gdef\GOfrench{\aeguill@GOfrench \aeguills}% \fi } \endinput %

Ignore Space Show notes View

APCCM_2017/sig-alternate-05-2015.cls 0 → 100644

% SIG-ALTERNATE.CLS - VERSION 2.8
% "COMPATIBLE" WITH THE "ACM_PROC_ARTICLE-SP.CLS" V3.2SP
% Gerald Murray - May 23rd 2012
% Boris Veytsman - April 23 2013
% Boris Veytsman - May 12 2013
% Boris Veytsman - June 09 2013
% Boris Veytsman - August 12 2013
%
% ---- Start of 'updates'  ----
% Added new permission/copyright statement - BV
% Changed $10 fee to $15 -- May 2012  --  Gerry
% Changed $5 fee to $10 --  April 2009 -- Gerry
% April 22nd. 2009 - Fixed 'Natbib' incompatibility problem - Gerry
% April 22nd. 2009 - Fixed 'Babel' incompatibility problem - Gerry
% April 22nd. 2009 - Inserted various bug-fixes and improvements - Gerry
%
% To produce Type 1 fonts in the document plus allow for 'normal LaTeX accenting' in the critical areas;
% title, author block, section-heads, confname, etc. etc. 
% i.e. the whole purpose of this version update is to NOT resort to 'inelegant accent patches'.
% After much research, three extra .sty packages were added to the the tail (ae, aecompl, aeguill) to solve,
% in particular, the accenting problem(s). We _could_ ask authors (via instructions/sample file) to 'include' these in
% the source .tex file - in the preamble - but if everything is already provided ('behind the scenes' - embedded IN the .cls)
% then this is less work for authors and also makes everything appear 'vanilla'.
% NOTE: all 'patchwork accenting" has been commented out (here) and is no longer 'used' in the sample .tex file (either).
% Gerry June 2007
%
% Patch for accenting in conference name/location. Gerry May 3rd. 2007
% Rule widths changed to .5, author count (>6) fixed, roll-back for Type 3 problem. Gerry March 20th. 2007
% Changes made to 'modernize' the fontnames but esp. for MikTeX users V2.4/2.5 - Nov. 30th. 2006
% Updated the \email definition to allow for its use inside of 'shared affiliations' - Nov. 30th. 2006
% Fixed the 'section number depth value' - Nov. 30th. 2006
%
% Footnotes inside table cells using \minipage (Oct. 2002)
% Georgia fixed bug in sub-sub-section numbering in paragraphs (July 29th. 2002)
% JS/GM fix to vertical spacing before Proofs (July 30th. 2002)
%
% Made the Permission Statement / Conference Info / Copyright Info
% 'user definable' in the source .tex file OR automatic if
% not specified.
%
% Allowance made to switch default fonts between those systems using
% normal/modern font names and those using 'Type 1' or 'Truetype' fonts.
% See LINE NUMBER 255 for details.
% Also provided for enumerated/annotated Corollaries 'surrounded' by
% enumerated Theorems (line 848).
% Gerry November 11th. 1999
%
% ---- End of 'updates' ----
%
\def\fileversion{v2.9}          % for ACM's tracking purposes
\def\filedate{August 12, 2013}    % Gerry Murray's tracking data
\def\docdate {\filedate}
\usepackage{epsfig}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% Need this for accents in Arial/Helvetica
%\usepackage[T1]{fontenc}  % Gerry March 12, 2007 - causes Type 3 problems (body text)
%\usepackage{textcomp}
%
% SIG-ALTERNATE DOCUMENT STYLE
% G.K.M. Tobin August-October 1999
%    adapted from ARTICLE document style by Ken Traub, Olin Shivers
%    also using elements of esub2acm.cls
% HEAVILY MODIFIED, SUBSEQUENTLY, BY GERRY MURRAY 2000
% ARTICLE DOCUMENT STYLE -- Released 16 March 1988
%    for LaTeX version 2.09
% Copyright (C) 1988 by Leslie Lamport
%
%
%%% sig-alternate.cls is an 'ALTERNATE' document style for producing
%%% two-column camera-ready pages for ACM conferences.
%%% THIS FILE DOES NOT STRICTLY ADHERE TO THE SIGS (BOARD-ENDORSED)
%%% PROCEEDINGS STYLE. It has been designed to produce a 'tighter'
%%% paper in response to concerns over page budgets.
%%% The main features of this style are:
%%%
%%% 1)  Two columns.
%%% 2)  Side and top margins of 4.5pc, bottom margin of 6pc, column gutter of
%%%     2pc, hence columns are 20pc wide and 55.5pc tall.  (6pc =3D 1in, approx)
%%% 3)  First page has title information, and an extra 6pc of space at the
%%%     bottom of the first column for the ACM copyright notice.
%%% 4)  Text is 9pt on 10pt baselines; titles (except main) are 9pt bold.
%%%
%%%
%%% There are a few restrictions you must observe:
%%%
%%% 1)  You cannot change the font size; ACM wants you to use 9pt.
%%% 3)  You must start your paper with the \maketitle command.  Prior to the
%%%     \maketitle you must have \title and \author commands.  If you have a
%%%     \date command it will be ignored; no date appears on the paper, since
%%%     the proceedings will have a date on the front cover.
%%% 4)  Marginal paragraphs, tables of contents, lists of figures and tables,
%%%     and page headings are all forbidden.
%%% 5)  The `figure' environment will produce a figure one column wide; if you
%%%     want one that is two columns wide, use `figure*'.
%%%
%
%%% Copyright Space:
%%% This style automatically reserves 1" blank space at the bottom of page 1/
%%% column 1.  This space can optionally be filled with some text using the
%%% \toappear{...} command.  If used, this command must be BEFORE the \maketitle
%%% command.  If this command is defined AND [preprint] is on, then the
%%% space is filled with the {...} text (at the bottom); otherwise, it is
%%% blank.  If you use \toappearbox{...} instead of \toappear{...} then a
%%% box will be drawn around the text (if [preprint] is on).
%%%
%%% A typical usage looks like this:
%%%     \toappear{To appear in the Ninth AES Conference on Medievil Lithuanian
%%%               Embalming Technique, June 1991, Alfaretta, Georgia.}
%%% This will be included in the preprint, and left out of the conference
%%% version.
%%%
%%% WARNING:
%%% Some dvi-ps converters heuristically allow chars to drift from their
%%% true positions a few pixels. This may be noticeable with the 9pt sans-serif
%%% bold font used for section headers.
%%% You may turn this hackery off via the -e option:
%%%     dvips -e 0 foo.dvi >foo.ps
%%%
\typeout{Document Class 'sig-alternate' <9th June '13>.  Modified by
  G.K.M. Tobin/Gerry Murray/Boris Veytsman}
\typeout{Based in part upon document Style `acmconf' <22 May 89>. Hacked 4/91 by}
\typeout{shivers@cs.cmu.edu, 4/93 by theobald@cs.mcgill.ca}
\typeout{Excerpts were taken from (Journal Style) 'esub2acm.cls'.}
\typeout{****** Bugs/comments/suggestions/technicalities to Gerry Murray -- murray@hq.acm.org ******}
\typeout{Questions on the style, SIGS policies, etc. to Adrienne Griscti griscti@acm.org}

% New option code by BV

\newcount\ACM@basesize
\ACM@basesize=9\relax
\DeclareOption{9pt}{\ACM@basesize=9\relax}
\DeclareOption{10pt}{\ACM@basesize=10\relax}
\DeclareOption{11pt}{\ClassError{sig-alternate}{The `11pt' option is
    not allowed}{sig-alternate now exists in 9pt and 10pt versions only}}
\DeclareOption{12pt}{\ClassError{sig-alternate}{The `12pt' option is
    not allowed}{sig-alternate now exists in 9pt and 10pt versions only}}

\ExecuteOptions{9pt}
\ProcessOptions

\let\@concepts\@empty
% Support for CCSXML file
\RequirePackage{comment}
\excludecomment{CCSXML}

% New concepts scheme
%
% The first argument is the significance, the
% second is the concept(s)
%
\newcommand\ccsdesc[2][100]{%
  \ccsdesc@parse#1~#2~}
%
% The parser of the expression Significance~General~Specific
%
\def\ccsdesc@parse#1~#2~#3~{%
  \expandafter\ifx\csname CCS@#2\endcsname\relax
    \expandafter\gdef\csname CCS@#2\endcsname{\textbullet\textbf{#2} $\to$ }%
  \g@addto@macro{\@concepts}{\csname CCS@#2\endcsname}\fi
  \expandafter\g@addto@macro\expandafter{\csname CCS@#2\endcsname}{%
    \ifnum#1>499\textbf{#3; }\else
    \ifnum#1>299\textit{#3; }\else
    #3; \fi\fi}}

\newcommand\printccsdesc{%
  \ifx\@concepts\@empty\else
  \if@twocolumn
    \section*{CCS Concepts}
    \@concepts
    \else \small
    \quotation{\@concepts}%
    \fi
    \fi}

\def\doi#1{\def\@doi{#1}}
\doi{http://dx.doi.org/10.1145/0000000.0000000}

\oddsidemargin 4.5pc
\evensidemargin 4.5pc
\advance\oddsidemargin by -1in  % Correct for LaTeX gratuitousness
\advance\evensidemargin by -1in % Correct for LaTeX gratuitousness
\marginparwidth 0pt             % Margin pars are not allowed.
\marginparsep 11pt              % Horizontal space between outer margin and
                                % marginal note

% Top of page:
\topmargin 4.5pc                % Nominal distance from top of page to top of
                                % box containing running head.
\advance\topmargin by -1in      % Correct for LaTeX gratuitousness
\headheight 0pt                 % Height of box containing running head.
\headsep 0pt                    % Space between running head and text.
                                % Bottom of page:
\footskip 30pt                  % Distance from baseline of box containing foot
                                % to baseline of last line of text.
\@ifundefined{footheight}{\newdimen\footheight}{}% this is for LaTeX2e
\footheight 12pt                % Height of box containing running foot.

%% Must redefine the top margin so there's room for headers and
%% page numbers if you are using the preprint option. Footers
%% are OK as is. Olin.
\advance\topmargin by -37pt     % Leave 37pt above text for headers
\headheight 12pt                % Height of box containing running head.
\headsep 25pt                   % Space between running head and text.

\textheight 666pt       % 9 1/4 column height
\textwidth 42pc         % Width of text line.
                        % For two-column mode:
\columnsep 2pc          %    Space between columns
\columnseprule 0pt      %    Width of rule between columns.
\hfuzz 1pt              % Allow some variation in column width, otherwise it's
                        % too hard to typeset in narrow columns.

\ifnum\ACM@basesize=9\relax
\footnotesep 5.6pt      % Height of strut placed at the beginning of every
                        % footnote =3D height of normal \footnotesize strut,
                        % so no extra space between footnotes.
\fi
\ifnum\ACM@basesize=10\relax
\footnotesep 6.22pt      % Height of strut placed at the beginning of every
                        % footnote =3D height of normal \footnotesize strut,
                        % so no extra space between footnotes.
\fi

\skip\footins 8.1pt plus 4pt minus 2pt  % Space between last line of text and
                                        % top of first footnote.
\floatsep 11pt plus 2pt minus 2pt       % Space between adjacent floats moved
                                        % to top or bottom of text page.
\textfloatsep 18pt plus 2pt minus 4pt   % Space between main text and floats
                                        % at top or bottom of page.
\intextsep 11pt plus 2pt minus 2pt      % Space between in-text figures and
                                        % text.
\@ifundefined{@maxsep}{\newdimen\@maxsep}{}% this is for LaTeX2e
\@maxsep 18pt                           % The maximum of \floatsep,
                                        % \textfloatsep and \intextsep (minus
                                        % the stretch and shrink).
\dblfloatsep 11pt plus 2pt minus 2pt    % Same as \floatsep for double-column
                                        % figures in two-column mode.
\dbltextfloatsep 18pt plus 2pt minus 4pt% \textfloatsep for double-column
                                        % floats.
\@ifundefined{@dblmaxsep}{\newdimen\@dblmaxsep}{}% this is for LaTeX2e
\@dblmaxsep 18pt                        % The maximum of \dblfloatsep and
                                        % \dbltexfloatsep.
\@fptop 0pt plus 1fil    % Stretch at top of float page/column. (Must be
                         % 0pt plus ...)
\@fpsep 8pt plus 2fil    % Space between floats on float page/column.
\@fpbot 0pt plus 1fil    % Stretch at bottom of float page/column. (Must be
                         % 0pt plus ... )
\@dblfptop 0pt plus 1fil % Stretch at top of float page. (Must be 0pt plus ...)
\@dblfpsep 8pt plus 2fil % Space between floats on float page.
\@dblfpbot 0pt plus 1fil % Stretch at bottom of float page. (Must be
                         % 0pt plus ... )
\marginparpush 5pt       % Minimum vertical separation between two marginal
                         % notes.

\parskip 0pt plus 1pt            % Extra vertical space between
                                % paragraphs.
\ifnum\ACM@basesize=9\relax
\parindent 9pt  % GM July 2000 / was 0pt - width of paragraph
                % indentation.
\fi
\ifnum\ACM@basesize=10\relax
\parindent 10pt  % GM July 2000 / was 0pt - width of paragraph
                % indentation.
\fi
\partopsep 2pt plus 1pt minus 1pt% Extra vertical space, in addition to
                                 % \parskip and \topsep, added when user
                                 % leaves blank line before environment.

\@lowpenalty   51       % Produced by \nopagebreak[1] or \nolinebreak[1]
\@medpenalty  151       % Produced by \nopagebreak[2] or \nolinebreak[2]
\@highpenalty 301       % Produced by \nopagebreak[3] or \nolinebreak[3]

\@beginparpenalty -\@lowpenalty % Before a list or paragraph environment.
\@endparpenalty   -\@lowpenalty % After a list or paragraph environment.
\@itempenalty     -\@lowpenalty % Between list items.

\RequirePackage{ifpdf}%
\ifpdf
\pdfpagewidth=8.5in
\pdfpageheight=11in
\fi

\lineskip 2pt           % \lineskip is 1pt for all font sizes.
\normallineskip 2pt
\def\baselinestretch{1}

\ifnum\ACM@basesize=9\relax
\abovedisplayskip 9pt plus2pt minus4.5pt%
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip  \z@ plus3pt%
\belowdisplayshortskip  5.4pt plus3pt minus3pt%
\let\@listi\@listI     % Setting of \@listi added 9 Jun 87

\def\small{\@setsize\small{9pt}\viiipt\@viiipt
\abovedisplayskip 7.6pt plus 3pt minus 4pt%
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus2pt%
\belowdisplayshortskip 3.6pt plus2pt minus 2pt
\def\@listi{\leftmargin\leftmargini %% Added 22 Dec 87
\topsep 4pt plus 2pt minus 2pt\parsep 2pt plus 1pt minus 1pt
\itemsep \parsep}}

\def\footnotesize{\@setsize\footnotesize{9pt}\ixpt\@ixpt
\abovedisplayskip 6.4pt plus 2pt minus 4pt%
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus 1pt%
\belowdisplayshortskip 2.7pt plus 1pt minus 2pt
\def\@listi{\leftmargin\leftmargini %% Added 22 Dec 87
\topsep 3pt plus 1pt minus 1pt\parsep 2pt plus 1pt minus 1pt
\itemsep \parsep}}
\fi

\ifnum\ACM@basesize=10\relax
\abovedisplayskip 10pt plus2pt minus4.5pt%
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip  \z@ plus3pt%
\belowdisplayshortskip  6pt plus3pt minus3pt%
\let\@listi\@listI     % Setting of \@listi added 9 Jun 87

\def\small{\@setsize\small{10pt}\ixpt\@ixpt
\abovedisplayskip 8.5pt plus 3pt minus 4pt%
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus2pt%
\belowdisplayshortskip 4pt plus2pt minus 2pt
\def\@listi{\leftmargin\leftmargini %% Added 22 Dec 87
\topsep 4.5pt plus 2pt minus 2pt\parsep 2pt plus 1pt minus 1pt
\itemsep \parsep}}

\def\footnotesize{\@setsize\footnotesize{10pt}\xpt\@xpt
\abovedisplayskip 7.6pt plus 2pt minus 4pt%
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip \z@ plus 1pt%
\belowdisplayshortskip 3.0pt plus 1pt minus 2pt
\def\@listi{\leftmargin\leftmargini %% Added 22 Dec 87
\topsep 3.2pt plus 1pt minus 1pt\parsep 2pt plus 1pt minus 1pt
\itemsep \parsep}}
\fi

\newcount\aucount
\newcount\originalaucount
\newdimen\auwidth
\auwidth=\textwidth
\newdimen\auskip
\newcount\auskipcount
\newdimen\auskip
\global\auskip=1pc
\newdimen\allauboxes
\allauboxes=\auwidth
\newtoks\addauthors
\newcount\addauflag
\global\addauflag=0 %Haven't shown additional authors yet

\newtoks\subtitletext
\gdef\subtitle#1{\subtitletext={#1}}

\gdef\additionalauthors#1{\addauthors={#1}}

\gdef\numberofauthors#1{\global\aucount=#1
\ifnum\aucount>3\global\originalaucount=\aucount \global\aucount=3\fi %g}  % 3 OK - Gerry March 2007
\global\auskipcount=\aucount\global\advance\auskipcount by 1
\global\multiply\auskipcount by 2
\global\multiply\auskip by \auskipcount
\global\advance\auwidth by -\auskip
\global\divide\auwidth by \aucount}

% \and was modified to count the number of authors.  GKMT 12 Aug 1999
\def\alignauthor{%                  % \begin{tabular}
\end{tabular}%
  \begin{tabular}[t]{p{\auwidth}}\centering}%

%  *** NOTE *** NOTE *** NOTE *** NOTE ***
%  If you have 'font problems' then you may need
%  to change these, e.g. 'arialb' instead of "arialbd".
%  Gerry Murray 11/11/1999
%  *** OR ** comment out block A and activate block B or vice versa.
% **********************************************
%
%  -- Start of block A -- (Type 1 or Truetype fonts)
%\newfont{\secfnt}{timesbd at 12pt} % was timenrb originally - now is timesbd
%\newfont{\secit}{timesbi at 12pt}   %13 Jan 00 gkmt
%\newfont{\subsecfnt}{timesi at 11pt} % was timenrri originally - now is timesi
%\newfont{\subsecit}{timesbi at 11pt} % 13 Jan 00 gkmt -- was times changed to timesbi gm 2/4/2000
%                         % because "normal" is italic, "italic" is Roman
%\newfont{\ttlfnt}{arialbd at 18pt} % was arialb originally - now is arialbd
%\newfont{\ttlit}{arialbi at 18pt}    % 13 Jan 00 gkmt
%\newfont{\subttlfnt}{arial at 14pt} % was arialr originally - now is arial
%\newfont{\subttlit}{ariali at 14pt} % 13 Jan 00 gkmt
%\newfont{\subttlbf}{arialbd at 14pt}  % 13 Jan 00 gkmt
%\newfont{\aufnt}{arial at 12pt} % was arialr originally - now is arial
%\newfont{\auit}{ariali at 12pt} % 13 Jan 00 gkmt
%\newfont{\affaddr}{arial at 10pt} % was arialr originally - now is arial
%\newfont{\affaddrit}{ariali at 10pt} %13 Jan 00 gkmt
%\newfont{\eaddfnt}{arial at 12pt} % was arialr originally - now is arial
%\newfont{\ixpt}{times at 9pt} % was timenrr originally - now is times
%\newfont{\confname}{timesi at 8pt} % was timenrri - now is timesi
%\newfont{\crnotice}{times at 8pt} % was timenrr originally - now is times
%\newfont{\ninept}{times at 9pt} % was timenrr originally - now is times

% *********************************************
%  -- End of block A --
%
%
% -- Start of block B -- UPDATED FONT NAMES
% *********************************************
% Gerry Murray 11/30/2006
% *********************************************
\ifnum\ACM@basesize=9\relax
\newfont{\secfnt}{ptmb8t at 12pt}
\newfont{\secit}{ptmbi8t at 12pt}    %13 Jan 00 gkmt
\newfont{\subsecfnt}{ptmri8t at 11pt}
\newfont{\subsecit}{ptmbi8t at 11pt}  % 
\newfont{\ttlfnt}{phvb8t at 18pt}
\newfont{\ttlit}{phvbo8t at 18pt}    % GM 2/4/2000
\newfont{\subttlfnt}{phvr8t at 14pt}
\newfont{\subttlit}{phvro8t at 14pt} % GM 2/4/2000
\newfont{\subttlbf}{phvb8t at 14pt}  % 13 Jan 00 gkmt
\newfont{\aufnt}{phvr8t at 12pt}
\newfont{\auit}{phvro8t at 12pt}     % GM 2/4/2000
\newfont{\affaddr}{phvr8t at 10pt}
\newfont{\affaddrit}{phvro8t at 10pt} % GM 2/4/2000
\newfont{\eaddfnt}{phvr8t at 12pt}
\newfont{\ixpt}{ptmr8t at 9pt}
\newfont{\confname}{ptmri8t at 8pt}
\newfont{\crnotice}{ptmr8t at 8pt}
\newfont{\ninept}{ptmr8t at 9pt}
\fi
\ifnum\ACM@basesize=10\relax
\newfont{\secfnt}{ptmb8t at 13pt}
\newfont{\secit}{ptmbi8t at 13pt}    %13 Jan 00 gkmt
\newfont{\subsecfnt}{ptmri8t at 12pt}
\newfont{\subsecit}{ptmbi8t at 12pt}  % 
\newfont{\ttlfnt}{phvb8t at 20pt}
\newfont{\ttlit}{phvbo8t at 20pt}    % GM 2/4/2000
\newfont{\subttlfnt}{phvr8t at 15pt}
\newfont{\subttlit}{phvro8t at 15pt} % GM 2/4/2000
\newfont{\subttlbf}{phvb8t at 15pt}  % 13 Jan 00 gkmt
\newfont{\aufnt}{phvr8t at 12pt}
\newfont{\auit}{phvro8t at 12pt}     % GM 2/4/2000
\newfont{\affaddr}{phvr8t at 11pt}
\newfont{\affaddrit}{phvro8t at 11pt} % GM 2/4/2000
\newfont{\eaddfnt}{phvr8t at 12pt}
\newfont{\ixpt}{ptmr8t at 10pt}
\newfont{\confname}{ptmri8t at 9pt}
\newfont{\crnotice}{ptmr8t at 9pt}
\newfont{\ninept}{ptmr8t at 10pt}
\fi
% +++++++++++++++++++++++++++++++++++++++++++++
% -- End of block B --

%\def\email#1{{{\eaddfnt{\vskip 4pt#1}}}}
% If we have an email, inside a "shared affiliation" then we need the following instead
\def\email#1{{{\eaddfnt{\par #1}}}}       % revised  - GM - 11/30/2006

\def\addauthorsection{\ifnum\originalaucount>6  % was 3 - Gerry March 2007
    \section{Additional Authors}\the\addauthors
  \fi}

\newcount\savesection
\newcount\sectioncntr
\global\sectioncntr=1

\setcounter{secnumdepth}{3}

\def\appendix{\par
\section*{APPENDIX}
\setcounter{section}{0}
 \setcounter{subsection}{0}
 \def\thesection{\Alph{section}} }

\leftmargini 22.5pt
\leftmarginii 19.8pt    % > \labelsep + width of '(m)'
\leftmarginiii 16.8pt   % > \labelsep + width of 'vii.'
\leftmarginiv 15.3pt    % > \labelsep + width of 'M.'
\leftmarginv 9pt
\leftmarginvi 9pt

\leftmargin\leftmargini
\labelsep 4.5pt
\labelwidth\leftmargini\advance\labelwidth-\labelsep

\def\@listI{\leftmargin\leftmargini \parsep 3.6pt plus 2pt minus 1pt%
\topsep 7.2pt plus 2pt minus 4pt%
\itemsep 3.6pt plus 2pt minus 1pt}

\let\@listi\@listI
\@listi

\def\@listii{\leftmargin\leftmarginii
   \labelwidth\leftmarginii\advance\labelwidth-\labelsep
   \topsep 3.6pt plus 2pt minus 1pt
   \parsep 1.8pt plus 0.9pt minus 0.9pt
   \itemsep \parsep}

\def\@listiii{\leftmargin\leftmarginiii
    \labelwidth\leftmarginiii\advance\labelwidth-\labelsep
    \topsep 1.8pt plus 0.9pt minus 0.9pt
    \parsep \z@ \partopsep 1pt plus 0pt minus 1pt
    \itemsep \topsep}

\def\@listiv{\leftmargin\leftmarginiv
     \labelwidth\leftmarginiv\advance\labelwidth-\labelsep}

\def\@listv{\leftmargin\leftmarginv
     \labelwidth\leftmarginv\advance\labelwidth-\labelsep}

\def\@listvi{\leftmargin\leftmarginvi
     \labelwidth\leftmarginvi\advance\labelwidth-\labelsep}

\def\labelenumi{\theenumi.}
\def\theenumi{\arabic{enumi}}

\def\labelenumii{(\theenumii)}
\def\theenumii{\alph{enumii}}
\def\p@enumii{\theenumi}

\def\labelenumiii{\theenumiii.}
\def\theenumiii{\roman{enumiii}}
\def\p@enumiii{\theenumi(\theenumii)}

\def\labelenumiv{\theenumiv.}
\def\theenumiv{\Alph{enumiv}}
\def\p@enumiv{\p@enumiii\theenumiii}

\def\labelitemi{$\bullet$}
\def\labelitemii{\bf --}
\def\labelitemiii{$\ast$}
\def\labelitemiv{$\cdot$}

\def\verse{\let\\=\@centercr
  \list{}{\itemsep\z@ \itemindent -1.5em\listparindent \itemindent
          \rightmargin\leftmargin\advance\leftmargin 1.5em}\item[]}
\let\endverse\endlist

\def\quotation{\list{}{\listparindent 1.5em
    \itemindent\listparindent
    \rightmargin\leftmargin \parsep 0pt plus 1pt}\item[]}
\let\endquotation=\endlist

\def\quote{\list{}{\rightmargin\leftmargin}\item[]}
\let\endquote=\endlist

\def\descriptionlabel#1{\hspace\labelsep \bf #1}
\def\description{\list{}{\labelwidth\z@ \itemindent-\leftmargin
       \let\makelabel\descriptionlabel}}

\let\enddescription\endlist

\def\theequation{\arabic{equation}}

\ifnum\ACM@basesize=9\relax
\arraycolsep 4.5pt   % Half the space between columns in an array environment.
\tabcolsep 5.4pt    % Half the space between columns in a tabular environment.
\arrayrulewidth .5pt % Width of rules in array and tabular environment. % (was .4) updated Gerry March 20 2007
\doublerulesep 1.8pt % Space between adjacent rules in array or tabular env.

\fi

\ifnum\ACM@basesize=10\relax
\arraycolsep 5pt   % Half the space between columns in an array environment.
\tabcolsep 6pt    % Half the space between columns in a tabular environment.
\arrayrulewidth .5pt % Width of rules in array and tabular environment. % (was .4) updated Gerry March 20 2007
\doublerulesep 1.8pt % Space between adjacent rules in array or tabular env.

\fi

\tabbingsep \labelsep   % Space used by the \' command.  (See LaTeX manual.)

\skip\@mpfootins =\skip\footins

\fboxsep =2.7pt      % Space left between box and text by \fbox and \framebox.
\fboxrule =.5pt      % Width of rules in box made by \fbox and \framebox. % (was .4) updated Gerry March 20 2007

\def\thepart{\Roman{part}} % Roman numeral part numbers.
\def\thesection       {\arabic{section}}
\def\thesubsection    {\thesection.\arabic{subsection}}
%\def\thesubsubsection {\thesubsection.\arabic{subsubsection}} % GM 7/30/2002
%\def\theparagraph     {\thesubsubsection.\arabic{paragraph}}  % GM 7/30/2002
\def\thesubparagraph  {\theparagraph.\arabic{subparagraph}}

\def\@pnumwidth{1.55em}
\def\@tocrmarg {2.55em}
\def\@dotsep{4.5}
\setcounter{tocdepth}{3}

%\def\tableofcontents{\@latexerr{\tableofcontents: Tables of contents are not
%  allowed in the `acmconf' document style.}\@eha}

\def\tableofcontents{\ClassError{%
    \string\tableofcontents\space is not allowed in the `acmconf' document	% January 2008
    style}\@eha}

\def\l@part#1#2{\addpenalty{\@secpenalty}
   \addvspace{2.25em plus 1pt}  % space above part line
   \begingroup
   \@tempdima 3em       % width of box holding part number, used by
     \parindent \z@ \rightskip \@pnumwidth      %% \numberline
     \parfillskip -\@pnumwidth
     {\large \bf        % set line in \large boldface
     \leavevmode        % TeX command to enter horizontal mode.
     #1\hfil \hbox to\@pnumwidth{\hss #2}}\par
     \nobreak           % Never break after part entry
   \endgroup}

\def\l@section#1#2{\addpenalty{\@secpenalty} % good place for page break
   \addvspace{1.0em plus 1pt}   % space above toc entry
   \@tempdima 1.5em             % width of box holding section number
   \begingroup
    \parindent  \z@ \rightskip \@pnumwidth
     \parfillskip -\@pnumwidth
     \bf                        % Boldface.
     \leavevmode                % TeX command to enter horizontal mode.
      \advance\leftskip\@tempdima %% added 5 Feb 88 to conform to
      \hskip -\leftskip           %% 25 Jan 88 change to \numberline
     #1\nobreak\hfil \nobreak\hbox to\@pnumwidth{\hss #2}\par
   \endgroup}

\def\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}}
\def\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}}
\def\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}}
\def\l@subparagraph{\@dottedtocline{5}{10em}{5em}}

%\def\listoffigures{\@latexerr{\listoffigures: Lists of figures are not
%  allowed in the `acmconf' document style.}\@eha}

\def\listoffigures{\ClassError{%
    \string\listoffigures\space is not allowed in the `acmconf' document	% January 2008
    style}\@eha}

\def\l@figure{\@dottedtocline{1}{1.5em}{2.3em}}

%\def\listoftables{\@latexerr{\listoftables: Lists of tables are not
%  allowed in the `acmconf' document style.}\@eha}
%\let\l@table\l@figure

\def\listoftables{\ClassError{%
    \string\listoftables\space is not allowed in the `acmconf' document		% January 2008
    style}\@eha}
 \let\l@table\l@figure

\def\footnoterule{\kern-3\p@
  \hrule width .5\columnwidth   % (was .4) updated Gerry March 20 2007
  \kern 2.6\p@}                 % The \hrule has default height of .4pt % (was .4) updated Gerry March 20 2007
% ------
\long\def\@makefntext#1{\noindent 
%\hbox to .5em{\hss$^{\@thefnmark}$}#1}   % original
\hbox to .5em{\hss\textsuperscript{\@thefnmark}}#1}  % C. Clifton / GM Oct. 2nd. 2002
% -------

\long\def\@maketntext#1{\noindent
#1}

\long\def\@maketitlenotetext#1#2{\noindent
            \hbox to 1.8em{\hss$^{#1}$}#2}

\setcounter{topnumber}{2}
\def\topfraction{.7}
\setcounter{bottomnumber}{1}
\def\bottomfraction{.3}
\setcounter{totalnumber}{3}
\def\textfraction{.2}
\def\floatpagefraction{.5}
\setcounter{dbltopnumber}{2}
\def\dbltopfraction{.7}
\def\dblfloatpagefraction{.5}

%
\long\def\@makecaption#1#2{
   \vskip \baselineskip
   \setbox\@tempboxa\hbox{\textbf{#1: #2}}
   \ifdim \wd\@tempboxa >\hsize % IF longer than one line:
       \textbf{#1: #2}\par               %   THEN set as ordinary paragraph.
     \else                      %   ELSE  center.
       \hbox to\hsize{\hfil\box\@tempboxa\hfil}\par
   \fi}

\long\def\@makecaption#1#2{
   \vskip 10pt
   \setbox\@tempboxa\hbox{\textbf{#1: #2}}
   \ifdim \wd\@tempboxa >\hsize % IF longer than one line:
       \textbf{#1: #2}\par                %   THEN set as ordinary paragraph.
     \else                      %   ELSE  center.
       \hbox to\hsize{\hfil\box\@tempboxa\hfil}
   \fi}

\@ifundefined{figure}{\newcounter {figure}} % this is for LaTeX2e

\def\fps@figure{tbp}
\def\ftype@figure{1}
\def\ext@figure{lof}
\def\fnum@figure{Figure \thefigure}
\def\figure{\@float{figure}}
%\let\endfigure\end@float
\def\endfigure{\end@float} 		% Gerry January 2008
\@namedef{figure*}{\@dblfloat{figure}}
\@namedef{endfigure*}{\end@dblfloat}

\@ifundefined{table}{\newcounter {table}} % this is for LaTeX2e

\def\fps@table{tbp}
\def\ftype@table{2}
\def\ext@table{lot}
\def\fnum@table{Table \thetable}
\def\table{\@float{table}}
%\let\endtable\end@float
\def\endtable{\end@float}		% Gerry January 2008
\@namedef{table*}{\@dblfloat{table}}
\@namedef{endtable*}{\end@dblfloat}

\newtoks\titleboxnotes
\newcount\titleboxnoteflag

\def\maketitle{\par
 \begingroup
   \def\thefootnote{\fnsymbol{footnote}}
   \def\@makefnmark{\hbox
       to 0pt{$^{\@thefnmark}$\hss}}
     \twocolumn[\@maketitle]
\@thanks
 \endgroup
 \setcounter{footnote}{0}
 \let\maketitle\relax
 \let\@maketitle\relax
 \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\gdef\@subtitle{}\let\thanks\relax
 \@copyrightspace}

%% CHANGES ON NEXT LINES
\newif\if@ll % to record which version of LaTeX is in use

\expandafter\ifx\csname LaTeXe\endcsname\relax % LaTeX2.09 is used
\else% LaTeX2e is used, so set ll to true
\global\@lltrue
\fi

\if@ll
  \NeedsTeXFormat{LaTeX2e}
  \ProvidesClass{sig-alternate} [2013/05/12 v2.7 based on acmproc.cls V1.3 <Nov. 30 '99>]
  \RequirePackage{latexsym}% QUERY: are these two really needed?
  \let\dooptions\ProcessOptions
\else
  \let\dooptions\@options
\fi
%% END CHANGES

\def\@height{height}
\def\@width{width}
\def\@minus{minus}
\def\@plus{plus}
\def\hb@xt@{\hbox to}
\newif\if@faircopy
\@faircopyfalse
\def\ds@faircopy{\@faircopytrue}

\def\ds@preprint{\@faircopyfalse}

\@twosidetrue
\@mparswitchtrue
\def\ds@draft{\overfullrule 5\p@}
%% CHANGE ON NEXT LINE
\dooptions

\lineskip \p@
\normallineskip \p@
\def\baselinestretch{1}
\def\@ptsize{0} %needed for amssymbols.sty

%% CHANGES ON NEXT LINES
\if@ll% allow use of old-style font change commands in LaTeX2e
\@maxdepth\maxdepth
%
\DeclareOldFontCommand{\rm}{\ninept\rmfamily}{\mathrm}
\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
\DeclareRobustCommand*{\cal}{\@fontswitch{\relax}{\mathcal}}
\DeclareRobustCommand*{\mit}{\@fontswitch{\relax}{\mathnormal}}
\fi
%
\if@ll
 \renewcommand{\rmdefault}{cmr}  % was 'ttm'
% Note! I have also found 'mvr' to work ESPECIALLY well.
% Gerry - October 1999
% You may need to change your LV1times.fd file so that sc is
% mapped to cmcsc - -for smallcaps -- that is if you decide
% to change {cmr} to {times} above. (Not recommended)
\ifnum\ACM@basesize=9\relax
  \renewcommand{\@ptsize}{}
  \renewcommand{\normalsize}{%
    \@setfontsize\normalsize\@ixpt{10.5\p@}%\ninept%
    \abovedisplayskip 6\p@ \@plus2\p@ \@minus\p@
    \belowdisplayskip \abovedisplayskip
    \abovedisplayshortskip 6\p@ \@minus 3\p@
    \belowdisplayshortskip 6\p@ \@minus 3\p@
    \let\@listi\@listI}
\fi
\ifnum\ACM@basesize=10\relax
  \renewcommand{\@ptsize}{}
  \renewcommand{\normalsize}{%
    \@setfontsize\normalsize\@xpt{11.5\p@}%\ninept%
    \abovedisplayskip 6.5\p@ \@plus2\p@ \@minus\p@
    \belowdisplayskip \abovedisplayskip
    \abovedisplayshortskip 6.5\p@ \@minus 3\p@
    \belowdisplayshortskip 6.5\p@ \@minus 3\p@
    \let\@listi\@listI}
\fi
\else
  \def\@normalsize{%changed next to 9 from 10
    \@setsize\normalsize{9\p@}\ixpt\@ixpt
   \abovedisplayskip 6\p@ \@plus2\p@ \@minus\p@
    \belowdisplayskip \abovedisplayskip
    \abovedisplayshortskip 6\p@ \@minus 3\p@
    \belowdisplayshortskip 6\p@ \@minus 3\p@
    \let\@listi\@listI
  }%
\fi
\if@ll
\ifnum\ACM@basesize=9\relax
  \newcommand\scriptsize{\@setfontsize\scriptsize\@viipt{8\p@}}
  \newcommand\tiny{\@setfontsize\tiny\@vpt{6\p@}}
  \newcommand\large{\@setfontsize\large\@xiipt{14\p@}}
  \newcommand\Large{\@setfontsize\Large\@xivpt{18\p@}}
  \newcommand\LARGE{\@setfontsize\LARGE\@xviipt{20\p@}}
  \newcommand\huge{\@setfontsize\huge\@xxpt{25\p@}}
  \newcommand\Huge{\@setfontsize\Huge\@xxvpt{30\p@}}
\fi
\ifnum\ACM@basesize=10\relax
  \newcommand\scriptsize{\@setfontsize\scriptsize\@viiipt{9\p@}}
  \newcommand\tiny{\@setfontsize\tiny\@vipt{7\p@}}
  \newcommand\large{\@setfontsize\large\@xiiipt{15\p@}}
  \newcommand\Large{\@setfontsize\Large\@xvpt{20\p@}}
  \newcommand\LARGE{\@setfontsize\LARGE\@xixpt{22\p@}}
  \newcommand\huge{\@setfontsize\huge\@xixpt{30\p@}}
  \newcommand\Huge{\@setfontsize\Huge30pt{36\p@}}
\fi
\else
  \def\scriptsize{\@setsize\scriptsize{8\p@}\viipt\@viipt}
  \def\tiny{\@setsize\tiny{6\p@}\vpt\@vpt}
  \def\large{\@setsize\large{14\p@}\xiipt\@xiipt}
  \def\Large{\@setsize\Large{18\p@}\xivpt\@xivpt}
  \def\LARGE{\@setsize\LARGE{20\p@}\xviipt\@xviipt}
  \def\huge{\@setsize\huge{25\p@}\xxpt\@xxpt}
  \def\Huge{\@setsize\Huge{30\p@}\xxvpt\@xxvpt}
\fi
\normalsize

% make aubox hsize/number of authors up to 3, less gutter
% then showbox gutter showbox gutter showbox -- GKMT Aug 99
\newbox\@acmtitlebox
\ifnum\ACM@basesize=9\relax
\def\@maketitle{\newpage
 \null
 \setbox\@acmtitlebox\vbox{%
\baselineskip 20pt
\vskip 2em                   % Vertical space above title.
   \begin{center}
    {\ttlfnt \@title\par}       % Title set in 18pt Helvetica (Arial) bold size.
    \vskip 1.5em                % Vertical space after title.
%This should be the subtitle.
{\subttlfnt \the\subtitletext\par}\vskip 1.25em%\fi
    {\baselineskip 16pt\aufnt   % each author set in \12 pt Arial, in a
     \lineskip .5em             % tabular environment
     \begin{tabular}[t]{c}\@author
     \end{tabular}\par}
    \vskip 1.5em               % Vertical space after author.
   \end{center}}
 \dimen0=\ht\@acmtitlebox
 \advance\dimen0 by -12.75pc\relax % Increased space for title box -- KBT
 \unvbox\@acmtitlebox
 \ifdim\dimen0<0.0pt\relax\vskip-\dimen0\fi}
\fi
\ifnum\ACM@basesize=10\relax
\def\@maketitle{\newpage
 \null
 \setbox\@acmtitlebox\vbox{%
\baselineskip 22pt
\vskip 2.2em                   % Vertical space above title.
   \begin{center}
    {\ttlfnt \@title\par}       % Title set in 18pt Helvetica (Arial) bold size.
    \vskip 2em                % Vertical space after title.
%This should be the subtitle.
{\subttlfnt \the\subtitletext\par}\vskip 1.25em%\fi
    {\baselineskip 18pt\aufnt   % each author set in \12 pt Arial, in a
     \lineskip .5em             % tabular environment
     \begin{tabular}[t]{c}\@author
     \end{tabular}\par}
    \vskip 2em               % Vertical space after author.
   \end{center}}
 \dimen0=\ht\@acmtitlebox
 \advance\dimen0 by -12.75pc\relax % Increased space for title box -- KBT
 \unvbox\@acmtitlebox
 \ifdim\dimen0<0.0pt\relax\vskip-\dimen0\fi}
\fi

\newcount\titlenotecount
\global\titlenotecount=0
\newtoks\tntoks
\newtoks\tntokstwo
\newtoks\tntoksthree
\newtoks\tntoksfour
\newtoks\tntoksfive

\def\abstract{
\ifnum\titlenotecount>0 % was =1
    \insert\footins{%
    \reset@font\footnotesize
        \interlinepenalty\interfootnotelinepenalty
        \splittopskip\footnotesep
        \splitmaxdepth \dp\strutbox \floatingpenalty \@MM
        \hsize\columnwidth \@parboxrestore
        \protected@edef\@currentlabel{%
        }%
        \color@begingroup
\ifnum\titlenotecount=1
      \@maketntext{%
         \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\@finalstrut\strutbox}%
\fi
\ifnum\titlenotecount=2
      \@maketntext{%
      \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\par\@finalstrut\strutbox}%
\@maketntext{%
         \raisebox{4pt}{$\dagger$}\rule\z@\footnotesep\ignorespaces\the\tntokstwo\@finalstrut\strutbox}%
\fi
\ifnum\titlenotecount=3
      \@maketntext{%
         \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\par\@finalstrut\strutbox}%
\@maketntext{%
         \raisebox{4pt}{$\dagger$}\rule\z@\footnotesep\ignorespaces\the\tntokstwo\par\@finalstrut\strutbox}%
\@maketntext{%
         \raisebox{4pt}{$\ddagger$}\rule\z@\footnotesep\ignorespaces\the\tntoksthree\@finalstrut\strutbox}%
\fi
\ifnum\titlenotecount=4
      \@maketntext{%
         \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\par\@finalstrut\strutbox}%
\@maketntext{%
         \raisebox{4pt}{$\dagger$}\rule\z@\footnotesep\ignorespaces\the\tntokstwo\par\@finalstrut\strutbox}%
\@maketntext{%
         \raisebox{4pt}{$\ddagger$}\rule\z@\footnotesep\ignorespaces\the\tntoksthree\par\@finalstrut\strutbox}%
\@maketntext{%
         \raisebox{4pt}{$\S$}\rule\z@\footnotesep\ignorespaces\the\tntoksfour\@finalstrut\strutbox}%
\fi
\ifnum\titlenotecount=5
      \@maketntext{%
         \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\par\@finalstrut\strutbox}%
\@maketntext{%
         \raisebox{4pt}{$\dagger$}\rule\z@\footnotesep\ignorespaces\the\tntokstwo\par\@finalstrut\strutbox}%
\@maketntext{%
         \raisebox{4pt}{$\ddagger$}\rule\z@\footnotesep\ignorespaces\the\tntoksthree\par\@finalstrut\strutbox}%
\@maketntext{%
         \raisebox{4pt}{$\S$}\rule\z@\footnotesep\ignorespaces\the\tntoksfour\par\@finalstrut\strutbox}%
\@maketntext{%
         \raisebox{4pt}{$\P$}\rule\z@\footnotesep\ignorespaces\the\tntoksfive\@finalstrut\strutbox}%
\fi
   \color@endgroup} %g}
\fi
\setcounter{footnote}{0}
\section*{ABSTRACT}\normalsize%\ninept
}

\def\endabstract{\if@twocolumn\else\endquotation\fi}

\def\keywords{\if@twocolumn
\section*{Keywords}
\else \small
\quotation
\fi}

\def\terms#1{%
%\if@twocolumn
%\section*{General Terms}
%\else \small
%\quotation
%\fi
}

% -- Classification needs to be a bit smart due to optionals - Gerry/Georgia November 2nd. 1999
\newcount\catcount
\global\catcount=1

\def\category#1#2#3{%
\ifnum\catcount=1
\section*{Categories and Subject Descriptors}
\advance\catcount by 1\else{\unskip; }\fi
    \@ifnextchar [{\@category{#1}{#2}{#3}}{\@category{#1}{#2}{#3}[]}%
}

\def\@category#1#2#3[#4]{%
    \begingroup
        \let\and\relax
            #1 [\textbf{#2}]%
            \if!#4!%
                \if!#3!\else : #3\fi
            \else
                :\space
                \if!#3!\else #3\kern\z@---\hskip\z@\fi
                \textit{#4}%
            \fi
    \endgroup
}
%

%%% This section (written by KBT) handles the 1" box in the lower left
%%% corner of the left column of the first page by creating a picture,
%%% and inserting the predefined string at the bottom (with a negative
%%% displacement to offset the space allocated for a non-existent
%%% caption).
%%%
\newtoks\copyrightnotice
\def\ftype@copyrightbox{8}
\def\@copyrightspace{
\@float{copyrightbox}[b]
\begin{center}
\setlength{\unitlength}{1pc}
\ifnum\ACM@basesize=9
\begin{picture}(20,6) %Space for copyright notice
\put(0,-0.95){\crnotice{\@toappear}}
\end{picture}
\fi
\ifnum\ACM@basesize=10
\begin{picture}(20,7) %Space for copyright notice
\put(0,-0.95){\crnotice{\@toappear}}
\end{picture}
\fi
\end{center}
\end@float}

\def\@toappear{} % Default setting blank - commands below change this.
\long\def\toappear#1{\def\@toappear{\parbox[b]{20pc}{\baselineskip 9pt#1}}}
\def\toappearbox#1{\def\@toappear{\raisebox{5pt}{\framebox[20pc]{\parbox[b]{19pc}{#1}}}}}

\newtoks\conf
\newtoks\confinfo
\def\conferenceinfo#1#2{\global\conf={#1}\global\confinfo{#2}}

%\def\marginpar{\@latexerr{The \marginpar command is not allowed in the
%  `acmconf' document style.}\@eha}

\def\marginpar{\ClassError{%
    \string\marginpar\space is not allowed in the `acmconf' document		% January 2008
    style}\@eha}

\mark{{}{}}     % Initializes TeX's marks

\def\today{\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\day, \number\year}

\def\@begintheorem#1#2{%
    \parskip 0pt % GM July 2000 (for tighter spacing)
    \trivlist
    \item[%
        \hskip 10\p@
        \hskip \labelsep
        {{\sc #1}\hskip 5\p@\relax#2.}%
    ]
    \it
}
\def\@opargbegintheorem#1#2#3{%
    \parskip 0pt % GM July 2000 (for tighter spacing)
    \trivlist
    \item[%
        \hskip 10\p@
        \hskip \labelsep
        {\sc #1\ #2\             % This mod by Gerry to enumerate corollaries
   \setbox\@tempboxa\hbox{(#3)}  % and bracket the 'corollary title'
        \ifdim \wd\@tempboxa>\z@ % and retain the correct numbering of e.g. theorems
            \hskip 5\p@\relax    % if they occur 'around' said corollaries.
            \box\@tempboxa       % Gerry - Nov. 1999.
        \fi.}%
    ]
    \it
}
\newif\if@qeded
\global\@qededfalse

% -- original
%\def\proof{%
%  \vspace{-\parskip} % GM July 2000 (for tighter spacing)
%    \global\@qededfalse
%    \@ifnextchar[{\@xproof}{\@proof}%
%}
% -- end of original

% (JSS) Fix for vertical spacing bug - Gerry Murray July 30th. 2002
\def\proof{%
\vspace{-\lastskip}\vspace{-\parsep}\penalty-51%
\global\@qededfalse
\@ifnextchar[{\@xproof}{\@proof}%
}

\def\endproof{%
    \if@qeded\else\qed\fi
    \endtrivlist
}
\def\@proof{%
    \trivlist
    \item[%
        \hskip 10\p@
        \hskip \labelsep
        {\sc Proof.}%
    ]
    \ignorespaces
}
\def\@xproof[#1]{%
    \trivlist
    \item[\hskip 10\p@\hskip \labelsep{\sc Proof #1.}]%
    \ignorespaces
}
\def\qed{%
    \unskip
    \kern 10\p@
    \begingroup
        \unitlength\p@
        \linethickness{.4\p@}%
        \framebox(6,6){}%
    \endgroup
    \global\@qededtrue
}

\def\newdef#1#2{%
    \expandafter\@ifdefinable\csname #1\endcsname
        {\@definecounter{#1}%
         \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}%
         \global\@namedef{#1}{\@defthm{#1}{#2}}%
         \global\@namedef{end#1}{\@endtheorem}%
    }%
}
\def\@defthm#1#2{%
    \refstepcounter{#1}%
    \@ifnextchar[{\@ydefthm{#1}{#2}}{\@xdefthm{#1}{#2}}%
}
\def\@xdefthm#1#2{%
    \@begindef{#2}{\csname the#1\endcsname}%
    \ignorespaces
}
\def\@ydefthm#1#2[#3]{%
    \trivlist
    \item[%
        \hskip 10\p@
        \hskip \labelsep
        {\it #2%
%         \savebox\@tempboxa{#3}%
         \saveb@x\@tempboxa{#3}%		% January 2008
         \ifdim \wd\@tempboxa>\z@
            \ \box\@tempboxa
         \fi.%
        }]%
    \ignorespaces
}
\def\@begindef#1#2{%
    \trivlist
    \item[%
        \hskip 10\p@
        \hskip \labelsep
        {\it #1\ \rm #2.}%
    ]%
}
\def\theequation{\arabic{equation}}

\newcounter{part}
\newcounter{section}
\newcounter{subsection}[section]
\newcounter{subsubsection}[subsection]
\newcounter{paragraph}[subsubsection]
\def\thepart{\Roman{part}}
\def\thesection{\arabic{section}}
\def\thesubsection{\thesection.\arabic{subsection}}
\def\thesubsubsection{\thesubsection.\arabic{subsubsection}} %removed \subsecfnt 29 July 2002 gkmt
\def\theparagraph{\thesubsubsection.\arabic{paragraph}} %removed \subsecfnt 29 July 2002 gkmt
\newif\if@uchead
\@ucheadfalse

%% CHANGES: NEW NOTE
%% NOTE: OK to use old-style font commands below, since they were
%% suitably redefined for LaTeX2e
%% END CHANGES
\setcounter{secnumdepth}{3}
\def\part{%
    \@startsection{part}{9}{\z@}{-10\p@ \@plus -4\p@ \@minus -2\p@}
        {4\p@}{\normalsize\@ucheadtrue}%
}
\def\section{%
    \@startsection{section}{1}{\z@}{-10\p@ \@plus -4\p@ \@minus -2\p@}% GM
    {4\p@}{\baselineskip 14pt\secfnt\@ucheadtrue}%
}

\def\subsection{%
    \@startsection{subsection}{2}{\z@}{-8\p@ \@plus -2\p@ \@minus -\p@}
    {4\p@}{\secfnt}%
}
\def\subsubsection{%
    \@startsection{subsubsection}{3}{\z@}{-8\p@ \@plus -2\p@ \@minus -\p@}%
    {4\p@}{\subsecfnt}%
}
%\def\paragraph{%
%    \vskip 12pt\@startsection{paragraph}{3}{\z@}{6\p@ \@plus \p@}% original
%    {-5\p@}{\subsecfnt}%
%}
%  If one wants sections, subsections and subsubsections numbered,
%  but not paragraphs, one usually sets secnumepth to 3.
%  For that, the "depth" of paragraphs must be given correctly
%  in the definition (``4'' instead of ``3'' as second argument
%  of @startsection):
\def\paragraph{%
    \vskip 12pt\@startsection{paragraph}{4}{\z@}{6\p@ \@plus \p@}%    % GM and Wolfgang May - 11/30/06
    {-5\p@}{\subsecfnt}%
}
\let\@period=.
\def\@startsection#1#2#3#4#5#6{%
        \if@noskipsec  %gkmt, 11 aug 99
        \global\let\@period\@empty
        \leavevmode
        \global\let\@period.%
    \fi
      \par %
    \@tempskipa #4\relax
    \@afterindenttrue
    \ifdim \@tempskipa <\z@
        \@tempskipa -\@tempskipa
        \@afterindentfalse
    \fi
    \if@nobreak
    \everypar{}%
    \else
        \addpenalty\@secpenalty
        \addvspace\@tempskipa
    \fi
\parskip=0pt % GM July 2000 (non numbered) section heads
    \@ifstar
        {\@ssect{#3}{#4}{#5}{#6}}
        {\@dblarg{\@sect{#1}{#2}{#3}{#4}{#5}{#6}}}%
}
\def\@sect#1#2#3#4#5#6[#7]#8{%
    \ifnum #2>\c@secnumdepth
        \let\@svsec\@empty
    \else
        \refstepcounter{#1}%
        \edef\@svsec{%
            \begingroup
                %\ifnum#2>2 \noexpand\rm \fi % changed to next 29 July 2002 gkmt
            \ifnum#2>2 \noexpand#6 \fi
                \csname the#1\endcsname
            \endgroup
            \ifnum #2=1\relax .\fi
            \hskip 1em
        }%
    \fi
    \@tempskipa #5\relax
    \ifdim \@tempskipa>\z@
        \begingroup
            #6\relax
            \@hangfrom{\hskip #3\relax\@svsec}%
            \begingroup
                \interlinepenalty \@M
                \if@uchead
                    \uppercase{#8}%
                \else
                    #8%
                \fi
                \par
            \endgroup
        \endgroup
        \csname #1mark\endcsname{#7}%
        \vskip -12pt  %gkmt, 11 aug 99 and GM July 2000 (was -14) - numbered section head spacing
\addcontentsline{toc}{#1}{%
            \ifnum #2>\c@secnumdepth \else
                \protect\numberline{\csname the#1\endcsname}%
            \fi
            #7%
        }%
    \else
        \def\@svsechd{%
            #6%
            \hskip #3\relax
            \@svsec
            \if@uchead
                \uppercase{#8}%
            \else
                #8%
            \fi
            \csname #1mark\endcsname{#7}%
            \addcontentsline{toc}{#1}{%
                \ifnum #2>\c@secnumdepth \else
                    \protect\numberline{\csname the#1\endcsname}%
                \fi
                #7%
            }%
        }%
    \fi
    \@xsect{#5}\hskip 1pt
    \par
}
\def\@xsect#1{%
    \@tempskipa #1\relax
    \ifdim \@tempskipa>\z@
        \par
        \nobreak
        \vskip \@tempskipa
        \@afterheading
    \else
        \global\@nobreakfalse
        \global\@noskipsectrue
        \everypar{%
            \if@noskipsec
                \global\@noskipsecfalse
                \clubpenalty\@M
                \hskip -\parindent
                \begingroup
                    \@svsechd
                    \@period
                \endgroup
                \unskip
                \@tempskipa #1\relax
                \hskip -\@tempskipa
            \else
                \clubpenalty \@clubpenalty
                \everypar{}%
            \fi
        }%
    \fi
    \ignorespaces
}
\def\@trivlist{%
    \@topsepadd\topsep
    \if@noskipsec
        \global\let\@period\@empty
        \leavevmode
        \global\let\@period.%
    \fi
    \ifvmode
        \advance\@topsepadd\partopsep
    \else
        \unskip
        \par
    \fi
    \if@inlabel
        \@noparitemtrue
        \@noparlisttrue
    \else
        \@noparlistfalse
        \@topsep\@topsepadd
    \fi
    \advance\@topsep \parskip
    \leftskip\z@skip
    \rightskip\@rightskip
    \parfillskip\@flushglue
    \@setpar{\if@newlist\else{\@@par}\fi}
    \global\@newlisttrue
    \@outerparskip\parskip
}

%%% Actually, 'abbrev' works just fine as the default
%%% Bibliography style.

\typeout{Using 'Abbrev' bibliography style}
\newcommand\bibyear[2]{%
    \unskip\quad\ignorespaces#1\unskip
    \if#2..\quad \else \quad#2 \fi
}
\newcommand{\bibemph}[1]{{\em#1}}
\newcommand{\bibemphic}[1]{{\em#1\/}}
\newcommand{\bibsc}[1]{{\sc#1}}
\def\@normalcite{%
    \def\@cite##1##2{[##1\if@tempswa , ##2\fi]}%
}
\def\@citeNB{%
    \def\@cite##1##2{##1\if@tempswa , ##2\fi}%
}
\def\@citeRB{%
    \def\@cite##1##2{##1\if@tempswa , ##2\fi]}%
}
\def\start@cite#1#2{%
    \edef\citeauthoryear##1##2##3{%
        ###1%
        \ifnum#2=\z@ \else\ ###2\fi
    }%
    \ifnum#1=\thr@@
        \let\@@cite\@citeyear
    \else
        \let\@@cite\@citenormal
    \fi
    \@ifstar{\@citeNB\@@cite}{\@normalcite\@@cite}%
}
%\def\cite{\start@cite23}
\DeclareRobustCommand\cite{\start@cite23}		% January 2008
\def\citeNP{\cite*}					% No Parentheses e.g. 5
%\def\citeA{\start@cite10}
\DeclareRobustCommand\citeA{\start@cite10}		% January 2008
\def\citeANP{\citeA*}
%\def\shortcite{\start@cite23}				
\DeclareRobustCommand\shortcite{\start@cite23}		% January 2008
\def\shortciteNP{\shortcite*}
%\def\shortciteA{\start@cite20}
\DeclareRobustCommand\shortciteA{\start@cite20}		% January 2008
\def\shortciteANP{\shortciteA*}
%\def\citeyear{\start@cite30}
\DeclareRobustCommand\citeyear{\start@cite30}		% January 2008
\def\citeyearNP{\citeyear*}
%\def\citeN{%
\DeclareRobustCommand\citeN{%				% January 2008
    \@citeRB
    \def\citeauthoryear##1##2##3{##1\ [##3%
        \def\reserved@a{##1}%
        \def\citeauthoryear####1####2####3{%
            \def\reserved@b{####1}%
            \ifx\reserved@a\reserved@b
                ####3%
            \else
                \errmessage{Package acmart Error: author mismatch
                         in \string\citeN^^J^^J%
                    See the acmart package documentation for explanation}%
            \fi
        }%
    }%
    \@ifstar\@citeyear\@citeyear
}
%\def\shortciteN{%
\DeclareRobustCommand\shortciteN{%			% January 2008
    \@citeRB
    \def\citeauthoryear##1##2##3{##2\ [##3%
        \def\reserved@a{##2}%
        \def\citeauthoryear####1####2####3{%
            \def\reserved@b{####2}%
            \ifx\reserved@a\reserved@b
                ####3%
            \else
                \errmessage{Package acmart Error: author mismatch
                         in \string\shortciteN^^J^^J%
                    See the acmart package documentation for explanation}%
            \fi
        }%
    }%
    \@ifstar\@citeyear\@citeyear  % GM July 2000
}

\def\@citenormal{%
    \@ifnextchar [{\@tempswatrue\@citex;}%
% original                 {\@tempswafalse\@citex,[]}% was ; Gerry 2/24/00
{\@tempswafalse\@citex[]}%  	% GERRY FIX FOR BABEL 3/20/2009
}

\def\@citeyear{%
    \@ifnextchar [{\@tempswatrue\@citex,}%
% original                  {\@tempswafalse\@citex,[]}%
{\@tempswafalse\@citex[]}%	%  GERRY FIX FOR BABEL 3/20/2009
}

\def\@citex#1[#2]#3{%
    \let\@citea\@empty
    \@cite{%
        \@for\@citeb:=#3\do{%
            \@citea
% original            \def\@citea{#1 }%
            \def\@citea{#1, }% 	% GERRY FIX FOR BABEL 3/20/2009 -- SO THAT YOU GET [1, 2] IN THE BODY TEXT
            \edef\@citeb{\expandafter\@iden\@citeb}%
            \if@filesw
                \immediate\write\@auxout{\string\citation{\@citeb}}%
            \fi
            \@ifundefined{b@\@citeb}{%
                {\bf ?}%
                \@warning{%
                    Citation `\@citeb' on page \thepage\space undefined%
                }%
            }%
            {\csname b@\@citeb\endcsname}%
        }%
    }{#2}%
}
%\let\@biblabel\@gobble   % Dec. 2008 - Gerry
% ----
\def\@biblabelnum#1{[#1]} % Gerry's solution #1 - for Natbib -- April 2009
\let\@biblabel=\@biblabelnum  % Gerry's solution #1 - for Natbib -- April 2009
\def\newblock{\relax} % Gerry Dec. 2008
% ---
\newdimen\bibindent
\setcounter{enumi}{1}
\bibindent=0em
\def\thebibliography#1{% 
\ifnum\addauflag=0\addauthorsection\global\addauflag=1\fi
     \section[References]{%    <=== OPTIONAL ARGUMENT ADDED HERE
        {References} % was uppercased but this affects pdf bookmarks (SP/GM October 2004)
          {\vskip -9pt plus 1pt} % GM Nov. 2006 / GM July 2000 (for somewhat tighter spacing) 
         \@mkboth{{\refname}}{{\refname}}%
     }%
     \list{[\arabic{enumi}]}{%
         \settowidth\labelwidth{[#1]}%
         \leftmargin\labelwidth
         \advance\leftmargin\labelsep
         \advance\leftmargin\bibindent
         \parsep=0pt\itemsep=1pt % GM July 2000
         \itemindent -\bibindent
         \listparindent \itemindent
         \usecounter{enumi}
     }%
     \let\newblock\@empty
     \raggedright % GM July 2000
     \sloppy
     \sfcode`\.=1000\relax
}

\gdef\balancecolumns
{\vfill\eject
\global\@colht=\textheight
\global\ht\@cclv=\textheight
}

\newcount\colcntr
\global\colcntr=0
%\newbox\savebox
\newbox\saveb@x				% January 2008

\gdef \@makecol {%
\global\advance\colcntr by 1
\ifnum\colcntr>2 \global\colcntr=1\fi
   \ifvoid\footins
     \setbox\@outputbox \box\@cclv
   \else
     \setbox\@outputbox \vbox{%
\boxmaxdepth \@maxdepth
       \@tempdima\dp\@cclv
       \unvbox \@cclv
       \vskip-\@tempdima
       \vskip \skip\footins
       \color@begingroup
         \normalcolor
         \footnoterule
         \unvbox \footins
       \color@endgroup
       }%
   \fi
   \xdef\@freelist{\@freelist\@midlist}%
   \global \let \@midlist \@empty
   \@combinefloats
   \ifvbox\@kludgeins
     \@makespecialcolbox
   \else
     \setbox\@outputbox \vbox to\@colht {%
\@texttop
       \dimen@ \dp\@outputbox
       \unvbox \@outputbox
   \vskip -\dimen@
       \@textbottom
       }%
   \fi
   \global \maxdepth \@maxdepth
}
\def\titlenote{\@ifnextchar[\@xtitlenote{\stepcounter\@mpfn
\global\advance\titlenotecount by 1
\ifnum\titlenotecount=1
    \raisebox{9pt}{$\ast$}
\fi
\ifnum\titlenotecount=2
    \raisebox{9pt}{$\dagger$}
\fi
\ifnum\titlenotecount=3
    \raisebox{9pt}{$\ddagger$}
\fi
\ifnum\titlenotecount=4
\raisebox{9pt}{$\S$}
\fi
\ifnum\titlenotecount=5
\raisebox{9pt}{$\P$}
\fi
         \@titlenotetext
}}

\long\def\@titlenotetext#1{\insert\footins{%
\ifnum\titlenotecount=1\global\tntoks={#1}\fi
\ifnum\titlenotecount=2\global\tntokstwo={#1}\fi
\ifnum\titlenotecount=3\global\tntoksthree={#1}\fi
\ifnum\titlenotecount=4\global\tntoksfour={#1}\fi
\ifnum\titlenotecount=5\global\tntoksfive={#1}\fi
    \reset@font\footnotesize
    \interlinepenalty\interfootnotelinepenalty
    \splittopskip\footnotesep
    \splitmaxdepth \dp\strutbox \floatingpenalty \@MM
    \hsize\columnwidth \@parboxrestore
    \protected@edef\@currentlabel{%
    }%
    \color@begingroup
   \color@endgroup}}

%%%%%%%%%%%%%%%%%%%%%%%%%
\ps@plain
\baselineskip=11pt
\let\thepage\relax % For NO page numbers - GM Nov. 30th. 1999 and July 2000
\def\setpagenumber#1{\global\setcounter{page}{#1}}
%\pagenumbering{arabic}  % Arabic page numbers GM July 2000
\twocolumn             % Double column.
\flushbottom           % Even bottom -- alas, does not balance columns at end of document
\pagestyle{plain}

% Need Copyright Year and Copyright Data to be user definable (in .tex file).
% Gerry Nov. 30th. 1999
\newtoks\copyrtyr
\newtoks\acmcopyr
\newtoks\boilerplate
\global\acmcopyr={X-XXXXX-XX-X/XX/XX}  % Default - 5/11/2001 *** Gerry
\global\copyrtyr={\the\year}                % Default - 3/3/2003 *** Gerry
\def\acmPrice#1{\gdef\@acmPrice{#1}}
\acmPrice{} %article price  % Changed to 15 - June 2012 - Gerry

\def\CopyrightYear#1{\global\copyrtyr{#1}}
\def\crdata#1{\global\acmcopyr{#1}}
\def\permission#1{\global\boilerplate{#1}}

% ISBN
%
\def\isbn#1{\global\acmcopyr={#1}}
\isbn{978-1-4503-2138-9}

\RequirePackage{url}
\urlstyle{rm}
\def\doi#1{\def\@doi{#1}}
\doi{10.1145/1235}
\def\printdoi#1{\url{#1}}

% Copyright
\RequirePackage{acmcopyright}
\setcopyright{none}

%
\global\boilerplate={\@copyrightpermission}

\newtoks\copyrightetc
\ifnum\ACM@basesize=9\relax
\global\copyrightetc{%
{\noindent\confname\ \the\conf\ \the\confinfo}\par\smallskip
  \if@printcopyright
    \copyright\ \the\copyrtyr\ \@copyrightowner
  \fi
  \if@acmowned ISBN \else\ifnum\acm@copyrightmode=2 ISBN \else \par\smallskip ACM ISBN \fi\fi
 \the\acmcopyr\ifx\@acmPrice\@empty.\else\dots\@acmPrice\fi\par\smallskip
{DOI: \small\expandafter\printdoi\expandafter{\@doi}}} 
\toappear{\fontsize{7pt}{8pt}\fontfamily{ptm}\selectfont
  \the\boilerplate\par\smallskip
 \the\copyrightetc}
\fi
\ifnum\ACM@basesize=10\relax
\global\copyrightetc{%
{\noindent\confname\ \the\conf\ \the\confinfo}\par\smallskip
  \if@printcopyright
    \copyright\ \the\copyrtyr\ \@copyrightowner
  \fi
  \if@acmowned ISBN \else\ifnum\acm@copyrightmode=2 ISBN \else \par\smallskip ACM ISBN \fi\fi
 \the\acmcopyr\ifx\@acmPrice\@empty.\else\dots\@acmPrice\fi\par\smallskip
{DOI: \small\expandafter\printdoi\expandafter{\@doi}}} 
\toappear{\fontsize{7.5pt}{8.5pt}\fontfamily{ptm}\selectfont
  \the\boilerplate\par\smallskip
 \the\copyrightetc}
\fi
%\DeclareFixedFont{\altcrnotice}{OT1}{tmr}{m}{n}{8}  % << patch needed for accenting e.g. Montreal - Gerry, May 2007
%\DeclareFixedFont{\altconfname}{OT1}{tmr}{m}{it}{8}  % << patch needed for accenting in italicized confname - Gerry, May 2007
%
%{\altconfname{{\the\conf}}} {\altcrnotice\the\confinfo\par} \the\copyrightetc.}  % << Gerry, May 2007
%
% The following section (i.e. 3 .sty inclusions) was added in May 2007 so as to fix the problems that many
% authors were having with accents. Sometimes accents would occur, but the letter-character would be of a different
% font. Conversely the letter-character font would be correct but, e.g. a 'bar' would appear superimposed on the
% character instead of, say, an unlaut/diaresis. Sometimes the letter-character would NOT appear at all.
% Using [T1]{fontenc} outright was not an option as this caused 99% of the authors to 'produce' a Type-3 (bitmapped)
% PDF file - useless for production. 
%
% For proper (font) accenting we NEED these packages to be part of the .cls file i.e. 'ae', 'aecompl' and 'aeguil' 
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%% This is file `ae.sty' 
\def\fileversion{1.3}
\def\filedate{2001/02/12}
\NeedsTeXFormat{LaTeX2e}
%\ProvidesPackage{ae}[\filedate\space\fileversion\space  % GM
% Almost European Computer Modern]                       % GM - keeping the log file clean(er)
\newif\if@ae@slides \@ae@slidesfalse
\DeclareOption{slides}{\@ae@slidestrue}
\ProcessOptions
\fontfamily{aer}
\RequirePackage[T1]{fontenc}
\if@ae@slides
    \renewcommand{\sfdefault}{laess}
    \renewcommand{\rmdefault}{laess} % no roman
    \renewcommand{\ttdefault}{laett}
\else
    \renewcommand{\sfdefault}{aess}
    \renewcommand{\rmdefault}{aer}
    \renewcommand{\ttdefault}{aett}
\fi
\endinput
%% 
%% End of file `ae.sty'.
%
%
\def\fileversion{0.9}
\def\filedate{1998/07/23}
\NeedsTeXFormat{LaTeX2e}
%\ProvidesPackage{aecompl}[\filedate\space\fileversion\space   % GM
%T1 Complements for AE fonts (D. Roegel)]                      % GM -- keeping the log file clean(er)
 
\def\@ae@compl#1{{\fontencoding{T1}\fontfamily{cmr}\selectfont\symbol{#1}}}
\def\guillemotleft{\@ae@compl{19}}
\def\guillemotright{\@ae@compl{20}}
\def\guilsinglleft{\@ae@compl{14}}
\def\guilsinglright{\@ae@compl{15}}
\def\TH{\@ae@compl{222}}
\def\NG{\@ae@compl{141}}
\def\ng{\@ae@compl{173}}
\def\th{\@ae@compl{254}}
\def\DJ{\@ae@compl{208}}
\def\dj{\@ae@compl{158}}
\def\DH{\@ae@compl{208}}
\def\dh{\@ae@compl{240}}
\def\@perthousandzero{\@ae@compl{24}}
\def\textperthousand{\%\@perthousandzero}
\def\textpertenthousand{\%\@perthousandzero\@perthousandzero}
\endinput
%
%
%% This is file `aeguill.sty' 
% This file gives french guillemets (and not guillemots!)
% built with the Polish CMR fonts (default), WNCYR fonts, the LASY fonts 
% or with the EC fonts. 
% This is useful in conjunction with the ae package
% (this package loads the ae package in case it has not been loaded)
%  and with or without the french(le) package.
%
% In order to get the guillemets, it is necessary to either type
% \guillemotleft and \guillemotright, or to use an 8 bit encoding
% (such as ISO-Latin1) which selects these two commands, 
% or, if you use the french package (but not the frenchle package), 
% to type << or >>.
%
% By default, you get the Polish CMR guillemets; if this package is loaded
% with the `cm' option, you get the LASY guillemets; with `ec,' you
% get the EC guillemets, and with `cyr,' you get the cyrillic guillemets.
%
% In verbatim mode, you always get the EC/TT guillemets.
%
% The default option is interesting in conjunction with PDF,
% because there is a Type 1 version of the Polish CMR fonts
% and these guillemets are very close in shape to the EC guillemets.
% There are no free Type 1 versions of the EC fonts.
%
% Support for Polish CMR guillemets was kindly provided by 
% Rolf Niepraschk <niepraschk@ptb.de> in version 0.99 (2000/05/22).
% Bernd Raichle provided extensive simplifications to the code
% for version 1.00.
%
% This package is released under the LPPL.
%
% Changes:
%   Date        version
%   2001/04/12  1.01    the frenchle and french package are now distinguished.
%
\def\fileversion{1.01}
\def\filedate{2001/04/12}
\NeedsTeXFormat{LaTeX2e}
%\ProvidesPackage{aeguill}[2001/04/12 1.01 %    % GM
%AE fonts with french guillemets (D. Roegel)]   % GM - keeping the log file clean(er)
%\RequirePackage{ae}  % GM May 2007 - already embedded here

\newcommand{\@ae@switch}[4]{#4}
\DeclareOption{ec}{\renewcommand\@ae@switch[4]{#1}}
\DeclareOption{cm}{\renewcommand\@ae@switch[4]{#2}}
\DeclareOption{cyr}{\renewcommand\@ae@switch[4]{#3}}
\DeclareOption{pl}{\renewcommand\@ae@switch[4]{#4}}

%
% Load necessary packages
%
\@ae@switch{% ec
  % do nothing
}{% cm
  \RequirePackage{latexsym}%  GM - May 2007 - already 'mentioned as required' up above
}{% cyr
  \RequirePackage[OT2,T1]{fontenc}%
}{% pl
  \RequirePackage[OT4,T1]{fontenc}%
}

% The following command will be compared to \frenchname,
% as defined in french.sty and frenchle.sty.
\def\aeguillfrenchdefault{french}%

\let\guill@verbatim@font\verbatim@font
\def\verbatim@font{\guill@verbatim@font\ecguills{cmtt}%
                   \let\guillemotleft\@oguills\let\guillemotright\@fguills}

\begingroup \catcode`\<=13 \catcode`\>=13
\def\x{\endgroup
 \def\ae@lfguill{<<}%
 \def\ae@rfguill{>>}%
}\x

\newcommand{\ecguills}[1]{%
  \def\selectguillfont{\fontencoding{T1}\fontfamily{#1}\selectfont}%
  \def\@oguills{{\selectguillfont\symbol{19}}}%
  \def\@fguills{{\selectguillfont\symbol{20}}}%
  }

\newcommand{\aeguills}{%
  \ae@guills
  % We redefine \guillemotleft and \guillemotright
  % in order to catch them when they are used 
  % with \DeclareInputText (in latin1.def for instance)
  % We use \auxWARNINGi as a safe indicator that french.sty is used.
  \gdef\guillemotleft{\ifx\auxWARNINGi\undefined
                         \@oguills % neither french.sty nor frenchle.sty
                      \else
                         \ifx\aeguillfrenchdefault\frenchname
                           \ae@lfguill  % french.sty
                         \else
                           \@oguills    % frenchle.sty
                         \fi
                      \fi}%
  \gdef\guillemotright{\ifx\auxWARNINGi\undefined
                         \@fguills % neither french.sty nor frenchle.sty
                       \else
                         \ifx\aeguillfrenchdefault\frenchname
                           \ae@rfguill  % french.sty
                         \else
                           \@fguills    % frenchle.sty
                         \fi
                       \fi}%
  }

%
% Depending on the class option
% define the internal command \ae@guills
\@ae@switch{% ec
  \newcommand{\ae@guills}{%
    \ecguills{cmr}}%
}{% cm
  \newcommand{\ae@guills}{%
    \def\selectguillfont{\fontencoding{U}\fontfamily{lasy}%
            \fontseries{m}\fontshape{n}\selectfont}%
    \def\@oguills{\leavevmode\nobreak
                \hbox{\selectguillfont (\kern-.20em(\kern.20em}\nobreak}%
    \def\@fguills{\leavevmode\nobreak
                \hbox{\selectguillfont \kern.20em)\kern-.2em)}%
                \ifdim\fontdimen\@ne\font>\z@\/\fi}}%
}{% cyr
  \newcommand{\ae@guills}{%
    \def\selectguillfont{\fontencoding{OT2}\fontfamily{wncyr}\selectfont}%
    \def\@oguills{{\selectguillfont\symbol{60}}}%
    \def\@fguills{{\selectguillfont\symbol{62}}}}
}{% pl
  \newcommand{\ae@guills}{%
    \def\selectguillfont{\fontencoding{OT4}\fontfamily{cmr}\selectfont}%
    \def\@oguills{{\selectguillfont\symbol{174}}}%
    \def\@fguills{{\selectguillfont\symbol{175}}}}
}

\AtBeginDocument{%
  \ifx\GOfrench\undefined
    \aeguills
  \else
    \let\aeguill@GOfrench\GOfrench
    \gdef\GOfrench{\aeguill@GOfrench \aeguills}%
  \fi
  }

\endinput
%

Show notes View sig-alternate.cls 100644 → 0
% SIG-ALTERNATE.CLS - VERSION 2.8 % "COMPATIBLE" WITH THE "ACM_PROC_ARTICLE-SP.CLS" V3.2SP % Gerald Murray - May 23rd 2012 % Boris Veytsman - April 23 2013 % Boris Veytsman - May 12 2013 % Boris Veytsman - June 09 2013 % Boris Veytsman - August 12 2013 % % ---- Start of 'updates' ---- % Added new permission/copyright statement - BV % Changed $10 fee to $15 -- May 2012 -- Gerry % Changed $5 fee to $10 -- April 2009 -- Gerry % April 22nd. 2009 - Fixed 'Natbib' incompatibility problem - Gerry % April 22nd. 2009 - Fixed 'Babel' incompatibility problem - Gerry % April 22nd. 2009 - Inserted various bug-fixes and improvements - Gerry % % To produce Type 1 fonts in the document plus allow for 'normal LaTeX accenting' in the critical areas; % title, author block, section-heads, confname, etc. etc. % i.e. the whole purpose of this version update is to NOT resort to 'inelegant accent patches'. % After much research, three extra .sty packages were added to the the tail (ae, aecompl, aeguill) to solve, % in particular, the accenting problem(s). We _could_ ask authors (via instructions/sample file) to 'include' these in % the source .tex file - in the preamble - but if everything is already provided ('behind the scenes' - embedded IN the .cls) % then this is less work for authors and also makes everything appear 'vanilla'. % NOTE: all 'patchwork accenting" has been commented out (here) and is no longer 'used' in the sample .tex file (either). % Gerry June 2007 % % Patch for accenting in conference name/location. Gerry May 3rd. 2007 % Rule widths changed to .5, author count (>6) fixed, roll-back for Type 3 problem. Gerry March 20th. 2007 % Changes made to 'modernize' the fontnames but esp. for MikTeX users V2.4/2.5 - Nov. 30th. 2006 % Updated the \email definition to allow for its use inside of 'shared affiliations' - Nov. 30th. 2006 % Fixed the 'section number depth value' - Nov. 30th. 2006 % % Footnotes inside table cells using \minipage (Oct. 2002) % Georgia fixed bug in sub-sub-section numbering in paragraphs (July 29th. 2002) % JS/GM fix to vertical spacing before Proofs (July 30th. 2002) % % Made the Permission Statement / Conference Info / Copyright Info % 'user definable' in the source .tex file OR automatic if % not specified. % % Allowance made to switch default fonts between those systems using % normal/modern font names and those using 'Type 1' or 'Truetype' fonts. % See LINE NUMBER 255 for details. % Also provided for enumerated/annotated Corollaries 'surrounded' by % enumerated Theorems (line 848). % Gerry November 11th. 1999 % % ---- End of 'updates' ---- % \def\fileversion{v2.9} % for ACM's tracking purposes \def\filedate{August 12, 2013} % Gerry Murray's tracking data \def\docdate {\filedate} \usepackage{epsfig} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % Need this for accents in Arial/Helvetica %\usepackage[T1]{fontenc} % Gerry March 12, 2007 - causes Type 3 problems (body text) %\usepackage{textcomp} % % SIG-ALTERNATE DOCUMENT STYLE % G.K.M. Tobin August-October 1999 % adapted from ARTICLE document style by Ken Traub, Olin Shivers % also using elements of esub2acm.cls % HEAVILY MODIFIED, SUBSEQUENTLY, BY GERRY MURRAY 2000 % ARTICLE DOCUMENT STYLE -- Released 16 March 1988 % for LaTeX version 2.09 % Copyright (C) 1988 by Leslie Lamport % % %%% sig-alternate.cls is an 'ALTERNATE' document style for producing %%% two-column camera-ready pages for ACM conferences. %%% THIS FILE DOES NOT STRICTLY ADHERE TO THE SIGS (BOARD-ENDORSED) %%% PROCEEDINGS STYLE. It has been designed to produce a 'tighter' %%% paper in response to concerns over page budgets. %%% The main features of this style are: %%% %%% 1) Two columns. %%% 2) Side and top margins of 4.5pc, bottom margin of 6pc, column gutter of %%% 2pc, hence columns are 20pc wide and 55.5pc tall. (6pc =3D 1in, approx) %%% 3) First page has title information, and an extra 6pc of space at the %%% bottom of the first column for the ACM copyright notice. %%% 4) Text is 9pt on 10pt baselines; titles (except main) are 9pt bold. %%% %%% %%% There are a few restrictions you must observe: %%% %%% 1) You cannot change the font size; ACM wants you to use 9pt. %%% 3) You must start your paper with the \maketitle command. Prior to the %%% \maketitle you must have \title and \author commands. If you have a %%% \date command it will be ignored; no date appears on the paper, since %%% the proceedings will have a date on the front cover. %%% 4) Marginal paragraphs, tables of contents, lists of figures and tables, %%% and page headings are all forbidden. %%% 5) The `figure' environment will produce a figure one column wide; if you %%% want one that is two columns wide, use `figure'. %%% % %%% Copyright Space: %%% This style automatically reserves 1" blank space at the bottom of page 1/ %%% column 1. This space can optionally be filled with some text using the %%% \toappear{...} command. If used, this command must be BEFORE the \maketitle %%% command. If this command is defined AND [preprint] is on, then the %%% space is filled with the {...} text (at the bottom); otherwise, it is %%% blank. If you use \toappearbox{...} instead of \toappear{...} then a %%% box will be drawn around the text (if [preprint] is on). %%% %%% A typical usage looks like this: %%% \toappear{To appear in the Ninth AES Conference on Medievil Lithuanian %%% Embalming Technique, June 1991, Alfaretta, Georgia.} %%% This will be included in the preprint, and left out of the conference %%% version. %%% %%% WARNING: %%% Some dvi-ps converters heuristically allow chars to drift from their %%% true positions a few pixels. This may be noticeable with the 9pt sans-serif %%% bold font used for section headers. %%% You may turn this hackery off via the -e option: %%% dvips -e 0 foo.dvi >foo.ps %%% \typeout{Document Class 'sig-alternate' <9th June '13>. Modified by G.K.M. Tobin/Gerry Murray/Boris Veytsman} \typeout{Based in part upon document Style `acmconf' <22 May 89>. Hacked 4/91 by} \typeout{shivers@cs.cmu.edu, 4/93 by theobald@cs.mcgill.ca} \typeout{Excerpts were taken from (Journal Style) 'esub2acm.cls'.} \typeout{*** Bugs/comments/suggestions/technicalities to Gerry Murray -- murray@hq.acm.org ***} \typeout{Questions on the style, SIGS policies, etc. to Adrienne Griscti griscti@acm.org} % New option code by BV \newcount\ACM@basesize \ACM@basesize=9\relax \DeclareOption{9pt}{\ACM@basesize=9\relax} \DeclareOption{10pt}{\ACM@basesize=10\relax} \DeclareOption{11pt}{\ClassError{sig-alternate}{The `11pt' option is not allowed}{sig-alternate now exists in 9pt and 10pt versions only}} \DeclareOption{12pt}{\ClassError{sig-alternate}{The `12pt' option is not allowed}{sig-alternate now exists in 9pt and 10pt versions only}} \ExecuteOptions{9pt} \ProcessOptions \let\@concepts\@empty % Support for CCSXML file \RequirePackage{comment} \excludecomment{CCSXML} % New concepts scheme % % The first argument is the significance, the % second is the concept(s) % \newcommand\ccsdesc[2][100]{% \ccsdesc@parse#1~#2~} % % The parser of the expression Significance~General~Specific % \def\ccsdesc@parse#1~#2~#3~{% \expandafter\ifx\csname CCS@#2\endcsname\relax \expandafter\gdef\csname CCS@#2\endcsname{\textbullet\textbf{#2} $\to$ }% \g@addto@macro{\@concepts}{\csname CCS@#2\endcsname}\fi \expandafter\g@addto@macro\expandafter{\csname CCS@#2\endcsname}{% \ifnum#1>499\textbf{#3; }\else \ifnum#1>299\textit{#3; }\else #3; \fi\fi}} \newcommand\printccsdesc{% \ifx\@concepts\@empty\else \if@twocolumn \section{CCS Concepts} \@concepts \else \small \quotation{\@concepts}% \fi \fi} \def\doi#1{\def\@doi{#1}} \doi{http://dx.doi.org/10.1145/0000000.0000000} \oddsidemargin 4.5pc \evensidemargin 4.5pc \advance\oddsidemargin by -1in % Correct for LaTeX gratuitousness \advance\evensidemargin by -1in % Correct for LaTeX gratuitousness \marginparwidth 0pt % Margin pars are not allowed. \marginparsep 11pt % Horizontal space between outer margin and % marginal note % Top of page: \topmargin 4.5pc % Nominal distance from top of page to top of % box containing running head. \advance\topmargin by -1in % Correct for LaTeX gratuitousness \headheight 0pt % Height of box containing running head. \headsep 0pt % Space between running head and text. % Bottom of page: \footskip 30pt % Distance from baseline of box containing foot % to baseline of last line of text. \@ifundefined{footheight}{\newdimen\footheight}{}% this is for LaTeX2e \footheight 12pt % Height of box containing running foot. %% Must redefine the top margin so there's room for headers and %% page numbers if you are using the preprint option. Footers %% are OK as is. Olin. \advance\topmargin by -37pt % Leave 37pt above text for headers \headheight 12pt % Height of box containing running head. \headsep 25pt % Space between running head and text. \textheight 666pt % 9 1/4 column height \textwidth 42pc % Width of text line. % For two-column mode: \columnsep 2pc % Space between columns \columnseprule 0pt % Width of rule between columns. \hfuzz 1pt % Allow some variation in column width, otherwise it's % too hard to typeset in narrow columns. \ifnum\ACM@basesize=9\relax \footnotesep 5.6pt % Height of strut placed at the beginning of every % footnote =3D height of normal \footnotesize strut, % so no extra space between footnotes. \fi \ifnum\ACM@basesize=10\relax \footnotesep 6.22pt % Height of strut placed at the beginning of every % footnote =3D height of normal \footnotesize strut, % so no extra space between footnotes. \fi \skip\footins 8.1pt plus 4pt minus 2pt % Space between last line of text and % top of first footnote. \floatsep 11pt plus 2pt minus 2pt % Space between adjacent floats moved % to top or bottom of text page. \textfloatsep 18pt plus 2pt minus 4pt % Space between main text and floats % at top or bottom of page. \intextsep 11pt plus 2pt minus 2pt % Space between in-text figures and % text. \@ifundefined{@maxsep}{\newdimen\@maxsep}{}% this is for LaTeX2e \@maxsep 18pt % The maximum of \floatsep, % \textfloatsep and \intextsep (minus % the stretch and shrink). \dblfloatsep 11pt plus 2pt minus 2pt % Same as \floatsep for double-column % figures in two-column mode. \dbltextfloatsep 18pt plus 2pt minus 4pt% \textfloatsep for double-column % floats. \@ifundefined{@dblmaxsep}{\newdimen\@dblmaxsep}{}% this is for LaTeX2e \@dblmaxsep 18pt % The maximum of \dblfloatsep and % \dbltexfloatsep. \@fptop 0pt plus 1fil % Stretch at top of float page/column. (Must be % 0pt plus ...) \@fpsep 8pt plus 2fil % Space between floats on float page/column. \@fpbot 0pt plus 1fil % Stretch at bottom of float page/column. (Must be % 0pt plus ... ) \@dblfptop 0pt plus 1fil % Stretch at top of float page. (Must be 0pt plus ...) \@dblfpsep 8pt plus 2fil % Space between floats on float page. \@dblfpbot 0pt plus 1fil % Stretch at bottom of float page. (Must be % 0pt plus ... ) \marginparpush 5pt % Minimum vertical separation between two marginal % notes. \parskip 0pt plus 1pt % Extra vertical space between % paragraphs. \ifnum\ACM@basesize=9\relax \parindent 9pt % GM July 2000 / was 0pt - width of paragraph % indentation. \fi \ifnum\ACM@basesize=10\relax \parindent 10pt % GM July 2000 / was 0pt - width of paragraph % indentation. \fi \partopsep 2pt plus 1pt minus 1pt% Extra vertical space, in addition to % \parskip and \topsep, added when user % leaves blank line before environment. \@lowpenalty 51 % Produced by \nopagebreak[1] or \nolinebreak[1] \@medpenalty 151 % Produced by \nopagebreak[2] or \nolinebreak[2] \@highpenalty 301 % Produced by \nopagebreak[3] or \nolinebreak[3] \@beginparpenalty -\@lowpenalty % Before a list or paragraph environment. \@endparpenalty -\@lowpenalty % After a list or paragraph environment. \@itempenalty -\@lowpenalty % Between list items. \RequirePackage{ifpdf}% \ifpdf \pdfpagewidth=8.5in \pdfpageheight=11in \fi \lineskip 2pt % \lineskip is 1pt for all font sizes. \normallineskip 2pt \def\baselinestretch{1} \ifnum\ACM@basesize=9\relax \abovedisplayskip 9pt plus2pt minus4.5pt% \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus3pt% \belowdisplayshortskip 5.4pt plus3pt minus3pt% \let\@listi\@listI % Setting of \@listi added 9 Jun 87 \def\small{\@setsize\small{9pt}\viiipt\@viiipt \abovedisplayskip 7.6pt plus 3pt minus 4pt% \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus2pt% \belowdisplayshortskip 3.6pt plus2pt minus 2pt \def\@listi{\leftmargin\leftmargini %% Added 22 Dec 87 \topsep 4pt plus 2pt minus 2pt\parsep 2pt plus 1pt minus 1pt \itemsep \parsep}} \def\footnotesize{\@setsize\footnotesize{9pt}\ixpt\@ixpt \abovedisplayskip 6.4pt plus 2pt minus 4pt% \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus 1pt% \belowdisplayshortskip 2.7pt plus 1pt minus 2pt \def\@listi{\leftmargin\leftmargini %% Added 22 Dec 87 \topsep 3pt plus 1pt minus 1pt\parsep 2pt plus 1pt minus 1pt \itemsep \parsep}} \fi \ifnum\ACM@basesize=10\relax \abovedisplayskip 10pt plus2pt minus4.5pt% \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus3pt% \belowdisplayshortskip 6pt plus3pt minus3pt% \let\@listi\@listI % Setting of \@listi added 9 Jun 87 \def\small{\@setsize\small{10pt}\ixpt\@ixpt \abovedisplayskip 8.5pt plus 3pt minus 4pt% \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus2pt% \belowdisplayshortskip 4pt plus2pt minus 2pt \def\@listi{\leftmargin\leftmargini %% Added 22 Dec 87 \topsep 4.5pt plus 2pt minus 2pt\parsep 2pt plus 1pt minus 1pt \itemsep \parsep}} \def\footnotesize{\@setsize\footnotesize{10pt}\xpt\@xpt \abovedisplayskip 7.6pt plus 2pt minus 4pt% \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus 1pt% \belowdisplayshortskip 3.0pt plus 1pt minus 2pt \def\@listi{\leftmargin\leftmargini %% Added 22 Dec 87 \topsep 3.2pt plus 1pt minus 1pt\parsep 2pt plus 1pt minus 1pt \itemsep \parsep}} \fi \newcount\aucount \newcount\originalaucount \newdimen\auwidth \auwidth=\textwidth \newdimen\auskip \newcount\auskipcount \newdimen\auskip \global\auskip=1pc \newdimen\allauboxes \allauboxes=\auwidth \newtoks\addauthors \newcount\addauflag \global\addauflag=0 %Haven't shown additional authors yet \newtoks\subtitletext \gdef\subtitle#1{\subtitletext={#1}} \gdef\additionalauthors#1{\addauthors={#1}} \gdef\numberofauthors#1{\global\aucount=#1 \ifnum\aucount>3\global\originalaucount=\aucount \global\aucount=3\fi %g} % 3 OK - Gerry March 2007 \global\auskipcount=\aucount\global\advance\auskipcount by 1 \global\multiply\auskipcount by 2 \global\multiply\auskip by \auskipcount \global\advance\auwidth by -\auskip \global\divide\auwidth by \aucount} % \and was modified to count the number of authors. GKMT 12 Aug 1999 \def\alignauthor{% % \begin{tabular} \end{tabular}% \begin{tabular}[t]{p{\auwidth}}\centering}% % * NOTE * NOTE * NOTE * NOTE * % If you have 'font problems' then you may need % to change these, e.g. 'arialb' instead of "arialbd". % Gerry Murray 11/11/1999 % * OR comment out block A and activate block B or vice versa. % ****************************************** % % -- Start of block A -- (Type 1 or Truetype fonts) %\newfont{\secfnt}{timesbd at 12pt} % was timenrb originally - now is timesbd %\newfont{\secit}{timesbi at 12pt} %13 Jan 00 gkmt %\newfont{\subsecfnt}{timesi at 11pt} % was timenrri originally - now is timesi %\newfont{\subsecit}{timesbi at 11pt} % 13 Jan 00 gkmt -- was times changed to timesbi gm 2/4/2000 % % because "normal" is italic, "italic" is Roman %\newfont{\ttlfnt}{arialbd at 18pt} % was arialb originally - now is arialbd %\newfont{\ttlit}{arialbi at 18pt} % 13 Jan 00 gkmt %\newfont{\subttlfnt}{arial at 14pt} % was arialr originally - now is arial %\newfont{\subttlit}{ariali at 14pt} % 13 Jan 00 gkmt %\newfont{\subttlbf}{arialbd at 14pt} % 13 Jan 00 gkmt %\newfont{\aufnt}{arial at 12pt} % was arialr originally - now is arial %\newfont{\auit}{ariali at 12pt} % 13 Jan 00 gkmt %\newfont{\affaddr}{arial at 10pt} % was arialr originally - now is arial %\newfont{\affaddrit}{ariali at 10pt} %13 Jan 00 gkmt %\newfont{\eaddfnt}{arial at 12pt} % was arialr originally - now is arial %\newfont{\ixpt}{times at 9pt} % was timenrr originally - now is times %\newfont{\confname}{timesi at 8pt} % was timenrri - now is timesi %\newfont{\crnotice}{times at 8pt} % was timenrr originally - now is times %\newfont{\ninept}{times at 9pt} % was timenrr originally - now is times % ***************************************** % -- End of block A -- % % % -- Start of block B -- UPDATED FONT NAMES % ***************************************** % Gerry Murray 11/30/2006 % ******************************************* \ifnum\ACM@basesize=9\relax \newfont{\secfnt}{ptmb8t at 12pt} \newfont{\secit}{ptmbi8t at 12pt} %13 Jan 00 gkmt \newfont{\subsecfnt}{ptmri8t at 11pt} \newfont{\subsecit}{ptmbi8t at 11pt} % \newfont{\ttlfnt}{phvb8t at 18pt} \newfont{\ttlit}{phvbo8t at 18pt} % GM 2/4/2000 \newfont{\subttlfnt}{phvr8t at 14pt} \newfont{\subttlit}{phvro8t at 14pt} % GM 2/4/2000 \newfont{\subttlbf}{phvb8t at 14pt} % 13 Jan 00 gkmt \newfont{\aufnt}{phvr8t at 12pt} \newfont{\auit}{phvro8t at 12pt} % GM 2/4/2000 \newfont{\affaddr}{phvr8t at 10pt} \newfont{\affaddrit}{phvro8t at 10pt} % GM 2/4/2000 \newfont{\eaddfnt}{phvr8t at 12pt} \newfont{\ixpt}{ptmr8t at 9pt} \newfont{\confname}{ptmri8t at 8pt} \newfont{\crnotice}{ptmr8t at 8pt} \newfont{\ninept}{ptmr8t at 9pt} \fi \ifnum\ACM@basesize=10\relax \newfont{\secfnt}{ptmb8t at 13pt} \newfont{\secit}{ptmbi8t at 13pt} %13 Jan 00 gkmt \newfont{\subsecfnt}{ptmri8t at 12pt} \newfont{\subsecit}{ptmbi8t at 12pt} % \newfont{\ttlfnt}{phvb8t at 20pt} \newfont{\ttlit}{phvbo8t at 20pt} % GM 2/4/2000 \newfont{\subttlfnt}{phvr8t at 15pt} \newfont{\subttlit}{phvro8t at 15pt} % GM 2/4/2000 \newfont{\subttlbf}{phvb8t at 15pt} % 13 Jan 00 gkmt \newfont{\aufnt}{phvr8t at 12pt} \newfont{\auit}{phvro8t at 12pt} % GM 2/4/2000 \newfont{\affaddr}{phvr8t at 11pt} \newfont{\affaddrit}{phvro8t at 11pt} % GM 2/4/2000 \newfont{\eaddfnt}{phvr8t at 12pt} \newfont{\ixpt}{ptmr8t at 10pt} \newfont{\confname}{ptmri8t at 9pt} \newfont{\crnotice}{ptmr8t at 9pt} \newfont{\ninept}{ptmr8t at 10pt} \fi % +++++++++++++++++++++++++++++++++++++++++++++ % -- End of block B -- %\def\email#1{{{\eaddfnt{\vskip 4pt#1}}}} % If we have an email, inside a "shared affiliation" then we need the following instead \def\email#1{{{\eaddfnt{\par #1}}}} % revised - GM - 11/30/2006 \def\addauthorsection{\ifnum\originalaucount>6 % was 3 - Gerry March 2007 \section{Additional Authors}\the\addauthors \fi} \newcount\savesection \newcount\sectioncntr \global\sectioncntr=1 \setcounter{secnumdepth}{3} \def\appendix{\par \section{APPENDIX} \setcounter{section}{0} \setcounter{subsection}{0} \def\thesection{\Alph{section}} } \leftmargini 22.5pt \leftmarginii 19.8pt % > \labelsep + width of '(m)' \leftmarginiii 16.8pt % > \labelsep + width of 'vii.' \leftmarginiv 15.3pt % > \labelsep + width of 'M.' \leftmarginv 9pt \leftmarginvi 9pt \leftmargin\leftmargini \labelsep 4.5pt \labelwidth\leftmargini\advance\labelwidth-\labelsep \def\@listI{\leftmargin\leftmargini \parsep 3.6pt plus 2pt minus 1pt% \topsep 7.2pt plus 2pt minus 4pt% \itemsep 3.6pt plus 2pt minus 1pt} \let\@listi\@listI \@listi \def\@listii{\leftmargin\leftmarginii \labelwidth\leftmarginii\advance\labelwidth-\labelsep \topsep 3.6pt plus 2pt minus 1pt \parsep 1.8pt plus 0.9pt minus 0.9pt \itemsep \parsep} \def\@listiii{\leftmargin\leftmarginiii \labelwidth\leftmarginiii\advance\labelwidth-\labelsep \topsep 1.8pt plus 0.9pt minus 0.9pt \parsep \z@ \partopsep 1pt plus 0pt minus 1pt \itemsep \topsep} \def\@listiv{\leftmargin\leftmarginiv \labelwidth\leftmarginiv\advance\labelwidth-\labelsep} \def\@listv{\leftmargin\leftmarginv \labelwidth\leftmarginv\advance\labelwidth-\labelsep} \def\@listvi{\leftmargin\leftmarginvi \labelwidth\leftmarginvi\advance\labelwidth-\labelsep} \def\labelenumi{\theenumi.} \def\theenumi{\arabic{enumi}} \def\labelenumii{(\theenumii)} \def\theenumii{\alph{enumii}} \def\p@enumii{\theenumi} \def\labelenumiii{\theenumiii.} \def\theenumiii{\roman{enumiii}} \def\p@enumiii{\theenumi(\theenumii)} \def\labelenumiv{\theenumiv.} \def\theenumiv{\Alph{enumiv}} \def\p@enumiv{\p@enumiii\theenumiii} \def\labelitemi{$\bullet$} \def\labelitemii{\bf --} \def\labelitemiii{$\ast$} \def\labelitemiv{$\cdot$} \def\verse{\let\\=\@centercr \list{}{\itemsep\z@ \itemindent -1.5em\listparindent \itemindent \rightmargin\leftmargin\advance\leftmargin 1.5em}\item[]} \let\endverse\endlist \def\quotation{\list{}{\listparindent 1.5em \itemindent\listparindent \rightmargin\leftmargin \parsep 0pt plus 1pt}\item[]} \let\endquotation=\endlist \def\quote{\list{}{\rightmargin\leftmargin}\item[]} \let\endquote=\endlist \def\descriptionlabel#1{\hspace\labelsep \bf #1} \def\description{\list{}{\labelwidth\z@ \itemindent-\leftmargin \let\makelabel\descriptionlabel}} \let\enddescription\endlist \def\theequation{\arabic{equation}} \ifnum\ACM@basesize=9\relax \arraycolsep 4.5pt % Half the space between columns in an array environment. \tabcolsep 5.4pt % Half the space between columns in a tabular environment. \arrayrulewidth .5pt % Width of rules in array and tabular environment. % (was .4) updated Gerry March 20 2007 \doublerulesep 1.8pt % Space between adjacent rules in array or tabular env. \fi \ifnum\ACM@basesize=10\relax \arraycolsep 5pt % Half the space between columns in an array environment. \tabcolsep 6pt % Half the space between columns in a tabular environment. \arrayrulewidth .5pt % Width of rules in array and tabular environment. % (was .4) updated Gerry March 20 2007 \doublerulesep 1.8pt % Space between adjacent rules in array or tabular env. \fi \tabbingsep \labelsep % Space used by the \' command. (See LaTeX manual.) \skip\@mpfootins =\skip\footins \fboxsep =2.7pt % Space left between box and text by \fbox and \framebox. \fboxrule =.5pt % Width of rules in box made by \fbox and \framebox. % (was .4) updated Gerry March 20 2007 \def\thepart{\Roman{part}} % Roman numeral part numbers. \def\thesection {\arabic{section}} \def\thesubsection {\thesection.\arabic{subsection}} %\def\thesubsubsection {\thesubsection.\arabic{subsubsection}} % GM 7/30/2002 %\def\theparagraph {\thesubsubsection.\arabic{paragraph}} % GM 7/30/2002 \def\thesubparagraph {\theparagraph.\arabic{subparagraph}} \def\@pnumwidth{1.55em} \def\@tocrmarg {2.55em} \def\@dotsep{4.5} \setcounter{tocdepth}{3} %\def\tableofcontents{\@latexerr{\tableofcontents: Tables of contents are not % allowed in the `acmconf' document style.}\@eha} \def\tableofcontents{\ClassError{% \string\tableofcontents\space is not allowed in the `acmconf' document % January 2008 style}\@eha} \def\l@part#1#2{\addpenalty{\@secpenalty} \addvspace{2.25em plus 1pt} % space above part line \begingroup \@tempdima 3em % width of box holding part number, used by \parindent \z@ \rightskip \@pnumwidth %% \numberline \parfillskip -\@pnumwidth {\large \bf % set line in \large boldface \leavevmode % TeX command to enter horizontal mode. #1\hfil \hbox to\@pnumwidth{\hss #2}}\par \nobreak % Never break after part entry \endgroup} \def\l@section#1#2{\addpenalty{\@secpenalty} % good place for page break \addvspace{1.0em plus 1pt} % space above toc entry \@tempdima 1.5em % width of box holding section number \begingroup \parindent \z@ \rightskip \@pnumwidth \parfillskip -\@pnumwidth \bf % Boldface. \leavevmode % TeX command to enter horizontal mode. \advance\leftskip\@tempdima %% added 5 Feb 88 to conform to \hskip -\leftskip %% 25 Jan 88 change to \numberline #1\nobreak\hfil \nobreak\hbox to\@pnumwidth{\hss #2}\par \endgroup} \def\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}} \def\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}} \def\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}} \def\l@subparagraph{\@dottedtocline{5}{10em}{5em}} %\def\listoffigures{\@latexerr{\listoffigures: Lists of figures are not % allowed in the `acmconf' document style.}\@eha} \def\listoffigures{\ClassError{% \string\listoffigures\space is not allowed in the `acmconf' document % January 2008 style}\@eha} \def\l@figure{\@dottedtocline{1}{1.5em}{2.3em}} %\def\listoftables{\@latexerr{\listoftables: Lists of tables are not % allowed in the `acmconf' document style.}\@eha} %\let\l@table\l@figure \def\listoftables{\ClassError{% \string\listoftables\space is not allowed in the `acmconf' document % January 2008 style}\@eha} \let\l@table\l@figure \def\footnoterule{\kern-3\p@ \hrule width .5\columnwidth % (was .4) updated Gerry March 20 2007 \kern 2.6\p@} % The \hrule has default height of .4pt % (was .4) updated Gerry March 20 2007 % ------ \long\def\@makefntext#1{\noindent %\hbox to .5em{\hss$^{\@thefnmark}$}#1} % original \hbox to .5em{\hss\textsuperscript{\@thefnmark}}#1} % C. Clifton / GM Oct. 2nd. 2002 % ------- \long\def\@maketntext#1{\noindent #1} \long\def\@maketitlenotetext#1#2{\noindent \hbox to 1.8em{\hss$^{#1}$}#2} \setcounter{topnumber}{2} \def\topfraction{.7} \setcounter{bottomnumber}{1} \def\bottomfraction{.3} \setcounter{totalnumber}{3} \def\textfraction{.2} \def\floatpagefraction{.5} \setcounter{dbltopnumber}{2} \def\dbltopfraction{.7} \def\dblfloatpagefraction{.5} % \long\def\@makecaption#1#2{ \vskip \baselineskip \setbox\@tempboxa\hbox{\textbf{#1: #2}} \ifdim \wd\@tempboxa >\hsize % IF longer than one line: \textbf{#1: #2}\par % THEN set as ordinary paragraph. \else % ELSE center. \hbox to\hsize{\hfil\box\@tempboxa\hfil}\par \fi} % \long\def\@makecaption#1#2{ \vskip 10pt \setbox\@tempboxa\hbox{\textbf{#1: #2}} \ifdim \wd\@tempboxa >\hsize % IF longer than one line: \textbf{#1: #2}\par % THEN set as ordinary paragraph. \else % ELSE center. \hbox to\hsize{\hfil\box\@tempboxa\hfil} \fi} \@ifundefined{figure}{\newcounter {figure}} % this is for LaTeX2e \def\fps@figure{tbp} \def\ftype@figure{1} \def\ext@figure{lof} \def\fnum@figure{Figure \thefigure} \def\figure{\@float{figure}} %\let\endfigure\end@float \def\endfigure{\end@float} % Gerry January 2008 \@namedef{figure}{\@dblfloat{figure}} \@namedef{endfigure}{\end@dblfloat} \@ifundefined{table}{\newcounter {table}} % this is for LaTeX2e \def\fps@table{tbp} \def\ftype@table{2} \def\ext@table{lot} \def\fnum@table{Table \thetable} \def\table{\@float{table}} %\let\endtable\end@float \def\endtable{\end@float} % Gerry January 2008 \@namedef{table}{\@dblfloat{table}} \@namedef{endtable}{\end@dblfloat} \newtoks\titleboxnotes \newcount\titleboxnoteflag \def\maketitle{\par \begingroup \def\thefootnote{\fnsymbol{footnote}} \def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}} \twocolumn[\@maketitle] \@thanks \endgroup \setcounter{footnote}{0} \let\maketitle\relax \let\@maketitle\relax \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\gdef\@subtitle{}\let\thanks\relax \@copyrightspace} %% CHANGES ON NEXT LINES \newif\if@ll % to record which version of LaTeX is in use \expandafter\ifx\csname LaTeXe\endcsname\relax % LaTeX2.09 is used \else% LaTeX2e is used, so set ll to true \global\@lltrue \fi \if@ll \NeedsTeXFormat{LaTeX2e} \ProvidesClass{sig-alternate} [2013/05/12 v2.7 based on acmproc.cls V1.3 <Nov. 30 '99>] \RequirePackage{latexsym}% QUERY: are these two really needed? \let\dooptions\ProcessOptions \else \let\dooptions\@options \fi %% END CHANGES \def\@height{height} \def\@width{width} \def\@minus{minus} \def\@plus{plus} \def\hb@xt@{\hbox to} \newif\if@faircopy \@faircopyfalse \def\ds@faircopy{\@faircopytrue} \def\ds@preprint{\@faircopyfalse} \@twosidetrue \@mparswitchtrue \def\ds@draft{\overfullrule 5\p@} %% CHANGE ON NEXT LINE \dooptions \lineskip \p@ \normallineskip \p@ \def\baselinestretch{1} \def\@ptsize{0} %needed for amssymbols.sty %% CHANGES ON NEXT LINES \if@ll% allow use of old-style font change commands in LaTeX2e \@maxdepth\maxdepth % \DeclareOldFontCommand{\rm}{\ninept\rmfamily}{\mathrm} \DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf} \DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt} \DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf} \DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit} \DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl} \DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc} \DeclareRobustCommand{\cal}{\@fontswitch{\relax}{\mathcal}} \DeclareRobustCommand{\mit}{\@fontswitch{\relax}{\mathnormal}} \fi % \if@ll \renewcommand{\rmdefault}{cmr} % was 'ttm' % Note! I have also found 'mvr' to work ESPECIALLY well. % Gerry - October 1999 % You may need to change your LV1times.fd file so that sc is % mapped to cmcsc - -for smallcaps -- that is if you decide % to change {cmr} to {times} above. (Not recommended) \ifnum\ACM@basesize=9\relax \renewcommand{\@ptsize}{} \renewcommand{\normalsize}{% \@setfontsize\normalsize\@ixpt{10.5\p@}%\ninept% \abovedisplayskip 6\p@ \@plus2\p@ \@minus\p@ \belowdisplayskip \abovedisplayskip \abovedisplayshortskip 6\p@ \@minus 3\p@ \belowdisplayshortskip 6\p@ \@minus 3\p@ \let\@listi\@listI} \fi \ifnum\ACM@basesize=10\relax \renewcommand{\@ptsize}{} \renewcommand{\normalsize}{% \@setfontsize\normalsize\@xpt{11.5\p@}%\ninept% \abovedisplayskip 6.5\p@ \@plus2\p@ \@minus\p@ \belowdisplayskip \abovedisplayskip \abovedisplayshortskip 6.5\p@ \@minus 3\p@ \belowdisplayshortskip 6.5\p@ \@minus 3\p@ \let\@listi\@listI} \fi \else \def\@normalsize{%changed next to 9 from 10 \@setsize\normalsize{9\p@}\ixpt\@ixpt \abovedisplayskip 6\p@ \@plus2\p@ \@minus\p@ \belowdisplayskip \abovedisplayskip \abovedisplayshortskip 6\p@ \@minus 3\p@ \belowdisplayshortskip 6\p@ \@minus 3\p@ \let\@listi\@listI }% \fi \if@ll \ifnum\ACM@basesize=9\relax \newcommand\scriptsize{\@setfontsize\scriptsize\@viipt{8\p@}} \newcommand\tiny{\@setfontsize\tiny\@vpt{6\p@}} \newcommand\large{\@setfontsize\large\@xiipt{14\p@}} \newcommand\Large{\@setfontsize\Large\@xivpt{18\p@}} \newcommand\LARGE{\@setfontsize\LARGE\@xviipt{20\p@}} \newcommand\huge{\@setfontsize\huge\@xxpt{25\p@}} \newcommand\Huge{\@setfontsize\Huge\@xxvpt{30\p@}} \fi \ifnum\ACM@basesize=10\relax \newcommand\scriptsize{\@setfontsize\scriptsize\@viiipt{9\p@}} \newcommand\tiny{\@setfontsize\tiny\@vipt{7\p@}} \newcommand\large{\@setfontsize\large\@xiiipt{15\p@}} \newcommand\Large{\@setfontsize\Large\@xvpt{20\p@}} \newcommand\LARGE{\@setfontsize\LARGE\@xixpt{22\p@}} \newcommand\huge{\@setfontsize\huge\@xixpt{30\p@}} \newcommand\Huge{\@setfontsize\Huge30pt{36\p@}} \fi \else \def\scriptsize{\@setsize\scriptsize{8\p@}\viipt\@viipt} \def\tiny{\@setsize\tiny{6\p@}\vpt\@vpt} \def\large{\@setsize\large{14\p@}\xiipt\@xiipt} \def\Large{\@setsize\Large{18\p@}\xivpt\@xivpt} \def\LARGE{\@setsize\LARGE{20\p@}\xviipt\@xviipt} \def\huge{\@setsize\huge{25\p@}\xxpt\@xxpt} \def\Huge{\@setsize\Huge{30\p@}\xxvpt\@xxvpt} \fi \normalsize % make aubox hsize/number of authors up to 3, less gutter % then showbox gutter showbox gutter showbox -- GKMT Aug 99 \newbox\@acmtitlebox \ifnum\ACM@basesize=9\relax \def\@maketitle{\newpage \null \setbox\@acmtitlebox\vbox{% \baselineskip 20pt \vskip 2em % Vertical space above title. \begin{center} {\ttlfnt \@title\par} % Title set in 18pt Helvetica (Arial) bold size. \vskip 1.5em % Vertical space after title. %This should be the subtitle. {\subttlfnt \the\subtitletext\par}\vskip 1.25em%\fi {\baselineskip 16pt\aufnt % each author set in \12 pt Arial, in a \lineskip .5em % tabular environment \begin{tabular}[t]{c}\@author \end{tabular}\par} \vskip 1.5em % Vertical space after author. \end{center}} \dimen0=\ht\@acmtitlebox \advance\dimen0 by -12.75pc\relax % Increased space for title box -- KBT \unvbox\@acmtitlebox \ifdim\dimen0<0.0pt\relax\vskip-\dimen0\fi} \fi \ifnum\ACM@basesize=10\relax \def\@maketitle{\newpage \null \setbox\@acmtitlebox\vbox{% \baselineskip 22pt \vskip 2.2em % Vertical space above title. \begin{center} {\ttlfnt \@title\par} % Title set in 18pt Helvetica (Arial) bold size. \vskip 2em % Vertical space after title. %This should be the subtitle. {\subttlfnt \the\subtitletext\par}\vskip 1.25em%\fi {\baselineskip 18pt\aufnt % each author set in \12 pt Arial, in a \lineskip .5em % tabular environment \begin{tabular}[t]{c}\@author \end{tabular}\par} \vskip 2em % Vertical space after author. \end{center}} \dimen0=\ht\@acmtitlebox \advance\dimen0 by -12.75pc\relax % Increased space for title box -- KBT \unvbox\@acmtitlebox \ifdim\dimen0<0.0pt\relax\vskip-\dimen0\fi} \fi \newcount\titlenotecount \global\titlenotecount=0 \newtoks\tntoks \newtoks\tntokstwo \newtoks\tntoksthree \newtoks\tntoksfour \newtoks\tntoksfive \def\abstract{ \ifnum\titlenotecount>0 % was =1 \insert\footins{% \reset@font\footnotesize \interlinepenalty\interfootnotelinepenalty \splittopskip\footnotesep \splitmaxdepth \dp\strutbox \floatingpenalty \@MM \hsize\columnwidth \@parboxrestore \protected@edef\@currentlabel{% }% \color@begingroup \ifnum\titlenotecount=1 \@maketntext{% \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\@finalstrut\strutbox}% \fi \ifnum\titlenotecount=2 \@maketntext{% \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\dagger$}\rule\z@\footnotesep\ignorespaces\the\tntokstwo\@finalstrut\strutbox}% \fi \ifnum\titlenotecount=3 \@maketntext{% \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\dagger$}\rule\z@\footnotesep\ignorespaces\the\tntokstwo\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\ddagger$}\rule\z@\footnotesep\ignorespaces\the\tntoksthree\@finalstrut\strutbox}% \fi \ifnum\titlenotecount=4 \@maketntext{% \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\dagger$}\rule\z@\footnotesep\ignorespaces\the\tntokstwo\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\ddagger$}\rule\z@\footnotesep\ignorespaces\the\tntoksthree\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\S$}\rule\z@\footnotesep\ignorespaces\the\tntoksfour\@finalstrut\strutbox}% \fi \ifnum\titlenotecount=5 \@maketntext{% \raisebox{4pt}{$\ast$}\rule\z@\footnotesep\ignorespaces\the\tntoks\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\dagger$}\rule\z@\footnotesep\ignorespaces\the\tntokstwo\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\ddagger$}\rule\z@\footnotesep\ignorespaces\the\tntoksthree\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\S$}\rule\z@\footnotesep\ignorespaces\the\tntoksfour\par\@finalstrut\strutbox}% \@maketntext{% \raisebox{4pt}{$\P$}\rule\z@\footnotesep\ignorespaces\the\tntoksfive\@finalstrut\strutbox}% \fi \color@endgroup} %g} \fi \setcounter{footnote}{0} \section{ABSTRACT}\normalsize%\ninept } \def\endabstract{\if@twocolumn\else\endquotation\fi} \def\keywords{\if@twocolumn \section{Keywords} \else \small \quotation \fi} \def\terms#1{% %\if@twocolumn %\section{General Terms} %\else \small %\quotation %\fi } % -- Classification needs to be a bit smart due to optionals - Gerry/Georgia November 2nd. 1999 \newcount\catcount \global\catcount=1 \def\category#1#2#3{% \ifnum\catcount=1 \section{Categories and Subject Descriptors} \advance\catcount by 1\else{\unskip; }\fi \@ifnextchar [{\@category{#1}{#2}{#3}}{\@category{#1}{#2}{#3}[]}% } \def\@category#1#2#3[#4]{% \begingroup \let\and\relax #1 [\textbf{#2}]% \if!#4!% \if!#3!\else : #3\fi \else :\space \if!#3!\else #3\kern\z@---\hskip\z@\fi \textit{#4}% \fi \endgroup } % %%% This section (written by KBT) handles the 1" box in the lower left %%% corner of the left column of the first page by creating a picture, %%% and inserting the predefined string at the bottom (with a negative %%% displacement to offset the space allocated for a non-existent %%% caption). %%% \newtoks\copyrightnotice \def\ftype@copyrightbox{8} \def\@copyrightspace{ \@float{copyrightbox}[b] \begin{center} \setlength{\unitlength}{1pc} \ifnum\ACM@basesize=9 \begin{picture}(20,6) %Space for copyright notice \put(0,-0.95){\crnotice{\@toappear}} \end{picture} \fi \ifnum\ACM@basesize=10 \begin{picture}(20,7) %Space for copyright notice \put(0,-0.95){\crnotice{\@toappear}} \end{picture} \fi \end{center} \end@float} \def\@toappear{} % Default setting blank - commands below change this. \long\def\toappear#1{\def\@toappear{\parbox[b]{20pc}{\baselineskip 9pt#1}}} \def\toappearbox#1{\def\@toappear{\raisebox{5pt}{\framebox[20pc]{\parbox[b]{19pc}{#1}}}}} \newtoks\conf \newtoks\confinfo \def\conferenceinfo#1#2{\global\conf={#1}\global\confinfo{#2}} %\def\marginpar{\@latexerr{The \marginpar command is not allowed in the % `acmconf' document style.}\@eha} \def\marginpar{\ClassError{% \string\marginpar\space is not allowed in the `acmconf' document % January 2008 style}\@eha} \mark{{}{}} % Initializes TeX's marks \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \def\@begintheorem#1#2{% \parskip 0pt % GM July 2000 (for tighter spacing) \trivlist \item[% \hskip 10\p@ \hskip \labelsep {{\sc #1}\hskip 5\p@\relax#2.}% ] \it } \def\@opargbegintheorem#1#2#3{% \parskip 0pt % GM July 2000 (for tighter spacing) \trivlist \item[% \hskip 10\p@ \hskip \labelsep {\sc #1\ #2\ % This mod by Gerry to enumerate corollaries \setbox\@tempboxa\hbox{(#3)} % and bracket the 'corollary title' \ifdim \wd\@tempboxa>\z@ % and retain the correct numbering of e.g. theorems \hskip 5\p@\relax % if they occur 'around' said corollaries. \box\@tempboxa % Gerry - Nov. 1999. \fi.}% ] \it } \newif\if@qeded \global\@qededfalse % -- original %\def\proof{% % \vspace{-\parskip} % GM July 2000 (for tighter spacing) % \global\@qededfalse % \@ifnextchar[{\@xproof}{\@proof}% %} % -- end of original % (JSS) Fix for vertical spacing bug - Gerry Murray July 30th. 2002 \def\proof{% \vspace{-\lastskip}\vspace{-\parsep}\penalty-51% \global\@qededfalse \@ifnextchar[{\@xproof}{\@proof}% } \def\endproof{% \if@qeded\else\qed\fi \endtrivlist } \def\@proof{% \trivlist \item[% \hskip 10\p@ \hskip \labelsep {\sc Proof.}% ] \ignorespaces } \def\@xproof[#1]{% \trivlist \item[\hskip 10\p@\hskip \labelsep{\sc Proof #1.}]% \ignorespaces } \def\qed{% \unskip \kern 10\p@ \begingroup \unitlength\p@ \linethickness{.4\p@}% \framebox(6,6){}% \endgroup \global\@qededtrue } \def\newdef#1#2{% \expandafter\@ifdefinable\csname #1\endcsname {\@definecounter{#1}% \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}% \global\@namedef{#1}{\@defthm{#1}{#2}}% \global\@namedef{end#1}{\@endtheorem}% }% } \def\@defthm#1#2{% \refstepcounter{#1}% \@ifnextchar[{\@ydefthm{#1}{#2}}{\@xdefthm{#1}{#2}}% } \def\@xdefthm#1#2{% \@begindef{#2}{\csname the#1\endcsname}% \ignorespaces } \def\@ydefthm#1#2[#3]{% \trivlist \item[% \hskip 10\p@ \hskip \labelsep {\it #2% % \savebox\@tempboxa{#3}% \saveb@x\@tempboxa{#3}% % January 2008 \ifdim \wd\@tempboxa>\z@ \ \box\@tempboxa \fi.% }]% \ignorespaces } \def\@begindef#1#2{% \trivlist \item[% \hskip 10\p@ \hskip \labelsep {\it #1\ \rm #2.}% ]% } \def\theequation{\arabic{equation}} \newcounter{part} \newcounter{section} \newcounter{subsection}[section] \newcounter{subsubsection}[subsection] \newcounter{paragraph}[subsubsection] \def\thepart{\Roman{part}} \def\thesection{\arabic{section}} \def\thesubsection{\thesection.\arabic{subsection}} \def\thesubsubsection{\thesubsection.\arabic{subsubsection}} %removed \subsecfnt 29 July 2002 gkmt \def\theparagraph{\thesubsubsection.\arabic{paragraph}} %removed \subsecfnt 29 July 2002 gkmt \newif\if@uchead \@ucheadfalse %% CHANGES: NEW NOTE %% NOTE: OK to use old-style font commands below, since they were %% suitably redefined for LaTeX2e %% END CHANGES \setcounter{secnumdepth}{3} \def\part{% \@startsection{part}{9}{\z@}{-10\p@ \@plus -4\p@ \@minus -2\p@} {4\p@}{\normalsize\@ucheadtrue}% } \def\section{% \@startsection{section}{1}{\z@}{-10\p@ \@plus -4\p@ \@minus -2\p@}% GM {4\p@}{\baselineskip 14pt\secfnt\@ucheadtrue}% } \def\subsection{% \@startsection{subsection}{2}{\z@}{-8\p@ \@plus -2\p@ \@minus -\p@} {4\p@}{\secfnt}% } \def\subsubsection{% \@startsection{subsubsection}{3}{\z@}{-8\p@ \@plus -2\p@ \@minus -\p@}% {4\p@}{\subsecfnt}% } %\def\paragraph{% % \vskip 12pt\@startsection{paragraph}{3}{\z@}{6\p@ \@plus \p@}% original % {-5\p@}{\subsecfnt}% %} % If one wants sections, subsections and subsubsections numbered, % but not paragraphs, one usually sets secnumepth to 3. % For that, the "depth" of paragraphs must be given correctly % in the definition (``4'' instead of ``3'' as second argument % of @startsection): \def\paragraph{% \vskip 12pt\@startsection{paragraph}{4}{\z@}{6\p@ \@plus \p@}% % GM and Wolfgang May - 11/30/06 {-5\p@}{\subsecfnt}% } \let\@period=. \def\@startsection#1#2#3#4#5#6{% \if@noskipsec %gkmt, 11 aug 99 \global\let\@period\@empty \leavevmode \global\let\@period.% \fi \par % \@tempskipa #4\relax \@afterindenttrue \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \@afterindentfalse \fi \if@nobreak \everypar{}% \else \addpenalty\@secpenalty \addvspace\@tempskipa \fi \parskip=0pt % GM July 2000 (non numbered) section heads \@ifstar {\@ssect{#3}{#4}{#5}{#6}} {\@dblarg{\@sect{#1}{#2}{#3}{#4}{#5}{#6}}}% } \def\@sect#1#2#3#4#5#6[#7]#8{% \ifnum #2>\c@secnumdepth \let\@svsec\@empty \else \refstepcounter{#1}% \edef\@svsec{% \begingroup %\ifnum#2>2 \noexpand\rm \fi % changed to next 29 July 2002 gkmt \ifnum#2>2 \noexpand#6 \fi \csname the#1\endcsname \endgroup \ifnum #2=1\relax .\fi \hskip 1em }% \fi \@tempskipa #5\relax \ifdim \@tempskipa>\z@ \begingroup #6\relax \@hangfrom{\hskip #3\relax\@svsec}% \begingroup \interlinepenalty \@M \if@uchead \uppercase{#8}% \else #8% \fi \par \endgroup \endgroup \csname #1mark\endcsname{#7}% \vskip -12pt %gkmt, 11 aug 99 and GM July 2000 (was -14) - numbered section head spacing \addcontentsline{toc}{#1}{% \ifnum #2>\c@secnumdepth \else \protect\numberline{\csname the#1\endcsname}% \fi #7% }% \else \def\@svsechd{% #6% \hskip #3\relax \@svsec \if@uchead \uppercase{#8}% \else #8% \fi \csname #1mark\endcsname{#7}% \addcontentsline{toc}{#1}{% \ifnum #2>\c@secnumdepth \else \protect\numberline{\csname the#1\endcsname}% \fi #7% }% }% \fi \@xsect{#5}\hskip 1pt \par } \def\@xsect#1{% \@tempskipa #1\relax \ifdim \@tempskipa>\z@ \par \nobreak \vskip \@tempskipa \@afterheading \else \global\@nobreakfalse \global\@noskipsectrue \everypar{% \if@noskipsec \global\@noskipsecfalse \clubpenalty\@M \hskip -\parindent \begingroup \@svsechd \@period \endgroup \unskip \@tempskipa #1\relax \hskip -\@tempskipa \else \clubpenalty \@clubpenalty \everypar{}% \fi }% \fi \ignorespaces } \def\@trivlist{% \@topsepadd\topsep \if@noskipsec \global\let\@period\@empty \leavevmode \global\let\@period.% \fi \ifvmode \advance\@topsepadd\partopsep \else \unskip \par \fi \if@inlabel \@noparitemtrue \@noparlisttrue \else \@noparlistfalse \@topsep\@topsepadd \fi \advance\@topsep \parskip \leftskip\z@skip \rightskip\@rightskip \parfillskip\@flushglue \@setpar{\if@newlist\else{\@@par}\fi} \global\@newlisttrue \@outerparskip\parskip } %%% Actually, 'abbrev' works just fine as the default %%% Bibliography style. \typeout{Using 'Abbrev' bibliography style} \newcommand\bibyear[2]{% \unskip\quad\ignorespaces#1\unskip \if#2..\quad \else \quad#2 \fi } \newcommand{\bibemph}[1]{{\em#1}} \newcommand{\bibemphic}[1]{{\em#1\/}} \newcommand{\bibsc}[1]{{\sc#1}} \def\@normalcite{% \def\@cite##1##2{[##1\if@tempswa , ##2\fi]}% } \def\@citeNB{% \def\@cite##1##2{##1\if@tempswa , ##2\fi}% } \def\@citeRB{% \def\@cite##1##2{##1\if@tempswa , ##2\fi]}% } \def\start@cite#1#2{% \edef\citeauthoryear##1##2##3{% ###1% \ifnum#2=\z@ \else\ ###2\fi }% \ifnum#1=\thr@@ \let\@@cite\@citeyear \else \let\@@cite\@citenormal \fi \@ifstar{\@citeNB\@@cite}{\@normalcite\@@cite}% } %\def\cite{\start@cite23} \DeclareRobustCommand\cite{\start@cite23} % January 2008 \def\citeNP{\cite} % No Parentheses e.g. 5 %\def\citeA{\start@cite10} \DeclareRobustCommand\citeA{\start@cite10} % January 2008 \def\citeANP{\citeA} %\def\shortcite{\start@cite23} \DeclareRobustCommand\shortcite{\start@cite23} % January 2008 \def\shortciteNP{\shortcite} %\def\shortciteA{\start@cite20} \DeclareRobustCommand\shortciteA{\start@cite20} % January 2008 \def\shortciteANP{\shortciteA} %\def\citeyear{\start@cite30} \DeclareRobustCommand\citeyear{\start@cite30} % January 2008 \def\citeyearNP{\citeyear} %\def\citeN{% \DeclareRobustCommand\citeN{% % January 2008 \@citeRB \def\citeauthoryear##1##2##3{##1\ [##3% \def\reserved@a{##1}% \def\citeauthoryear####1####2####3{% \def\reserved@b{####1}% \ifx\reserved@a\reserved@b ####3% \else \errmessage{Package acmart Error: author mismatch in \string\citeN^^J^^J% See the acmart package documentation for explanation}% \fi }% }% \@ifstar\@citeyear\@citeyear } %\def\shortciteN{% \DeclareRobustCommand\shortciteN{% % January 2008 \@citeRB \def\citeauthoryear##1##2##3{##2\ [##3% \def\reserved@a{##2}% \def\citeauthoryear####1####2####3{% \def\reserved@b{####2}% \ifx\reserved@a\reserved@b ####3% \else \errmessage{Package acmart Error: author mismatch in \string\shortciteN^^J^^J% See the acmart package documentation for explanation}% \fi }% }% \@ifstar\@citeyear\@citeyear % GM July 2000 } \def\@citenormal{% \@ifnextchar [{\@tempswatrue\@citex;}% % original {\@tempswafalse\@citex,[]}% was ; Gerry 2/24/00 {\@tempswafalse\@citex[]}% % GERRY FIX FOR BABEL 3/20/2009 } \def\@citeyear{% \@ifnextchar [{\@tempswatrue\@citex,}% % original {\@tempswafalse\@citex,[]}% {\@tempswafalse\@citex[]}% % GERRY FIX FOR BABEL 3/20/2009 } \def\@citex#1[#2]#3{% \let\@citea\@empty \@cite{% \@for\@citeb:=#3\do{% \@citea % original \def\@citea{#1 }% \def\@citea{#1, }% % GERRY FIX FOR BABEL 3/20/2009 -- SO THAT YOU GET [1, 2] IN THE BODY TEXT \edef\@citeb{\expandafter\@iden\@citeb}% \if@filesw \immediate\write\@auxout{\string\citation{\@citeb}}% \fi \@ifundefined{b@\@citeb}{% {\bf ?}% \@warning{% Citation `\@citeb' on page \thepage\space undefined% }% }% {\csname b@\@citeb\endcsname}% }% }{#2}% } %\let\@biblabel\@gobble % Dec. 2008 - Gerry % ---- \def\@biblabelnum#1{[#1]} % Gerry's solution #1 - for Natbib -- April 2009 \let\@biblabel=\@biblabelnum % Gerry's solution #1 - for Natbib -- April 2009 \def\newblock{\relax} % Gerry Dec. 2008 % --- \newdimen\bibindent \setcounter{enumi}{1} \bibindent=0em \def\thebibliography#1{% \ifnum\addauflag=0\addauthorsection\global\addauflag=1\fi \section[References]{% <=== OPTIONAL ARGUMENT ADDED HERE {References} % was uppercased but this affects pdf bookmarks (SP/GM October 2004) {\vskip -9pt plus 1pt} % GM Nov. 2006 / GM July 2000 (for somewhat tighter spacing) \@mkboth{{\refname}}{{\refname}}% }% \list{[\arabic{enumi}]}{% \settowidth\labelwidth{[#1]}% \leftmargin\labelwidth \advance\leftmargin\labelsep \advance\leftmargin\bibindent \parsep=0pt\itemsep=1pt % GM July 2000 \itemindent -\bibindent \listparindent \itemindent \usecounter{enumi} }% \let\newblock\@empty \raggedright % GM July 2000 \sloppy \sfcode`\.=1000\relax } \gdef\balancecolumns {\vfill\eject \global\@colht=\textheight \global\ht\@cclv=\textheight } \newcount\colcntr \global\colcntr=0 %\newbox\savebox \newbox\saveb@x % January 2008 \gdef \@makecol {% \global\advance\colcntr by 1 \ifnum\colcntr>2 \global\colcntr=1\fi \ifvoid\footins \setbox\@outputbox \box\@cclv \else \setbox\@outputbox \vbox{% \boxmaxdepth \@maxdepth \@tempdima\dp\@cclv \unvbox \@cclv \vskip-\@tempdima \vskip \skip\footins \color@begingroup \normalcolor \footnoterule \unvbox \footins \color@endgroup }% \fi \xdef\@freelist{\@freelist\@midlist}% \global \let \@midlist \@empty \@combinefloats \ifvbox\@kludgeins \@makespecialcolbox \else \setbox\@outputbox \vbox to\@colht {% \@texttop \dimen@ \dp\@outputbox \unvbox \@outputbox \vskip -\dimen@ \@textbottom }% \fi \global \maxdepth \@maxdepth } \def\titlenote{\@ifnextchar[\@xtitlenote{\stepcounter\@mpfn \global\advance\titlenotecount by 1 \ifnum\titlenotecount=1 \raisebox{9pt}{$\ast$} \fi \ifnum\titlenotecount=2 \raisebox{9pt}{$\dagger$} \fi \ifnum\titlenotecount=3 \raisebox{9pt}{$\ddagger$} \fi \ifnum\titlenotecount=4 \raisebox{9pt}{$\S$} \fi \ifnum\titlenotecount=5 \raisebox{9pt}{$\P$} \fi \@titlenotetext }} \long\def\@titlenotetext#1{\insert\footins{% \ifnum\titlenotecount=1\global\tntoks={#1}\fi \ifnum\titlenotecount=2\global\tntokstwo={#1}\fi \ifnum\titlenotecount=3\global\tntoksthree={#1}\fi \ifnum\titlenotecount=4\global\tntoksfour={#1}\fi \ifnum\titlenotecount=5\global\tntoksfive={#1}\fi \reset@font\footnotesize \interlinepenalty\interfootnotelinepenalty \splittopskip\footnotesep \splitmaxdepth \dp\strutbox \floatingpenalty \@MM \hsize\columnwidth \@parboxrestore \protected@edef\@currentlabel{% }% \color@begingroup \color@endgroup}} %%%%%%%%%%%%%%%%%%%%%%%%% \ps@plain \baselineskip=11pt \let\thepage\relax % For NO page numbers - GM Nov. 30th. 1999 and July 2000 \def\setpagenumber#1{\global\setcounter{page}{#1}} %\pagenumbering{arabic} % Arabic page numbers GM July 2000 \twocolumn % Double column. \flushbottom % Even bottom -- alas, does not balance columns at end of document \pagestyle{plain} % Need Copyright Year and Copyright Data to be user definable (in .tex file). % Gerry Nov. 30th. 1999 \newtoks\copyrtyr \newtoks\acmcopyr \newtoks\boilerplate \global\acmcopyr={X-XXXXX-XX-X/XX/XX} % Default - 5/11/2001 * Gerry \global\copyrtyr={\the\year} % Default - 3/3/2003 * Gerry \def\acmPrice#1{\gdef\@acmPrice{#1}} \acmPrice{} %article price % Changed to 15 - June 2012 - Gerry \def\CopyrightYear#1{\global\copyrtyr{#1}} \def\crdata#1{\global\acmcopyr{#1}} \def\permission#1{\global\boilerplate{#1}} % ISBN % \def\isbn#1{\global\acmcopyr={#1}} \isbn{978-1-4503-2138-9} \RequirePackage{url} \urlstyle{rm} \def\doi#1{\def\@doi{#1}} \doi{10.1145/1235} \def\printdoi#1{\url{#1}} % Copyright \RequirePackage{acmcopyright} \setcopyright{none} % \global\boilerplate={\@copyrightpermission} \newtoks\copyrightetc \ifnum\ACM@basesize=9\relax \global\copyrightetc{% {\noindent\confname\ \the\conf\ \the\confinfo}\par\smallskip \if@printcopyright \copyright\ \the\copyrtyr\ \@copyrightowner \fi \if@acmowned ISBN \else\ifnum\acm@copyrightmode=2 ISBN \else \par\smallskip ACM ISBN \fi\fi \the\acmcopyr\ifx\@acmPrice\@empty.\else\dots\@acmPrice\fi\par\smallskip {DOI: \small\expandafter\printdoi\expandafter{\@doi}}} \toappear{\fontsize{7pt}{8pt}\fontfamily{ptm}\selectfont \the\boilerplate\par\smallskip \the\copyrightetc} \fi \ifnum\ACM@basesize=10\relax \global\copyrightetc{% {\noindent\confname\ \the\conf\ \the\confinfo}\par\smallskip \if@printcopyright \copyright\ \the\copyrtyr\ \@copyrightowner \fi \if@acmowned ISBN \else\ifnum\acm@copyrightmode=2 ISBN \else \par\smallskip ACM ISBN \fi\fi \the\acmcopyr\ifx\@acmPrice\@empty.\else\dots\@acmPrice\fi\par\smallskip {DOI: \small\expandafter\printdoi\expandafter{\@doi}}} \toappear{\fontsize{7.5pt}{8.5pt}\fontfamily{ptm}\selectfont \the\boilerplate\par\smallskip \the\copyrightetc} \fi %\DeclareFixedFont{\altcrnotice}{OT1}{tmr}{m}{n}{8} % << patch needed for accenting e.g. Montreal - Gerry, May 2007 %\DeclareFixedFont{\altconfname}{OT1}{tmr}{m}{it}{8} % << patch needed for accenting in italicized confname - Gerry, May 2007 % %{\altconfname{{\the\conf}}} {\altcrnotice\the\confinfo\par} \the\copyrightetc.} % << Gerry, May 2007 % % The following section (i.e. 3 .sty inclusions) was added in May 2007 so as to fix the problems that many % authors were having with accents. Sometimes accents would occur, but the letter-character would be of a different % font. Conversely the letter-character font would be correct but, e.g. a 'bar' would appear superimposed on the % character instead of, say, an unlaut/diaresis. Sometimes the letter-character would NOT appear at all. % Using [T1]{fontenc} outright was not an option as this caused 99% of the authors to 'produce' a Type-3 (bitmapped) % PDF file - useless for production. % % For proper (font) accenting we NEED these packages to be part of the .cls file i.e. 'ae', 'aecompl' and 'aeguil' % ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ %% This is file `ae.sty' \def\fileversion{1.3} \def\filedate{2001/02/12} \NeedsTeXFormat{LaTeX2e} %\ProvidesPackage{ae}[\filedate\space\fileversion\space % GM % Almost European Computer Modern] % GM - keeping the log file clean(er) \newif\if@ae@slides \@ae@slidesfalse \DeclareOption{slides}{\@ae@slidestrue} \ProcessOptions \fontfamily{aer} \RequirePackage[T1]{fontenc} \if@ae@slides \renewcommand{\sfdefault}{laess} \renewcommand{\rmdefault}{laess} % no roman \renewcommand{\ttdefault}{laett} \else \renewcommand{\sfdefault}{aess} \renewcommand{\rmdefault}{aer} \renewcommand{\ttdefault}{aett} \fi \endinput %% %% End of file `ae.sty'. % % \def\fileversion{0.9} \def\filedate{1998/07/23} \NeedsTeXFormat{LaTeX2e} %\ProvidesPackage{aecompl}[\filedate\space\fileversion\space % GM %T1 Complements for AE fonts (D. Roegel)] % GM -- keeping the log file clean(er) \def\@ae@compl#1{{\fontencoding{T1}\fontfamily{cmr}\selectfont\symbol{#1}}} \def\guillemotleft{\@ae@compl{19}} \def\guillemotright{\@ae@compl{20}} \def\guilsinglleft{\@ae@compl{14}} \def\guilsinglright{\@ae@compl{15}} \def\TH{\@ae@compl{222}} \def\NG{\@ae@compl{141}} \def\ng{\@ae@compl{173}} \def\th{\@ae@compl{254}} \def\DJ{\@ae@compl{208}} \def\dj{\@ae@compl{158}} \def\DH{\@ae@compl{208}} \def\dh{\@ae@compl{240}} \def\@perthousandzero{\@ae@compl{24}} \def\textperthousand{\%\@perthousandzero} \def\textpertenthousand{\%\@perthousandzero\@perthousandzero} \endinput % % %% This is file `aeguill.sty' % This file gives french guillemets (and not guillemots!) % built with the Polish CMR fonts (default), WNCYR fonts, the LASY fonts % or with the EC fonts. % This is useful in conjunction with the ae package % (this package loads the ae package in case it has not been loaded) % and with or without the french(le) package. % % In order to get the guillemets, it is necessary to either type % \guillemotleft and \guillemotright, or to use an 8 bit encoding % (such as ISO-Latin1) which selects these two commands, % or, if you use the french package (but not the frenchle package), % to type << or >>. % % By default, you get the Polish CMR guillemets; if this package is loaded % with the `cm' option, you get the LASY guillemets; with `ec,' you % get the EC guillemets, and with `cyr,' you get the cyrillic guillemets. % % In verbatim mode, you always get the EC/TT guillemets. % % The default option is interesting in conjunction with PDF, % because there is a Type 1 version of the Polish CMR fonts % and these guillemets are very close in shape to the EC guillemets. % There are no free Type 1 versions of the EC fonts. % % Support for Polish CMR guillemets was kindly provided by % Rolf Niepraschk <niepraschk@ptb.de> in version 0.99 (2000/05/22). % Bernd Raichle provided extensive simplifications to the code % for version 1.00. % % This package is released under the LPPL. % % Changes: % Date version % 2001/04/12 1.01 the frenchle and french package are now distinguished. % \def\fileversion{1.01} \def\filedate{2001/04/12} \NeedsTeXFormat{LaTeX2e} %\ProvidesPackage{aeguill}[2001/04/12 1.01 % % GM %AE fonts with french guillemets (D. Roegel)] % GM - keeping the log file clean(er) %\RequirePackage{ae} % GM May 2007 - already embedded here \newcommand{\@ae@switch}[4]{#4} \DeclareOption{ec}{\renewcommand\@ae@switch[4]{#1}} \DeclareOption{cm}{\renewcommand\@ae@switch[4]{#2}} \DeclareOption{cyr}{\renewcommand\@ae@switch[4]{#3}} \DeclareOption{pl}{\renewcommand\@ae@switch[4]{#4}} % % Load necessary packages % \@ae@switch{% ec % do nothing }{% cm \RequirePackage{latexsym}% GM - May 2007 - already 'mentioned as required' up above }{% cyr \RequirePackage[OT2,T1]{fontenc}% }{% pl \RequirePackage[OT4,T1]{fontenc}% } % The following command will be compared to \frenchname, % as defined in french.sty and frenchle.sty. \def\aeguillfrenchdefault{french}% \let\guill@verbatim@font\verbatim@font \def\verbatim@font{\guill@verbatim@font\ecguills{cmtt}% \let\guillemotleft\@oguills\let\guillemotright\@fguills} \begingroup \catcode`\<=13 \catcode`\>=13 \def\x{\endgroup \def\ae@lfguill{<<}% \def\ae@rfguill{>>}% }\x \newcommand{\ecguills}[1]{% \def\selectguillfont{\fontencoding{T1}\fontfamily{#1}\selectfont}% \def\@oguills{{\selectguillfont\symbol{19}}}% \def\@fguills{{\selectguillfont\symbol{20}}}% } \newcommand{\aeguills}{% \ae@guills % We redefine \guillemotleft and \guillemotright % in order to catch them when they are used % with \DeclareInputText (in latin1.def for instance) % We use \auxWARNINGi as a safe indicator that french.sty is used. \gdef\guillemotleft{\ifx\auxWARNINGi\undefined \@oguills % neither french.sty nor frenchle.sty \else \ifx\aeguillfrenchdefault\frenchname \ae@lfguill % french.sty \else \@oguills % frenchle.sty \fi \fi}% \gdef\guillemotright{\ifx\auxWARNINGi\undefined \@fguills % neither french.sty nor frenchle.sty \else \ifx\aeguillfrenchdefault\frenchname \ae@rfguill % french.sty \else \@fguills % frenchle.sty \fi \fi}% } % % Depending on the class option % define the internal command \ae@guills \@ae@switch{% ec \newcommand{\ae@guills}{% \ecguills{cmr}}% }{% cm \newcommand{\ae@guills}{% \def\selectguillfont{\fontencoding{U}\fontfamily{lasy}% \fontseries{m}\fontshape{n}\selectfont}% \def\@oguills{\leavevmode\nobreak \hbox{\selectguillfont (\kern-.20em(\kern.20em}\nobreak}% \def\@fguills{\leavevmode\nobreak \hbox{\selectguillfont \kern.20em)\kern-.2em)}% \ifdim\fontdimen\@ne\font>\z@\/\fi}}% }{% cyr \newcommand{\ae@guills}{% \def\selectguillfont{\fontencoding{OT2}\fontfamily{wncyr}\selectfont}% \def\@oguills{{\selectguillfont\symbol{60}}}% \def\@fguills{{\selectguillfont\symbol{62}}}} }{% pl \newcommand{\ae@guills}{% \def\selectguillfont{\fontencoding{OT4}\fontfamily{cmr}\selectfont}% \def\@oguills{{\selectguillfont\symbol{174}}}% \def\@fguills{{\selectguillfont\symbol{175}}}} } \AtBeginDocument{% \ifx\GOfrench\undefined \aeguills \else \let\aeguill@GOfrench\GOfrench \gdef\GOfrench{\aeguill@GOfrench \aeguills}% \fi } \endinput %

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