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1 parent 315b9c2 commit 92eaf208d4a498266c07f6efd447a58311df0a6a Nigel Stanger authored on 8 Feb 2017

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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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Presentation/View_Updating.tex 0 → 100644

\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
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% TikZ setup
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\usetikzlibrary{calc}
\usetikzlibrary{arrows}

% fontspec setup
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\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}}}},
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    input delete/.style={structure!40,thick,dashed},
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    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}}}}
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\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}}}}
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% Constraints.
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% will insert too much vertical space. Put a \tinyskip at the end of the line instead.
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% Bold-face text using the structure colour.
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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}

\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}

Ignore Space Show notes View 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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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]}}

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

\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]{$\mid$}}}

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

\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),
            };

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

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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