candidates.tex 10.1 KB
 Michael Banken committed Oct 16, 2017 1 \chapter{Candidates}  Michael Banken committed Oct 25, 2017 2   Michael Banken committed Oct 16, 2017 3 4 \section{Types of optimization candidates} \subsection{Simply redundant inclusion}  Michael Banken committed Oct 25, 2017 5 6 An inclusion of a theory $A\hookrightarrow{}C$ is \underline{simply redundant}, if $C$ includes a theory $B$, such that $B\hookrightarrow{}C$ (see \autoref{fig:redundantbasic}).  Michael Banken committed Oct 16, 2017 7 8 9 \providecommand\myxscale{3.9} \providecommand\myyscale{2.2} \providecommand\myfontsize{\footnotesize}  Michael Banken committed Oct 25, 2017 10 11 \begin{figure}[!htb] \begin{tikzpicture}[node distance=3cm]\footnotesize  Michael Banken committed Oct 16, 2017 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 \node[thy] (bottom) {\begin{tabular}{l} \textsf{bottom}\\\hline ...\\\hline ... \end{tabular}}; \node[thy, above of = bottom] (middle) {\begin{tabular}{l} \textsf{middle}\\\hline ...\\\hline ... \end{tabular}}; \node[thy, above of = middle] (top) {\begin{tabular}{l} \textsf{top}\\\hline ...\\\hline ... \end{tabular}}; \draw[include] (bottom) -- (middle); \draw[include, bend left] (bottom) edge (top); \draw[include] (middle) -- (top); \end{tikzpicture}  Michael Banken committed Oct 25, 2017 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 \caption{Simply redundant inclusion example} \label{fig:redundantbasic} \end{figure} Simply redundant inclusions can be safely optimized by simply removing the redundant inclusion, as seen in \autoref{fig:redundantoptimized}, without changing the flattened graph due to the transitive nature of inclusions. \begin{figure}[!htb] \begin{tikzpicture}[node distance=3cm]\footnotesize \node[thy] (bottom) {\begin{tabular}{l} \textsf{bottom}\\\hline ...\\\hline ... \end{tabular}}; \node[thy, above of = bottom] (middle) {\begin{tabular}{l} \textsf{middle}\\\hline ...\\\hline ... \end{tabular}}; \node[thy, above of = middle] (top) {\begin{tabular}{l} \textsf{top}\\\hline ...\\\hline ... \end{tabular}}; \draw[include] (bottom) -- (middle); \draw[include] (middle) -- (top); \end{tikzpicture} \caption{Example of simply redundant inclusion optimized} \label{fig:redundantoptimized} \end{figure}  Michael Banken committed Oct 16, 2017 60 \subsection{Superfluous Inclusion}  Michael Banken committed Oct 25, 2017 61 62 An inclusion of a theory $A\hookrightarrow{}B$ is \underline{superfluous}, if $B$ uses none of the constants declared in $A$. Such an inclusion can be \underline{purely superfluous} (\autoref{sec:puresi}) if it can be entirely removed, or \underline{partially superfluous} if it can be reduced to a subset of the theory inclusions in $A$ \autoref{sec:partiallysi}.\\ In both of these cases the inclusion can be replaced or removed entirely, while still yielding a well formed theory. However this changes the resulting flattened theory graph and can invalidate theories that include the optimized theory.\\  Michael Banken committed Oct 16, 2017 63 \subsubsection{Purely Superfluous Inclusion}  Michael Banken committed Oct 25, 2017 64 65 66 67 An inclusion of a theory $A\hookrightarrow{}B$ is \underline{purely superfluous}, if $B$ uses none of the constants in $A$, not even if they were declared in a theory $C$ such that $C\hookrightarrow{}A$.\\ \label{sec:puresi} \begin{figure}[!htb] \begin{tikzpicture}[node distance=3cm]\footnotesize  Michael Banken committed Oct 16, 2017 68 69 70 71 72 73 74 75 76 77 78 79 \node[thy] (bottom) {\begin{tabular}{l} \textsf{bottom}\\\hline X\\\hline ... \end{tabular}}; \node[thy, above of = bottom] (top) {\begin{tabular}{l} \textsf{top}\\\hline ...\\\hline no X \end{tabular}}; \draw[include] (bottom) -- (top); \end{tikzpicture}  Michael Banken committed Oct 25, 2017 80 81 82 83 84 85 86 \caption{Purely superfluous inclusion example} \label{fig:purelysuperfluousbasic} \end{figure} Purely superfluous inclusions can be removed while still retaining a valid theory, however this will change the resulting theory graph. These changes may or may not be what is desired. An example for such an optimization can be seen in \autoref{fig:purelysuperfluousoptimized}. \begin{figure}[!htb] \begin{tikzpicture}[node distance=3cm]\footnotesize  Michael Banken committed Oct 16, 2017 87 \node[thy] (bottom) {\begin{tabular}{l}  Michael Banken committed Oct 16, 2017 88  \textsf{bottom}\\\hline  Michael Banken committed Oct 16, 2017 89 90  X\\\hline ...  Michael Banken committed Oct 16, 2017 91  \end{tabular}};  Michael Banken committed Oct 25, 2017 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 \node[thy, above of = bottom] (top) {\begin{tabular}{l} \textsf{top}\\\hline ...\\\hline no X \end{tabular}}; \end{tikzpicture} \caption{Example of purely superfluous inclusion optimized} \label{fig:purelysuperfluousoptimized} \end{figure} \subsubsection{Partially Superfluous Inclusion} \label{sec:partiallysi} An inclusion of a theory $A\hookrightarrow{}B$ is \underline{partially superfluous}, if $B$ uses none of the constants in $A$, but uses some declarations that were declared in one or more theory $C$ such that C is (transitively) included in A.\\ \begin{figure}[!htb] \begin{tikzpicture}[node distance=3cm]\footnotesize \node[thy] (bottom) {\begin{tabular}{l} \textsf{bottom}\\\hline $X_1$\\\hline ... \end{tabular}}; \node[right of = bottom] (dots) {...}; \node[thy, right of = dots] (bottomn) {\begin{tabular}{l} \textsf{$bottom_n$}\\\hline $X_n$\\\hline ... \end{tabular}};  Michael Banken committed Oct 16, 2017 118 \node[thy, above of = bottom] (middle) {\begin{tabular}{l}  Michael Banken committed Oct 16, 2017 119 120 121 122 123 124  \textsf{middle}\\\hline Y\\\hline ... \end{tabular}}; \node[thy, above of = middle] (top) {\begin{tabular}{l} \textsf{top}\\\hline  Michael Banken committed Oct 16, 2017 125  ...\\\hline  Michael Banken committed Oct 25, 2017 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143  $X\subseteq\bigcup{}\{X_1,...,X_n\}$, but not Y \end{tabular}}; \draw[include] (bottom) -- (middle); \draw[include] (bottomn) -- (middle); \draw[include] (middle) -- (top); \end{tikzpicture} \caption{Partially superfluous inclusion example} \label{fig:partiallysuperfluousbasic} \end{figure} Partially superfluous inclusions can be optimized by identifying the used subset of (transitive) inclusions and replacing the superfluous inclusion with these inclusions. As seen in \autoref{fig:partiallysuperfluousoptimized}. \begin{figure}[!htb] \begin{tikzpicture}[node distance=3cm]\footnotesize \node[thy] (bottom) {\begin{tabular}{l} \textsf{bottom}\\\hline $X_1$\\\hline ...  Michael Banken committed Oct 16, 2017 144  \end{tabular}};  Michael Banken committed Oct 16, 2017 145 146 147 148 149 150 \node[right of = bottom] (dots) {...}; \node[thy, right of = dots] (bottomn) {\begin{tabular}{l} \textsf{$bottom_n$}\\\hline $X_n$\\\hline ... \end{tabular}};  Michael Banken committed Oct 25, 2017 151 152 153 154 155 156 157 158 159 160 \node[thy, above of = bottom] (middle) {\begin{tabular}{l} \textsf{middle}\\\hline Y\\\hline ... \end{tabular}}; \node[thy, above of = middle] (top) {\begin{tabular}{l} \textsf{top}\\\hline ...\\\hline $X\subseteq\bigcup{}\{X_1,...,X_n\}$, but not Y \end{tabular}};  Michael Banken committed Oct 16, 2017 161 \draw[include] (bottom) -- (middle);  Michael Banken committed Oct 25, 2017 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 \draw[include] (bottomn) -- (middle); \draw[include] (bottomn) -- (top); \draw[include, bend left] (bottom) edge (top); \end{tikzpicture} \caption{Example of partially superfluous inclusion optimized} \label{fig:partiallysuperfluousoptimized} \end{figure} \section{Viability and dangers of candidate optimizations} The candidates discussed earlier are cases in which inclusions can be changed, not necessarily where they should be changed.\\ There are multiple reasons for not \subsection{Simply redundant inclusion} As noted in \autoref{sec:puresi} there is very little that speaks against removing simply redundant inclusions, since the flattened graph is preserved. However there are some cases where they might do more harm than good, particularly when they overlap with superfluous inclusions, as seen in \autoref{fig:redundantoverlap}.\\ Removing the redundant edge will not change the flattened graph, but it will complicate the changes needed to remove the superfluous edge between middle and top. A simple solution to avoiding this problem is to relegate the removal of simple redundancies until after other optimizations have been performed on the theory.\\ \begin{figure}[h] \begin{tikzpicture}[node distance=3cm]\footnotesize \node[thy] (bottom) {\begin{tabular}{l} \textsf{bottom}\\\hline X\\\hline ... \end{tabular}}; \node[thy, above of = bottom] (middle) {\begin{tabular}{l} \textsf{middle}\\\hline Y\\\hline ... \end{tabular}}; \node[thy, above of = middle] (top) {\begin{tabular}{l} \textsf{top}\\\hline ...\\\hline X, but not Y \end{tabular}}; \draw[include] (bottom) -- (middle); \draw[include, bend left] (bottom) edge (top);  Michael Banken committed Oct 16, 2017 194 195 \draw[include] (middle) -- (top); \end{tikzpicture}  Michael Banken committed Oct 25, 2017 196 197 198 199 200 201 202 203 \caption{Example of overlap between simply redundant inclusion and superfluous inclusion} \label{fig:redundantoverlap} \end{figure} \subsection{Purely superfluous inclusion} \subsection{Partially superfluous inclusion} \section{Structures}