133 lines
4.0 KiB
TeX
133 lines
4.0 KiB
TeX
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\section{Classification System}\seclabel{class}
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\index{classification}
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A \gls{classification} is a user-defined abstraction that describes
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(``classifies'') arbitrary data.
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Classifications can be used as predicates, generating functions, and can be
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composed into more complex classifications.
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Nearly all conditions in \tame{} are specified using classifications.
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\index{first-order logic!sentence}
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\index{classification!coupling}
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All classifications represent \emph{first-order sentences}---%
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that is,
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they contain no \glspl{free variable}.
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Intuitively,
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this means that all variables within a~classification are
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\emph{tightly coupled} to the classification itself.
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This limitation is mitigated through use of the template system.
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For example,
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consider the following classification \tameclass{cost-exceeded}.
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Let~\tameparam{cost} be a scalar parameter.
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\index{classification!classify@\xmlnode{classify}}
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\begin{lstlisting}
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<classify as="cost-exceeded" desc="Cost of item is too expensive">
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<t:match-gt on="cost" value="100.00" />
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</classify>
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\end{lstlisting}
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\noindent
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is then equivalent to the proposition
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\begin{equation*}
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\tameclass{cost-exceeded} \equiv \tameparam{cost} > 100.00.
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\end{equation*}
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\index{classification!domain}
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A classification is either \glssymbol{true} or~\glssymbol{false}.
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Let $\tameparam{cost}=150.00$.
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Then,
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\begin{align*}
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\tameclass{cost-exceeded} & \equiv \tameparam{cost} > 100.00 \\
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& \equiv 150.00 > 100.00 \\
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& \equiv \true.
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\end{align*}
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Each \xmlnode{match} of a classification is a~\gls{predicate}.
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Multiple predicates are by default joined by \gls{conjunction}:
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\begin{lstlisting}
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<classify as="pool-hazard" desc="Hazardous pool">
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<match on="diving_board" />
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<t:match-lt on="pool_depth_ft" value="8" />
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</classify>
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\end{lstlisting}
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\noindent
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is equivalent to the proposition
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\begin{equation*}
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\tameclass{pool-hazard} \equiv \tameparam{diving\_board}
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\logand \tameparam{pool\_depth\_ft} < 8.
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\end{equation*}
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\index{classification!universal}
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\begin{definition}[Universal Classification]\dfnlabel{classu}
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A classification~$c$ by default performs \gls{conjunction} on its match
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expressions $M_0\ldots M_n$.
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\begin{alignat*}{2}
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&\xml{<classify as="}&&c\xml{" desc="$\ldots$">} \\
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&\quad M_0 \\
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&\quad \vdots \\
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&\quad M_n \\
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&\xml{</classify>}
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&&\equiv c\in\Bool \\
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& &&\equiv \exists\left( M_0 \logand \ldots \logand M_n \right).
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\end{alignat*}
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\end{definition}
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\index{classification!existential}
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\begin{definition}[Existential Classification]\dfnlabel{classe}
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A classification~$c$ with the attribute \xpath{@any="true"} performs
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\gls{disjunction} on its match expressions $M_0\ldots M_n$.
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\begin{alignat*}{2}
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&\xml{<classify as="} &&c\xml{" any="true" desc="$\ldots$">} \\
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&\quad M_0 \\
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&\quad \vdots \\
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&\quad M_n \\
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&\texttt{</classify>}
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&&\equiv c\in\Bool \\
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& &&\equiv \exists\left( M_0 \logor \ldots \logor M_n \right).
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\end{alignat*}
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\end{definition}
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\subsection{Matches}
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\begin{definition}[Match Equality]
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\begin{equation*}
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\xml{<match on="$x$" value="$y$" />} \equiv x = y.
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\end{equation*}
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\end{definition}
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\begin{definition}[Match Equality Short Form]
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\begin{equation*}
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\xml{<match on="$x$" />}
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\equiv \xml{<match on="$x$" value="TRUE" />}.
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\end{equation*}
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\end{definition}
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\begin{definition}[Match Equality Long Form]
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\begin{alignat*}{2}
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\xml{<match on="$x$" value="$y$" />}
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&\equiv {}&&\xml{<match on="$x$">} \\
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& &&\quad \xml{<c:eq>} \\
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& &&\quad\quad \xml{<c:value-of name="$y$">} \\
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& &&\quad \xml{</c:eq>} \\
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& &&\xml{</match>} \\
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&\equiv {}&&\xml{<t:match-eq on="$x$" value="$y$" />}.
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\end{alignat*}
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\end{definition}
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\begin{definition}[Match Membership Equivalence]
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When $T$ is a type defined with \xmlnode{typedef},
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\begin{equation*}
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\xml{<match on="$x$" anyOf="$T$" />} \equiv x \in T.
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\end{equation*}
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\end{definition}
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