\section{Classification System}\seclabel{class}
\index{classification}
A \emph{classification} is a user-defined abstraction that describes
  (``classifies'') arbitrary data.
Classifications can be used as predicates, generating functions, and can be
  composed into more complex classifications.
Nearly all conditions in \tame{} are specified using classifications.

\index{first-order logic!sentence}
\index{classification!coupling}
All classifications represent \emph{first-order sentences}---%
  that is,
    they contain no \emph{free variables}.
Intuitively,
  this means that all variables within a~classification are
  \emph{tightly coupled} to the classification itself.
This limitation is mitigated through use of the template system.

\todo{$\Classify$ itself has not yet been defined.}
\index{classification!conjunctive}
\begin{definition}[\logand-Classification]\dfnlabel{classu}
  A conjunctive\footnote{%
    Conjunctive and disjunctive classifications used to be referred to,
      respectively,
      as \emph{universal} and \emph{existential},
        referring to fact that
          $\forall\Set{a_0,\ldots,a_n}(a) \equiv a_0\logand\ldots\logand a_n$,
            and similarly for $\exists$.
    This terminology has changed since all classifications are in fact
      existential over their matches' index sets,
        and so the terminology would otherwise lead to confusion.
    }
    classification~$c$ performs conjunction on its match expressions
      $M_0\ldots M_n$.

  \begin{align*}
    &\xml{<classify as="$c$" yields="$\gamma$" desc="$\ldots$">} \\
    &\quad M_0 \\
    &\quad \vdots \\
    &\quad M_n \\
    &\xml{</classify>}
      \equiv \Classify^c_\gamma M_0 \logand \ldots \logand M_n.
  \end{align*}
\end{definition}

\index{classification!disjunctive}
\begin{definition}[\logor-Classification]\dfnlabel{classe}
  A disjunctive classification~$d$ with \xpath{@any="true"}
    performs disjunction on its match expressions
    $M_0\ldots M_n$.\footnote{%
      It is notationally convenient that~$c$ is a common prefix for both
        \underline{c}lassification \emph{and} \underline{c}onjunction,
          and also that~$d$ happens to follow~$c$ \emph{and} be the prefix for
          \underline{d}isjunction.
      This notation will only be used where such a distinction is relevant,
        and $c$~will otherwise refer generically to any type of
        classification.}

  \begin{align*}
    &\xml{<classify as="$d$" yields="$\gamma$" any="true" desc="$\ldots$">} \\
    &\quad M_0 \\
    &\quad \vdots \\
    &\quad M_n \\
    &\texttt{</classify>}
      \equiv \Classify^d_\gamma M_0 \logor \ldots \logor M_n.
  \end{align*}
\end{definition}

\mremark{Note that these illustrate \emph{scalar} values only.}
For example,
  consider the following classification $\Classify^\texttt{cost-exceeded}$.
Let~\tameparam{cost} be a scalar parameter.

\index{classification!classify@\xmlnode{classify}}
\begin{lstlisting}
  <classify as="cost-exceeded" desc="Cost of item is too expensive">
    <t:match-gt on="cost" value="100.00" />
  </classify>
\end{lstlisting}

\noindent
is then equivalent to the proposition

\begin{equation*}
  \tameclass{cost-exceeded} \equiv \tameparam{cost} > 100.00.
\end{equation*}

\index{classification!domain}
A classification is either \true or~\false.
Let $\tameparam{cost}=150.00$.
Then,

\begin{align*}
  \tameclass{cost-exceeded} & \equiv \tameparam{cost} > 100.00 \\
                            & \equiv 150.00 > 100.00 \\
                            & \equiv \true.
\end{align*}

Each \xmlnode{match} of a classification is a~\emph{predicate}.
Multiple predicates are by default joined by conjunction:

\begin{lstlisting}
  <classify as="pool-hazard" desc="Hazardous pool">
    <match on="diving_board" />
    <t:match-lt on="pool_depth_ft" value="8" />
  </classify>
\end{lstlisting}

\noindent
is equivalent to the proposition

\begin{equation*}
  \tameclass{pool-hazard} \equiv \tameparam{diving\_board}
    \logand \tameparam{pool\_depth\_ft} < 8.
\end{equation*}


\subsection{Matches}
\todo{Non-scalar values.}
\begin{definition}[Match Equality]
  \begin{equation*}
    \xml{<match on="$x$" value="$y$" />} \equiv x = y.
  \end{equation*}
\end{definition}

\begin{definition}[Match Equality Short Form]
  \begin{equation*}
    \xml{<match on="$x$" />}
      \equiv \xml{<match on="$x$" value="TRUE" />}.
  \end{equation*}
\end{definition}

\begin{definition}[Match Equality Long Form]
  \begin{alignat*}{2}
    \xml{<match on="$x$" value="$y$" />}
      &\equiv {}&&\xml{<match on="$x$">} \\
      &         &&\quad \xml{<c:eq>} \\
      &         &&\quad\quad \xml{<c:value-of name="$y$">} \\
      &         &&\quad \xml{</c:eq>} \\
      &         &&\xml{</match>} \\
      &\equiv {}&&\xml{<t:match-eq on="$x$" value="$y$" />}.
  \end{alignat*}
\end{definition}

\begin{definition}[Match Membership Equivalence]
  When $T$ is a type defined with \xmlnode{typedef},

  \begin{equation*}
    \xml{<match on="$x$" anyOf="$T$" />} \equiv x \in T.
  \end{equation*}
\end{definition}