784 lines
26 KiB
TeX
784 lines
26 KiB
TeX
% The TAME Programming Language Classification System
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%
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% Copyright (C) 2021 Ryan Specialty, LLC.
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%
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% Licensed under the Creative Commons Attribution-ShareAlike 4.0
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% International License.
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%%
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\section{Classification System}\seclabel{class}
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\index{classification|textbf}
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A \dfn{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 \dfn{first-order sentences}---%
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that is,
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they contain no \dfn{free variables}.
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Intuitively,
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this means that all variables within a~classification are
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\dfn{tightly coupled} to the classification itself.
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This limitation is mitigated through use of the template system.
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\begin{axiom}[Classification Introduction]\axmlabel{class-intro}
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\indexsym\Classify{classification}
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\indexsym\gamma{classification, yield}
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\index{classification!index set}
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\index{index set!classification}
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\index{classification!classify@\xmlnode{classify}}
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\index{classification!as@\xmlattr{as}}
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\index{classification!yields@\xmlattr{yields}}
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\todo{Symbol in place of $=$ here ($\equiv$ not appropriate).}
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\begin{subequations}
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\begin{gather}
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\begin{alignedat}{3}
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&\xml{<classify as="$c$" }&&\xml{yields="$\gamma$" desc}&&\xml{="$\_$"
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$\alpha$>}\label{eq:xml-classify} \\
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&\quad \MFam{M^0}jJkK &&\VFam{v^0}jJ &&\quad s^0 \\[-4mm]
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&\quad \quad\vdots &&\quad\vdots &&\quad \vdots \\
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&\quad \MFam{M^l}jJkK &&\VFam{v^m}jJ &&\quad s^n \\[-3mm]
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&\xml{</classify>}
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% NB: This -50mu needs adjustment if you change the alignment above!
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&&\mspace{-50mu}= \Classify^c_\gamma\left(\odot,M,v,s\right),
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\end{alignedat}
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\end{gather}
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\noindent
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where
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\indexsym\emptystr{empty string}
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\index{empty string (\ensuremath\emptystr)}
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\begin{align}
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J &\subset\Int \neq\emptyset, \\
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\forall{j\in J}\Big(K_j &\subset\Int \neq\emptyset\Big), \\
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\forall{k}\Big(M^k &: J \rightarrow K_{j\in J} \rightarrow \Bool\Big),
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\label{eq:class-matrix} \\
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\forall{k}\Big(v^k &: J \rightarrow \Bool\Big), \\
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\forall{k}\Big(s^k &\in\Bool\Big), \\
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\alpha &\in\Set{\emptystr,\, \texttt{any="true"}}, \label{eq:xml-any-domain}
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\end{align}
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\noindent
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and the monoid~$\odot$ is defined as
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\indexsym\odot{classification, monoid}
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\index{classification!any@\xmlattr{any}}
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\index{classification!monoid|(}
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\begin{equation}\label{eq:classify-rel}
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\odot = \begin{cases}
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\Monoid\Bool\land\true &\alpha = \emptystr,\\
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\Monoid\Bool\lor\false &\alpha = \texttt{any="true"}.
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\end{cases}
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\end{equation}
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\end{subequations}
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\end{axiom}
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% This TODO was the initial motivation for this paper!
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\todo{Emphasize index sets, both relationships and nonempty.}
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We use a $4$-tuple $\Classify\left(\odot,M,v,s\right)$ to represent a
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$\odot_1$-classification
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(a classification with the binary operation $\land$ or~$\lor$)
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consisting of a combination of matrix~($M$), vector~($v$), and
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scalar~($s$) matches,
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rendered above in columns.\footnote{%
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The symbol~$\odot$ was chosen since the binary operation for a monoid
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is~$\monoidop$
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(see \secref{monoids})
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and~$\odot$ looks vaguely like~$(\monoidop)$,
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representing a portion of the monoid triple.}
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A $\land$-classification is pronounced ``conjunctive classification'',
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and $\lor$ ``disjunctive''.\footnote{%
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\index{classification!terminology history}
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Conjunctive and disjunctive classifications used to be referred to,
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respectively,
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as \dfn{universal} and \dfn{existential},
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referring to fact that
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$\forall\Set{a_0,\ldots,a_n}(a) \equiv a_0\land\ldots\land a_n$,
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and similarly for $\exists$.
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This terminology has changed since all classifications are in fact
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existential over their matches' index sets,
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and so the terminology would otherwise lead to confusion.}
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The variables~$c$ and~$\gamma$ are required in~\tame{} but are both optional
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in our notation~$\Classify^c_\gamma$,
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and can be used to identify the two different data representations of
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the classification.\footnote{%
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\xpath{classify/@yields} is optional in the grammar of \tame{},
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but the compiler will generate one for us if one is not provided.
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As such,
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we will for simplicity consider it to be required here.}
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$\alpha$~serves as a placeholder for an optional \xml{any="true"},
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with $\emptystr$~representing the empty string in~\eqref{eq:xml-any-domain}.
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Note the wildcard variable matching \xmlattr{desc}---%
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its purpose is only to provide documentation.
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\begin{corollary}[$\odot$ Commutative Monoid]\corlabel{odot-monoid}
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\index{classification!commutativity|(}
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$\odot$ is a commutative monoid in \axmref{class-intro}.
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\end{corollary}
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\begin{proof}
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By \axmref{class-intro},
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$\odot$ must be a monoid.
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Assume $\alpha=\emptystr$.
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Then,
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$\odot = \Monoid\Bool\land\true$,
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which is proved by \lemref{monoid-land}.
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Next, assume $\alpha=\texttt{any="true"}$.
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Then,
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$\odot = \Monoid\Bool\lor\false$,
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which is proved by \lemref{monoid-land}.
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\end{proof}
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While \axmref{class-intro} seems to imply an ordering to matches,
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users of the language are free to specify matches in any order
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and the compiler will rearrange matches as it sees fit.
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\index{compiler!classification commutativity}
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This is due to the commutativity of~$\odot$ as proved by
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\corref{odot-monoid},
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and not only affords great ease of use to users of~\tame{},
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but also great flexibility to compiler writers.
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\index{classification!commutativity|)}
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For notational convenience,
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we will let
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\index{classification!monoid|)}
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\begin{equation}
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\begin{aligned}
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\Classifyland(M,v,s)
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&= \Classify\left(\Monoid\Bool\land\true,M,v,s\right), \\
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\Classifylor(M,v,s)
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&= \Classify\left(\Monoid\Bool\lor\true,M,v,s\right). \\
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\end{aligned}
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\end{equation}
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\def\cpredmatseq{{M^0_j}_k \monoidops {M^l_j}_k}
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\def\cpredvecseq{v^0_j\monoidops v^m_j}
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\def\cpredscalarseq{s^0\monoidops s^n}
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\begin{axiom}[Classification-Predicate Equivalence]\axmlabel{class-pred}
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\index{classification!as predicate}
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Let $\Classify^c_\gamma\left(\Monoid\Bool\monoidop e,M,v,s\right)$ be a
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classification by~\axmref{class-intro}.
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We then have the first-order sentence
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\begin{equation*}
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c \equiv
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{} \Exists{j\in J}{\Exists{k\in K_j}\cpredmatseq\monoidop\cpredvecseq}
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\monoidop\cpredscalarseq.
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\end{equation*}
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\end{axiom}
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\begin{axiom}[Classification Yield]\axmlabel{class-yield}
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\indexsym\Gamma{classification, yield}
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\index{classification!yield (\ensuremath\gamma, \ensuremath\Gamma)}
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Let $\Classify^c_\gamma\left(\Monoid\Bool\monoidop e,M,v,s\right)$ be a
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classification by~\axmref{class-intro}.
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Then,
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\begin{subequations}
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\begin{align}
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r &= \begin{cases}
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2 &M\neq\emptyset, \\
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1 &M=\emptyset \land v\neq\emptyset, \\
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0 &M\union v = \emptyset,
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\end{cases} \\
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\displaybreak[0]
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\exists{j\in J}\Big(\exists{k\in K_j}\Big(
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\Gamma^2_{j_k} &= \cpredmatseq\monoidop\cpredvecseq\monoidop\cpredscalarseq
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\Big)\Big), \\
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%
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\exists{j\in J}\Big(
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\Gamma^1_j &= \cpredvecseq\monoidop\cpredscalarseq
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\Big), \\
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%
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\Gamma^0 &= \cpredscalarseq. \\
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%
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\gamma &= \Gamma^r.
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\end{align}
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\end{subequations}
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\end{axiom}
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\begin{theorem}[Classification Composition]\thmlabel{class-compose}
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\index{classification!composition|(}
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Classifications may be composed to create more complex classifications
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using the classification yield~$\gamma$ as in~\axmref{class-yield}.
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This interpretation is equivalent to \axmref{class-pred} by
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\begin{equation}
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c \equiv \Exists{j\in J}{
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\Exists{k\in K_j}{\Gamma^2_{j_k}}
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\monoidop \Gamma^1_j
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}
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\monoidop \Gamma^0.
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\end{equation}
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\end{theorem}
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\def\eejJ{\equiv \exists{j\in J}\Big(}
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\begin{proof}
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Expanding each~$\Gamma$ in \axmref{class-yield},
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we have
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\begin{alignat*}{3}
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c &\eejJ\Exists{k\in K_j}{\Gamma^2_{j_k}}
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\monoidop \Gamma^1_j
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\Big)
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\monoidop \Gamma^0
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&&\text{by \axmref{class-yield}} \\
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%
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&\eejJ\exists{k\in K_j}\Big(
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\cpredmatseq \monoidop \cpredvecseq \monoidop \cpredscalarseq
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\Big) \\
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&\hphantom{\eejJ}\;\cpredvecseq \monoidop \cpredscalarseq \Big)
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\monoidop \cpredscalarseq, \\
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%
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&\eejJ\exists{k\in K_j}\Big(\cpredmatseq\Big)
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\monoidop \cpredvecseq \monoidop \cpredscalarseq \\
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&\hphantom{\eejJ}\;\cpredvecseq \monoidop \cpredscalarseq \Big)
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\monoidop \cpredscalarseq,
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&&\text{by \dfnref{quant-conn}} \\
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%
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&\eejJ\exists{k\in K_j}\Big(\cpredmatseq\Big)
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&&\text{by \dfnref{prop-taut}} \\
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&\hphantom{\eejJ}\;\cpredvecseq \monoidop \cpredscalarseq \Big)
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\monoidop \cpredscalarseq, \\
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%
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&\eejJ\exists{k\in K_j}\Big(\cpredmatseq\Big)
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&&\text{by \dfnref{quant-conn}} \\
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&\hphantom{\eejJ}\;\cpredvecseq\Big) \monoidop \cpredscalarseq
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\monoidop \cpredscalarseq, \\
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%
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&\eejJ\exists{k\in K_j}\Big(\cpredmatseq\Big)
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&&\text{by \dfnref{prop-taut}} \\
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&\hphantom{\eejJ}\;\cpredvecseq\Big)
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\monoidop \cpredscalarseq.
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\tag*{\qedhere} \\
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\end{alignat*}
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\end{proof}
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\index{classification!composition|)}
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\begin{lemma}[Classification Predicate Vacuity]\lemlabel{class-pred-vacu}
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\index{classification!vacuity|(}
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Let $\Classify^c_\gamma\left(\Monoid\Bool\monoidop e,\emptyset,\emptyset,\emptyset\right)$
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be a classification by~\axmref{class-intro}.
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$\odot$ is a monoid by \corref{odot-monoid}.
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Then $c \equiv \gamma \equiv e$.
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\end{lemma}
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\begin{proof}
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First consider $c$.
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\begin{alignat*}{3}
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c &\equiv \Exists{j\in J}{\Exists{k}{e}\monoidop e} \monoidop e
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\qquad&&\text{by \dfnref{monoid-seq}} \label{p:cri-c} \\
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&\equiv \Exists{j\in J}{e \monoidop e} \monoidop e
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&&\text{by \dfnref{quant-elim}} \\
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&\equiv \Exists{j\in J}{e} \monoidop e
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&&\text{by \ref{eq:monoid-identity}} \\
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&\equiv e \monoidop e
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&&\text{by \dfnref{quant-elim}} \\
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&\equiv e.
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&&\text{by \ref{eq:monoid-identity}}
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\end{alignat*}
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For $\gamma$,
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we have $r=0$ by \axmref{class-yield},
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and so by similar steps as~$c$,
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$\gamma=\Gamma^r=e$.
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Therefore $c\equiv e$.
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\end{proof}
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\begin{figure}[ht]
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\begin{alignat*}{3}
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\begin{aligned}
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\xml{<classify }&\xml{as="always" yields="alwaysTrue"} \xmlnl
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&\xml{desc="Always true" />}
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\end{aligned}
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\quad&=\quad
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\Classifyland^\texttt{always}_\texttt{alwaysTrue}
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&&\left(\emptyset,\emptyset,\emptyset\right). \\
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%
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\begin{aligned}
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\xml{<classify }&\xml{as="never" yields="neverTrue"} \xmlnl
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&\xml{any="true"} \xmlnl
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&\xml{desc="Never true" />}
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\end{aligned}
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\quad&=\quad
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\Classifylor^\texttt{never}_\texttt{neverTrue}
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&&\left(\emptyset,\emptyset,\emptyset\right).
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\end{alignat*}
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\caption{\tameclass{always} and \tameclass{never} from package
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\tamepkg{core/base}.}
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\label{fig:always-never}
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\end{figure}
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\spref{fig:always-never} demonstrates \lemref{class-pred-vacu} in the
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definitions of the classifications \tameclass{always} and
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\tameclass{never}.
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These classifications are typically referenced directly for clarity rather
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than creating other vacuous classifications,
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encapsulating \lemref{class-pred-vacu}.
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\index{classification!vacuity|)}
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\begin{theorem}[Classification Rank Independence]\thmlabel{class-rank-indep}
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\index{classification!rank|(}
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Let $\odot=\Monoid\Bool\monoidop e$.
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Then,
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\begin{equation}
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\Classify_\gamma\left(\odot,M,v,s\right)
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\equiv \Classify\left(
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\odot,
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\Classify_{\gamma'''}\left(\odot,M,\emptyset,\emptyset\right),
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\Classify_{\gamma''}\left(\odot,\emptyset,v,\emptyset\right),
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\Classify_{\gamma'}\left(\odot,\emptyset,\emptyset,s\right)
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\right).
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\end{equation}
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\end{theorem}
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\begin{proof}
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First,
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by \axmref{class-yield},
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observe these special cases following from \lemref{class-pred-vacu}:
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\begin{equation}
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\begin{alignedat}{3}
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\Gamma'''^2 &= \cpredmatseq, \qquad&&\text{assuming $v\union s=\emptyset$} \\
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\Gamma''^1 &= \cpredvecseq, &&\text{assuming $M\union s=\emptyset$} \\
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\Gamma'^0 &= \cpredscalarseq. &&\text{assuming $M\union v=\emptyset$}
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\end{alignedat}
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\end{equation}
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By \thmref{class-compose},
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we must prove
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\begin{multline}\label{eq:rank-indep-goal}
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\Exists{j\in J}{
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\Exists{k\in K_j}{\cpredmatseq}
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\monoidop \cpredvecseq
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}
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\monoidop \cpredscalarseq \\
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\equiv c \equiv
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\Exists{j\in J}{
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\Exists{k\in K_j}{\gamma'''_{j_k}}
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\monoidop \gamma''_j
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}
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\monoidop \gamma'.
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\end{multline}
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By \axmref{class-yield},
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we have $r'''=2$, $r''=1$, and $r'=0$,
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and so $\gamma'''=\Gamma'''^2$,
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$\gamma''=\Gamma''^1$,
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and $\gamma'=\Gamma'^0$.
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By substituting these values in~\ref{eq:rank-indep-goal},
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the theorem is proved.
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\end{proof}
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\index{classification!rank|)}
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These definitions may also be used as a form of pattern matching to look up
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a corresponding variable.
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For example,
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if we have $\Classify^\texttt{foo}$ and want to know its \xmlattr{yields},
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we can write~$\Classify^\texttt{foo}_\gamma$ to bind the
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\xmlattr{yields} to~$\gamma$.\footnote{%
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This is conceptually like a symbol table lookup in the compiler.}
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\subsection{Matches}
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A classification consists of a set of binary predicates called
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\emph{matches}.
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Matches may reference any values,
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including the results of other classifications
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(as in \thmpref{class-compose}),
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allowing for the construction of complex abstractions over the data being
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classified.
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Matches are intended to act intuitively across inputs of different ranks---%
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that is,
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one can match on any combination of matrix, vector, and scalar values.
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\begin{axiom}[Match Input Translation]\axmlabel{match-input}
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Let $j$ and $k$ be free variables intended to be bound in the
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context of \axmref{class-pred}.
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Let $J$ and $K$ be defined by \axmref{class-intro}.
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Given some input~$x$,
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\begin{equation*}
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\varsub x =
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\begin{cases}
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x_{j_k} &\rank{x} = 2; \\
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x_j &\rank{x} = 1; \\
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x &\rank{x} = 0,
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\end{cases}
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\qquad\qquad
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\begin{aligned}
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j&\in J, \\
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k&\in K_j.
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\end{aligned}
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\end{equation*}
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\end{axiom}
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\begin{axiom}[Match Rank]\axmlabel{match-rank}
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Let~$\sim{} : \Real\times\Real\rightarrow\Real$ be some binary relation.
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Then,
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\begin{equation*}
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\rank{\varsub x \sim \varsub y} =
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\begin{cases}
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\rank{x} &\rank{x} \geq \rank{y}, \\
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\rank{y} &\text{otherwise}.
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\end{cases}
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\end{equation*}
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\end{axiom}
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\def\xyequivish{\varsub x\equivish \varsub y}
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\begin{axiom}[Element-Wise Equivalence ($\equivish$)]
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\indexsym\equivish{equivalence, element-wise}
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\index{equivalence!element-wise (\ensuremath\equivish)}
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\begin{align*}
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\rank{\varsub x}=\rank{\varsub y}=2,\,
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(\xyequivish) &\infer \Forall{j,k}{x_{j_k} \equiv y_{j_k}}, \\
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\rank{\varsub x}=\rank{\varsub y}=1,\,
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(\xyequivish) &\infer \Forall{j}{x_j \equiv y_j}, \\
|
||
\rank{\varsub x}=\rank{\varsub y}=0,\,
|
||
(\xyequivish) &\infer (x\equiv y).
|
||
\end{align*}
|
||
\end{axiom}
|
||
|
||
|
||
\index{package!core/match@\tamepkg{core/match}}
|
||
Matches are represented by \xmlnode{match} nodes in \tame{}.
|
||
Since the primitive is rather verbose,
|
||
\tamepkg{core/match} also defines templates providing a more concise
|
||
notation
|
||
(\xmlnode{t:match-$\zeta$} below).
|
||
|
||
\index{classification!match@\xmlnode{match}}
|
||
\begin{axiom}[Match Introduction]\axmlabel{match-intro}
|
||
\begin{alignat*}{2}
|
||
\begin{aligned}[b]
|
||
\xml{<t:match-$\zeta$ }&\xml{on="$x$"} \xmlnll
|
||
&\xml{value="$y$" />}
|
||
\end{aligned}
|
||
{}&\equivish{}
|
||
\begin{aligned}
|
||
&\xml{<match on="$x$">} \xmlnll
|
||
&\quad \xml{<c:$\zeta$>} \xmlnll
|
||
&\quad\quad \xml{<c:value-of name="$y$">} \xmlnll
|
||
&\quad \xml{</c:$\zeta$>} \xmlnll
|
||
&\xml{</match>}
|
||
\end{aligned}
|
||
\qquad
|
||
\sim{} = \smash{\begin{cases}
|
||
= &\zeta=\xml{eq}, \\
|
||
< &\zeta=\xml{lt}, \\
|
||
> &\zeta=\xml{gt}, \\
|
||
\leq &\zeta=\xml{leq}, \\
|
||
\geq &\zeta=\xml{geq}.
|
||
\end{cases}} \\
|
||
&\equivish \varsub x \sim \varsub y,
|
||
\end{alignat*}
|
||
\end{axiom}
|
||
|
||
\begin{axiom}[Match Equality Short Form]
|
||
\begin{equation*}
|
||
\xml{<match on="$x$" />}
|
||
\equivish \xml{<match on="$x$" value="TRUE" />}.
|
||
\end{equation*}
|
||
\end{axiom}
|
||
|
||
\todo{Define types and \xml{typedef}.}
|
||
\begin{axiom}[Match Membership]
|
||
When $T$ is a type defined with \xmlnode{typedef},
|
||
\begin{equation*}
|
||
\xml{<match on="$x$" anyOf="$T$" />} \equivish \varsub x \in T.
|
||
\end{equation*}
|
||
\end{axiom}
|
||
|
||
\begin{theorem}[Classification Match Element-Wise Binary Relation]
|
||
\thmlabel{class-match}
|
||
Within the context of \axmref{class-pred},
|
||
all \xmlnode{match} forms represent binary relations
|
||
$\Real\times\Real\rightarrow\Bool$
|
||
ranging over individual elements of all index sets $J$ and $K_j\in K$.
|
||
\end{theorem}
|
||
\begin{proof}
|
||
First,
|
||
observe that each of $=$, $<$, $>$, $\leq$, $\geq$, and $\in$
|
||
have type $\Real\times\Real\rightarrow\Bool$.
|
||
We must then prove that $\varsub x$ and $\varsub y$ are able to be
|
||
interpreted as~$\Real$ within the context of \axmref{class-pred}.
|
||
|
||
When $x,y\in\Real$,
|
||
we have the trivial case $\varsub x=x\in\Real$ and $\varsub y=y\in\Real$
|
||
by \axmref{match-input}.
|
||
Otherwise,
|
||
variables $j$ and $k$ are free.
|
||
|
||
Consider $\rank{\varsub x \sim \varsub y} = 2$;
|
||
then $\rank{\varsub x \sim \varsub y} \in\Matrices$ by \dfnref{rank},
|
||
and so by \thmref{class-rank-indep} we have
|
||
\begin{equation}\label{p:match-rel}
|
||
\Forall{j\in J}{\Forall{k\in K_j}{\cpredmatseq}}
|
||
\equiv
|
||
\Forall{j\in J}{\Forall{k\in K_j}{\varsub x \sim \varsub y}},
|
||
\end{equation}
|
||
which binds $j$ and $k$ to the variables of their respective quantifiers.
|
||
Proceed similarly for $\rank{\varsub x \sim \varsub y} = 1$ and observe that
|
||
$j$ becomes bound.
|
||
|
||
Assume $x\in\Matrices$;
|
||
then $x_{j_k}\in\Real$ by \dfnref{matrix}.
|
||
Assume $y\in\Vectors^\Real$;
|
||
then $y_j\in\Real$ by \dfnref{vec}.
|
||
Finally,
|
||
observe that $j$ ranges over $J$ in \ref{p:match-rel},
|
||
and $k$ over $K_j$.
|
||
\end{proof}
|
||
|
||
\thmref{class-match} is responsible for proving that matches range over each
|
||
individual index.
|
||
More subtly,
|
||
it also shows that matches work with any combination of rank.
|
||
\spref{f:ex:class-match-all-ranks} demonstrates a complete translation of
|
||
source \tame{}~XML using all ranks.
|
||
|
||
\begin{figure}[ht]
|
||
\begin{align}
|
||
&\begin{aligned}
|
||
&\xml{<classify as="fullrank" desc="Example of all ranks">} \xmlnl
|
||
&\quad\begin{aligned}
|
||
&\xml{<match on="$A$" value="$u$" />}
|
||
\quad&&\equivish \varsub A = \varsub u \\[-2mm]
|
||
&\xml{<match on="$A$" value="$t$" />}
|
||
\quad&&\equivish \varsub A = \varsub t \xmlnll
|
||
&\xml{<match on="$u$" value="$t$" />}
|
||
\quad&&\equivish \varsub u = \varsub t \xmlnll
|
||
&\xml{<match on="$t$" />}
|
||
\quad&&\equivish \varsub t = \true
|
||
\end{aligned} \xmlnll
|
||
&\xml{</classify>}
|
||
\end{aligned}
|
||
&\text{by \axmref{match-intro}} \\
|
||
&= \Classifyland^\texttt{fullrank}\left(
|
||
\Big(\left({A_j}_k = u_j\right),
|
||
\left({A_j}_k = t \right)
|
||
\Big),
|
||
\left(u_j = t\right),
|
||
t
|
||
\right)
|
||
&\text{by \axmref{class-intro}} \\
|
||
&\equiv \Exists{j\in J}{
|
||
\Exists*{k\in K_j}{\Big(
|
||
\left({A_j}_k = u_j\right)
|
||
\land \left({A_j}_k = t \right)
|
||
\Big)}
|
||
\land u_j = t
|
||
}
|
||
\land t.
|
||
&\text{by \thmref{class-match}}.
|
||
\end{align}
|
||
\caption{Example demonstrating \thmref{class-match} using all ranks.}
|
||
\label{f:ex:class-match-all-ranks}
|
||
\end{figure}
|
||
|
||
Visually,
|
||
the one-dimensional construction of \axmref{class-pred} does not lend
|
||
itself well to how intuitive the behavior of the system actually is.
|
||
We therefore establish a relationship to the notation of linear algebra
|
||
to emphasize the relationship between each of the inputs.
|
||
|
||
\newcommand\matseqsup[1]{%
|
||
\begin{bmatrix}
|
||
M^{#1}_{0_0} & \dots & M^{#1}_{0_k} \\
|
||
\vdots & \ddots & \vdots \\
|
||
M^{#1}_{j_0} & \dots & M^{#1}_{j_k} \\
|
||
\end{bmatrix}%
|
||
}
|
||
\newcommand\vecseqsup[1]{%
|
||
\begin{bmatrix}
|
||
v^{#1}_0 \\
|
||
\vdots \\
|
||
v^{#1}_j \\
|
||
\end{bmatrix}%
|
||
}
|
||
|
||
% This must be an axiom because it defines how the connectives operate; see
|
||
% the remark.
|
||
\index{classification!matrix notation}
|
||
\begin{axiom}[Classification Matrix Notation]\axmlabel{class-mat-not}
|
||
Let $\Gamma^2$ be defined by \axmref{class-yield}.
|
||
Then,
|
||
\begin{equation*}
|
||
\Gamma^2 =
|
||
\matseqsup{0}\monoidops\matseqsup{l}
|
||
\monoidop
|
||
\vecseqsup{0}\monoidops\vecseqsup{m}
|
||
\monoidop
|
||
s^0\monoidops s^n,
|
||
\end{equation*}
|
||
from which $\Gamma^1$, $\Gamma^0$, and $\gamma$ can be derived.
|
||
\end{axiom}
|
||
|
||
\begin{remark}[Logical Connectives With Matrix Notation]
|
||
From the definition of \axmref{class-mat-not},
|
||
it should be clear that the logical connective $\monoidop$ necessarily
|
||
acts like a Hadamard product\cite{wp:hadamard-product} with respect to
|
||
how individual elements are combined.
|
||
\end{remark}
|
||
|
||
\index{classification!intuition}
|
||
\axmref{class-mat-not} makes it easy to visualize classification
|
||
operations simply by drawing horizontal boxes across the predicates,
|
||
as demonstrated by \spref{f:class-mat-boxes}.
|
||
This visualization helps to show intuitively how the classification system
|
||
is intended to function,
|
||
with matrices serving as higher-resolution vectors.\footnote{%
|
||
For example,
|
||
with insurance,
|
||
one may have a vector of data by risk location,
|
||
and a matrix of chosen class codes by location.
|
||
Consequently,
|
||
we expect $M_j$ to be the set of class codes associated with
|
||
location~$j$ so that it can be easily matched against location-level
|
||
data~$v_j$.}
|
||
|
||
% NB: Give this formatting extra attention if the document's formatting is
|
||
% substantially changed, since it's not exactly responsible with it's
|
||
% hard-coded units.
|
||
\begingroup
|
||
\begin{figure}[ht]
|
||
\def\classmatraise#1{%
|
||
\begin{aligned}
|
||
#1 \\ {} \\ #1
|
||
\end{aligned}
|
||
}
|
||
\def\classmateq{%
|
||
\matseqsup{0}
|
||
\classmatraise{\monoidop\cdots\monoidop}
|
||
\matseqsup{l}
|
||
\classmatraise\monoidop
|
||
\vecseqsup{0}
|
||
\classmatraise{\monoidop\cdots\monoidop}
|
||
\vecseqsup{m}
|
||
\classmatraise{%
|
||
{}\monoidop s^0\monoidop\cdots\monoidop s^n%
|
||
}
|
||
}
|
||
\def\classmatlines#1{%
|
||
\begin{alignedat}{2}
|
||
\Big( &M^0_{{#1}_0} \monoidops {}&&M^l_{{#1}_0} \Big)
|
||
\monoidop
|
||
v^0_{#1} \monoidops v^m_{#1}
|
||
\monoidop
|
||
s^0 \monoidops s^n \\
|
||
&\quad\!\vdots &&\quad\!\vdots \\
|
||
\Big( &M^0_{{#1}_k} \monoidops {}&&M^l_{{#1}_k} \Big)
|
||
\monoidop
|
||
v^0_{#1} \monoidops v^m_{#1}
|
||
\monoidop
|
||
s^0 \monoidops s^n
|
||
\end{alignedat}
|
||
}
|
||
|
||
\begin{align*}
|
||
&\quad\raisebox{-11mm}[0mm]{%
|
||
\begin{turn}{45}
|
||
$\equiv$
|
||
\end{turn}%
|
||
}\; \classmatlines{0} &\Gamma^2_0 \\[-2mm]
|
||
&\fbox{\raisebox{0mm}[0mm][6mm]{\hphantom{$\classmateq$}}} \\[-8mm]
|
||
%
|
||
&\classmateq &\vdots\; \\[-10mm]
|
||
%
|
||
&\fbox{\raisebox{0mm}[0mm][6mm]{\hphantom{$\classmateq$}}} \\
|
||
&\quad\raisebox{11mm}[0mm]{%
|
||
\begin{turn}{-45}
|
||
$\equiv$
|
||
\end{turn}%
|
||
}\; \classmatlines{j} &\Gamma^2_j
|
||
\end{align*}
|
||
\caption{Visual interpretation of classification by \axmref{class-mat-not}.
|
||
For each boxed row of the matrix notation there is an equivalence
|
||
to the first-order logic of \thmref{class-compose}.}
|
||
\label{f:class-mat-boxes}
|
||
\end{figure}
|
||
\endgroup
|
||
|
||
\index{classification!as proposition|(}
|
||
\begin{lemma}[Match As Proposition]\lemlabel{match-prop}
|
||
Matches can be represented using propositional logic provided that
|
||
binary operators of \axmref{match-intro} are restricted to $\cbif\Bool$.
|
||
\end{lemma}
|
||
\begin{proof}
|
||
\begin{alignat*}{4}
|
||
x = \true &\equiv x, &&\quad= &&: \cbif\Bool; \\
|
||
x = \false &\equiv \neg x, &&\quad= &&: \cbif\Bool; \\
|
||
x < y &\equiv \neg x \land y, &&\quad< &&: \cbif\Bool; \\
|
||
x > y &\equiv x \land \neg y, &&\quad> &&: \cbif\Bool; \\
|
||
x \leq y &\equiv \neg x \lor y, &&\quad\leq &&: \cbif\Bool; \\
|
||
x \geq y &\equiv x \lor \neg y, &&\quad\geq &&: \cbif\Bool; \\
|
||
x \in\Bool &\equiv \true, &&\quad\in &&: \cbif\Bool.\tag*{\qedhere}
|
||
\end{alignat*}
|
||
\end{proof}
|
||
|
||
\begin{theorem}[Classification As Proposition]
|
||
\index{classification!as proposition|(}
|
||
Classifications with either $M\union v=\emptyset$ or with constant index
|
||
sets can be represented by propositional logic provided that the domains
|
||
of the binary operators of \axmref{match-intro} are restricted to
|
||
$\cbif\Bool$.
|
||
\end{theorem}
|
||
\begin{proof}
|
||
Propositional logic does not include quantifiers or relations.
|
||
Matches of the domain $\cbif\Bool$ are proved to be propositions by
|
||
\lemref{match-prop}.
|
||
Having eliminated relations,
|
||
we must now eliminate quantifiers.
|
||
|
||
Assume $M\union v=\emptyset$.
|
||
By \thmref{class-rank-indep},
|
||
|
||
\begin{align*}
|
||
c &\equiv \cpredscalarseq,
|
||
\end{align*}
|
||
|
||
\noindent
|
||
which is a propositional formula.
|
||
|
||
Similarly,
|
||
if we define our index set~$J$ to be constant,
|
||
we are then able to eliminate existential quantification over~$J$
|
||
as follows:
|
||
\begin{equation}\label{eq:prop-vec}
|
||
\begin{aligned}
|
||
c &\equiv \Exists{j\in J}{\cpredvecseq}, \\
|
||
&\equiv \left(v^0_0\monoidops v^m_0\right)
|
||
\lor\cdots\lor
|
||
\left(v^0_{|J|-1}\monoidops v^m_{|J|-1}\right),
|
||
\end{aligned}
|
||
\end{equation}
|
||
which is a propositional formula.
|
||
Similarly,
|
||
for matrices,
|
||
|
||
\begin{align*}
|
||
c &\equiv \Exists{j\in J}{\Exists{k\in K_j}{\cpredmatseq}}, \nonumber\\
|
||
&\equiv \Exists{j\in J}{
|
||
\left({M^0_j}_0\monoidops{M^0_j}_{|K_j|-1}\right)
|
||
\lor\cdots\lor
|
||
\left({M^l_j}_0\monoidops{M^l_j}_{|K_j|-1}\right)
|
||
},
|
||
\end{align*}
|
||
and then proceed as in~\ref{eq:prop-vec}.
|
||
\end{proof}
|
||
\index{classification!as proposition|)} |