% The TAME Programming Language Classification System % % Copyright (C) 2021 Ryan Specialty Group, LLC. % % Licensed under the Creative Commons Attribution-ShareAlike 4.0 % International License. %% \section{Classification System}\seclabel{class} \index{classification|textbf} A \dfn{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 \dfn{first-order sentences}---% that is, they contain no \dfn{free variables}. Intuitively, this means that all variables within a~classification are \dfn{tightly coupled} to the classification itself. This limitation is mitigated through use of the template system. \begin{axiom}[Classification Introduction]\axmlabel{class-intro} \indexsym\Classify{classification} \indexsym\gamma{classification, yield} \index{classification!index set} \index{index set!classification} \index{classification!classify@\xmlnode{classify}} \index{classification!as@\xmlattr{as}} \index{classification!yields@\xmlattr{yields}} \todo{Symbol in place of $=$ here ($\equiv$ not appropriate).} \begin{alignat}{3} &\xml{}\label{eq:xml-classify} \\ &\quad \MFam{M^0}jJkK &&\VFam{v^0}jJ &&\quad s^0 \nonumber\\[-4mm] &\quad \quad\vdots &&\quad\vdots &&\quad \vdots \nonumber\\ &\quad \MFam{M^l}jJkK &&\VFam{v^m}jJ &&\quad s^n \nonumber\\[-3mm] &\xml{} % NB: This -50mu needs adjustment if you change the alignment above! &&\mspace{-50mu}= \Classify^c_\gamma\left(\odot,M,v,s\right), \nonumber \end{alignat} \noindent where \indexsym\emptystr{empty string} \index{empty string (\ensuremath\emptystr)} \begin{align} J &\subset\Int \neq\emptyset, \\ \forall{j\in J}\Big(K_j &\subset\Int \neq\emptyset\Big), \\ \forall{k}\Big(M^k &: J \rightarrow K_{j\in J} \rightarrow \Bool\Big), \label{eq:class-matrix} \\ \forall{k}\Big(v^k &: J \rightarrow \Bool\Big), \\ \forall{k}\Big(s^k &\in\Bool\Big), \\ \alpha &\in\Set{\emptystr,\, \texttt{any="true"}}, \label{eq:xml-any-domain} \end{align} \noindent and the monoid~$\odot$ is defined as \indexsym\odot{classification, monoid} \index{classification!any@\xmlattr{any}} \index{classification!monoid|(} \begin{equation}\label{eq:classify-rel} \odot = \begin{cases} \Monoid\Bool\land\true &\alpha = \emptystr,\\ \Monoid\Bool\lor\false &\alpha = \texttt{any="true"}. \end{cases} \end{equation} \end{axiom} % This TODO was the initial motivation for this paper! \todo{Emphasize index sets, both relationships and nonempty.} We use a $4$-tuple $\Classify\left(\odot,M,v,s\right)$ to represent a $\odot_1$-classification (a classification with the binary operation $\land$ or~$\lor$) consisting of a combination of matrix~($M$), vector~($v$), and scalar~($s$) matches, rendered above in columns.\footnote{% The symbol~$\odot$ was chosen since the binary operation for a monoid is~$\bullet$ (see \secref{monoids}) and~$\odot$ looks vaguely like~$(\bullet)$, representing a portion of the monoid triple.} A $\land$-classification is pronounced ``conjunctive classification'', and $\lor$ ``disjunctive''.\footnote{% \index{classification!terminology history} Conjunctive and disjunctive classifications used to be referred to, respectively, as \dfn{universal} and \dfn{existential}, referring to fact that $\forall\Set{a_0,\ldots,a_n}(a) \equiv a_0\land\ldots\land a_n$, 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.} The variables~$c$ and~$\gamma$ are required in~\tame{} but are both optional in our notation~$\Classify^c_\gamma$, and can be used to identify the two different data representations of the classification.\footnote{% \xpath{classify/@yields} is optional in the grammar of \tame{}, but the compiler will generate one for us if one is not provided. As such, we will for simplicity consider it to be required here.} $\alpha$~serves as a placeholder for an optional \xml{any="true"}, with $\emptystr$~representing the empty string in~\eqref{eq:xml-any-domain}. Note the wildcard variable matching \xmlattr{desc}---% its purpose is only to provide documentation. \begin{corollary}[$\odot$ Commutative Monoid]\corlabel{odot-monoid} \index{classification!commutativity|(} $\odot$ is a commutative monoid in \axmref{class-intro}. \end{corollary} \begin{proof} By \axmref{class-intro}, $\odot$ must be a monoid. Assume $\alpha=\emptystr$. Then, $\odot = \Monoid\Bool\land\true$, which is proved by \lemref{monoid-land}. Next, assume $\alpha=\texttt{any="true"}$. Then, $\odot = \Monoid\Bool\lor\false$, which is proved by \lemref{monoid-land}. \end{proof} While \axmref{class-intro} seems to imply an ordering to matches, users of the language are free to specify matches in any order and the compiler will rearrange matches as it sees fit. \index{compiler!classification commutativity} This is due to the commutativity of~$\odot$ as proved by \corref{odot-monoid}, and not only affords great ease of use to users of~\tame{}, but also great flexibility to compiler writers. \index{classification!commutativity|)} For notational convenience, we will let \index{classification!monoid|)} \begin{align} \odot^\land &= \Monoid\Bool\land\true, \\ \odot^\lor &= \Monoid\Bool\lor\false. \end{align} \def\cpredmatseq{{M^0_j}_k \bullet\cdots\bullet {M^l_j}_k} \def\cpredvecseq{v^0_j\bullet\cdots\bullet v^m_j} \def\cpredscalarseq{s^0\bullet\cdots\bullet s^n} \begin{axiom}[Classification-Predicate Equivalence]\axmlabel{class-pred} \index{classification!as predicate} Let $\Classify^c_\gamma\left(\Monoid\Bool\bullet e,M,v,s\right)$ be a classification by~\axmref{class-intro}. We then have the first-order sentence \begin{equation*} c \equiv {} \Exists{j\in J}{\Exists{k\in K_j}\cpredmatseq\bullet\cpredvecseq} \bullet\cpredscalarseq. \end{equation*} \end{axiom} \begin{axiom}[Classification Yield]\axmlabel{class-yield} \indexsym\Gamma{classification, yield} \index{classification!yield (\ensuremath\gamma, \ensuremath\Gamma)} Let $\Classify^c_\gamma\left(\Monoid\Bool\bullet e,M,v,s\right)$ be a classification by~\axmref{class-intro}. Then, \begin{align} r &= \begin{cases} 2 &M\neq\emptyset, \\ 1 &M=\emptyset \land v\neq\emptyset, \\ 0 &M\union v = \emptyset, \end{cases} \\ \exists{j\in J}\Big(\exists{k\in K_j}\Big( \Gamma^2_{j_k} &= \cpredmatseq\bullet\cpredvecseq\bullet\cpredscalarseq \Big)\Big), \\ % \exists{j\in J}\Big( \Gamma^1_j &= \cpredvecseq\bullet\cpredscalarseq \Big), \\ % \Gamma^0 &= \cpredscalarseq. \\ % \gamma &= \Gamma^r. \end{align} \end{axiom} \begin{theorem}[Classification Composition]\thmlabel{class-compose} \index{classification!composition|(} Classifications may be composed to create more complex classifications using the classification yield~$\gamma$ as in~\axmref{class-yield}. This interpretation is equivalent to \axmref{class-pred} by \begin{equation} c \equiv \Exists{j\in J}{ \Exists{k\in K_j}{\Gamma^2_{j_k}} \bullet \Gamma^1_j } \bullet \Gamma^0. \end{equation} \end{theorem} \def\eejJ{\equiv \exists{j\in J}\Big(} \begin{proof} Expanding each~$\Gamma$ in \axmref{class-yield}, we have \begin{alignat*}{3} c &\eejJ\Exists{k\in K_j}{\Gamma^2_{j_k}} \bullet \Gamma^1_j \Big) \bullet \Gamma^0 &&\text{by \axmref{class-yield}} \\ % &\eejJ\exists{k\in K_j}\Big( \cpredmatseq \bullet \cpredvecseq \bullet \cpredscalarseq \Big) \\ &\hphantom{\eejJ}\;\cpredvecseq \bullet \cpredscalarseq \Big) \bullet \cpredscalarseq, \\ % &\eejJ\exists{k\in K_j}\Big(\cpredmatseq\Big) \bullet \cpredvecseq \bullet \cpredscalarseq \\ &\hphantom{\eejJ}\;\cpredvecseq \bullet \cpredscalarseq \Big) \bullet \cpredscalarseq, &&\text{by \dfnref{quant-conn}} \\ % &\eejJ\exists{k\in K_j}\Big(\cpredmatseq\Big) &&\text{by \dfnref{prop-taut}} \\ &\hphantom{\eejJ}\;\cpredvecseq \bullet \cpredscalarseq \Big) \bullet \cpredscalarseq, \\ % &\eejJ\exists{k\in K_j}\Big(\cpredmatseq\Big) &&\text{by \dfnref{quant-conn}} \\ &\hphantom{\eejJ}\;\cpredvecseq\Big) \bullet \cpredscalarseq \bullet \cpredscalarseq, \\ % &\eejJ\exists{k\in K_j}\Big(\cpredmatseq\Big) &&\text{by \dfnref{prop-taut}} \\ &\hphantom{\eejJ}\;\cpredvecseq\Big) \bullet \cpredscalarseq. \tag*{\qedhere} \\ \end{alignat*} \end{proof} \index{classification!composition|)} \begin{lemma}[Classification Predicate Vacuity]\lemlabel{class-pred-vacu} \index{classification!vacuity|(} Let $\Classify^c_\gamma\left(\Monoid\Bool\bullet e,\emptyset,\emptyset,\emptyset\right)$ be a classification by~\axmref{class-intro}. $\odot$ is a monoid by \corref{odot-monoid}. Then $c \equiv \gamma \equiv e$. \end{lemma} \begin{proof} First consider $c$. \begin{alignat}{3} c &\equiv \Exists{j\in J}{\Exists{k}{e}\bullet e} \bullet e \qquad&&\text{by \dfnref{monoid-seq}} \label{p:cri-c} \\ &\equiv \Exists{j\in J}{e \bullet e} \bullet e &&\text{by \dfnref{quant-elim}} \\ &\equiv \Exists{j\in J}{e} \bullet e &&\text{by \ref{eq:monoid-identity}} \\ &\equiv e \bullet e &&\text{by \dfnref{quant-elim}} \\ &\equiv e. &&\text{by \ref{eq:monoid-identity}} \end{alignat} For $\gamma$, we have $r=0$ by \axmref{class-yield}, and so by similar steps as~$c$, $\gamma=\Gamma^r=e$. Therefore $c\equiv e$. \end{proof} \begin{figure}[h]\label{fig:always-never} \begin{alignat*}{3} \begin{aligned} \xml{} \end{aligned} \quad&=\quad \Classify^\texttt{always}_\texttt{alwaysTrue} &&\left(\odot^\land,\emptyset,\emptyset,\emptyset\right). \\ % \begin{aligned} \xml{} \end{aligned} \quad&=\quad \Classify^\texttt{never}_\texttt{neverTrue} &&\left(\odot^\lor,\emptyset,\emptyset,\emptyset\right). \end{alignat*} \caption{\tameclass{always} and \tameclass{never} from package \tamepkg{core/base}.} \end{figure} Figure~\ref{fig:always-never} demonstrates \lemref{class-pred-vacu} in the definitions of the classifications \tameclass{always} and \tameclass{never}. These classifications are typically referenced directly for clarity rather than creating other vacuous classifications, encapsulating \lemref{class-pred-vacu}. \index{classification!vacuity|)} \begin{theorem}[Classification Rank Independence]\thmlabel{class-rank-indep} \index{classification!rank|(} Let $\odot=\Monoid\Bool\bullet e$. Then, \begin{equation} \Classify_\gamma\left(\odot,M,v,s\right) \equiv \Classify\left( \odot, \Classify_{\gamma'''}\left(\odot,M,\emptyset,\emptyset\right), \Classify_{\gamma''}\left(\odot,\emptyset,v,\emptyset\right), \Classify_{\gamma'}\left(\odot,\emptyset,\emptyset,s\right) \right). \end{equation} \end{theorem} \begin{proof} First, by \axmref{class-yield}, observe these special cases following from \lemref{class-pred-vacu}: \begin{alignat}{3} \Gamma'''^2 &= \cpredmatseq, \qquad&&\text{assuming $v\union s=\emptyset$} \\ \Gamma''^1 &= \cpredvecseq, &&\text{assuming $M\union s=\emptyset$}\\ \Gamma'^0 &= \cpredscalarseq. &&\text{assuming $M\union v=\emptyset$} \end{alignat} By \thmref{class-compose}, we must prove \begin{align} \Exists{j\in J}{ \Exists{k\in K_j}{\cpredmatseq} \bullet \cpredvecseq } \bullet \cpredscalarseq \nonumber\\ \equiv c \equiv \Exists{j\in J}{ \Exists{k\in K_j}{\gamma'''_{j_k}} \bullet \gamma''_j } \bullet \gamma'. \label{eq:rank-indep-goal} \end{align} By \axmref{class-yield}, we have $r'''=2$, $r''=1$, and $r'=0$, and so $\gamma'''=\Gamma'''^2$, $\gamma''=\Gamma''^1$, and $\gamma'=\Gamma'^0$. By substituting these values in~\ref{eq:rank-indep-goal}, the theorem is proved. \end{proof} \index{classification!rank|)} \begin{corollary}[Classification As Proposition] \index{classification!as proposition|(} Classifications with $M\union v=\emptyset$ or with constant index sets can be represented by propositional logic (that is---without first-order logic). \end{corollary} \begin{proof} 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 (such that it is known at compile-time)\footnote{% Alternatively, we could set an upper bound for~$J$, always expand into that upper bound, and then let undefined values of $v^m_j$ be~$e$.}, we are then able to eliminate existential quantification over~$J$ as follows: Then, \begin{align}\label{eq:prop-vec} c &\equiv \Exists{j\in J}{\cpredvecseq}, \nonumber\\ &\equiv \left(v^0_0\bullet\cdots\bullet v^m_0\right) \lor\cdots\lor \left(v^0_{|J|-1}\bullet\cdots\bullet v^m_{|J|-1}\right), \end{align} \noindent 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\bullet\cdots\bullet{M^0_j}_{|K_j|-1}\right) \lor\cdots\lor \left({M^l_j}_0\bullet\cdots\bullet{M^l_j}_{|K_j|-1}\right) }, \end{align} \noindent and then proceed as in~\ref{eq:prop-vec}. \end{proof} \index{classification!as proposition|)} \INCOMPLETE{The classification definitions are incomplete!} These definitions may also be used as a form of pattern matching to look up a corresponding variable. For example, if we have $\Classify^\texttt{foo}$ and want to know its \xmlattr{yields}, we can write~$\Classify^\texttt{foo}_\gamma$ to bind the \xmlattr{yields} to~$\gamma$.\footnote{% This is conceptually like a symbol table lookup in the compiler.} \mremark{Note that these illustrate \emph{scalar} values only.} Consider the following classification $\Classify^\texttt{cost-exceeded}$. Let~\tameparam{cost} be a scalar parameter. \begin{lstlisting} \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~\dfn{predicate}. Multiple predicates are by default joined by conjunction: \begin{lstlisting} \end{lstlisting} \noindent is equivalent to the proposition \begin{equation*} \tameclass{pool-hazard} \equiv \tameparam{diving\_board} \land \tameparam{pool\_depth\_ft} < 8. \end{equation*} \subsection{Matches} \todo{Non-scalar values.} \begin{definition}[Match Equality] \begin{equation*} \xml{} \equiv x = y. \end{equation*} \end{definition} \begin{definition}[Match Equality Short Form] \begin{equation*} \xml{} \equiv \xml{}. \end{equation*} \end{definition} \begin{definition}[Match Equality Long Form] \begin{alignat*}{2} \xml{} &\equiv {}&&\xml{} \\ & &&\quad \xml{} \\ & &&\quad\quad \xml{} \\ & &&\quad \xml{} \\ & &&\xml{} \\ &\equiv {}&&\xml{}. \end{alignat*} \end{definition} \begin{definition}[Match Membership Equivalence] When $T$ is a type defined with \xmlnode{typedef}, \begin{equation*} \xml{} \equiv x \in T. \end{equation*} \end{definition}