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design/tpl: Beginnings of classifications in terms of first-order logic 

This is going to evolve a great deal, and note that the yield definition is
completely absent.

It may be time to switch to natural deduction (Gentzen-style).
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Mike Gerwitz 2021-05-14 12:05:17 -04:00
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195
design/tpl/sec/class.tex
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@ -17,10 +17,59 @@ Intuitively,
\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}[$\land$-Classification]\dfnlabel{classu}
A conjunctive\footnote{%
\begin{definition}[Classification Introduction]
\todo{Symbol in place of $=$ here ($\equiv$ not appropriate).}
\begin{alignat}{3}
&\xml{<classify as="$c$" }&&\xml{yields="$\gamma$" desc}&&\xml{="$\_$"
$\alpha$>}\label{eq:xml-classify} \\
&\quad \VFam{M^0}jJ &&\VFam{v^0}jJ &&\quad s^0 \nonumber\\
&\quad \quad\vdots &&\quad\vdots &&\quad \vdots \nonumber\\
&\quad \VFam{M^l}jJ &&\VFam{v^m}jJ &&\quad s^n \nonumber\\
&\xml{</classify>}
% 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
\begin{align}
J &\subset\Int, \\
\forall{k}\Big(M^k &: J \rightarrow \Int \rightarrow \Real\Big),
\label{eq:class-matrix} \\
\forall{k}\Big(v^k &: J \rightarrow \Real\Big), \\
\forall{k}\Big(s^k &\in\Real\Big), \\
l+m+n &\geq 0, \label{eq:mvs-opt} \\
\alpha &\in\Set{\epsilon,\, \texttt{any="true"}}, \label{eq:xml-any-domain}
\end{align}
\end{definition}
\noindent
and the monoid~$\odot$ is defined as
\begin{equation}\label{eq:classify-rel}
\odot = \begin{cases}
\Monoid\Bool\land\true &\alpha = \epsilon,\\
\Monoid\Bool\lor\false &\alpha = \texttt{any="true"}.
\end{cases}
\end{equation}
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.}
The sum~\eqref{eq:mvs-opt} emphasizes that there may be zero or more of each
(see also \remref{vacuous-truth}).
A $\land$-classification is pronounced ``conjunctive classification'',
and $\lor$ ``disjunctive''.\footnote{%
Conjunctive and disjunctive classifications used to be referred to,
respectively,
as \emph{universal} and \emph{existential},
@ -29,47 +78,115 @@ This limitation is mitigated through use of the template system.
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
$M_0\ldots M_n$.
and so the terminology would otherwise lead to confusion.}
\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 \land \ldots \land M_n.
\end{align*}
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.}
% This TODO was the initial motivation for this paper!
\todo{Describe what \tame{} does when this is not the case.}
Each $M^k$ and~$v^k$ share the same index set~$J$.
Intuitively,
this means that all vectors must be the same length,
that their length must match the number of rows in each matrix,
and that each matrix must have the same number of rows.
But it leaves \emph{unbounded} the lengths of each inner vector of a matrix
(its columns; see \secref{vec}),
as emphasized by~\eqref{eq:class-matrix}.
This is further constrained by~\dfnref{class-pred}.
$\alpha$~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.
\def\cpredscalar{s^0\bullet\cdots\bullet s^n}
\begin{definition}[Classification-Predicate Equivalence]\dfnlabel{class-pred}
Let $\Classify^c_\gamma\left(\Monoid\Bool\bullet e,M,v,s\right)$ be a
classification by~\eqref{eq:xml-classify}.
We then have the first-order sentence
\begin{equation*}
c \equiv
{} \Exists{j\in J}{
\Exists{k}{{M^0_j}_k \bullet\cdots\bullet {M^l_j}_k}
\bullet v^0_j \bullet\cdots\bullet v^m_j}
\bullet \cpredscalar.
\end{equation*}
\end{definition}
\index{classification!disjunctive}
\begin{definition}[$\lor$-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{corollary}[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}
These facts follow clearly from \dfnref{class-pred}.
Since the length of vectors and matrices are unknown until runtime,
first-order logic is required to quantify over their index sets.
If, however, we restrict ourselves to scalar inputs,
we have
\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 \lor \ldots \lor M_n.
\end{align*}
\end{definition}
\begin{align*}
c &\equiv \Exists{j\in J}{\Exists{k}{e}\bullet e}
\bullet\cpredscalar \\
&\equiv \Exists{j\in J}{e \bullet e}
\bullet\cpredscalar \\
&\equiv \Exists{j\in J}{e}
\bullet\cpredscalar \\
&\equiv e \bullet\cpredscalar \\
&\equiv \cpredscalar,
\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~$\false$.},
we are then able to eliminate existential quantification over~$J$
as follows:
Let $j=|J|-1$.
Then,
% Better to have this on another page than to have the sentence that
% follows be a widow.
\goodbreak
\begin{equation}\label{eq:class-vexpand}
c \equiv (v^0_0 \lor\cdots\lor v^0_j)
\bullet\cdots\bullet
(v^m_0 \lor\cdots\lor v^0_j),
\end{equation}
\noindent
which is a propositional formula.
We can extend the same concept to the index set of the inner vector of
each matrix and proceed inductively on~\eqref{eq:class-vexpand} above
with the inner vector's index set in place of~$J$.
\end{proof}
\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.}
For example,
consider the following classification $\Classify^\texttt{cost-exceeded}$.
Consider the following classification $\Classify^\texttt{cost-exceeded}$.
Let~\tameparam{cost} be a scalar parameter.
\index{classification!classify@\xmlnode{classify}}

4
design/tpl/sec/notation.tex
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@ -151,7 +151,7 @@ $\forall$ denotes first-order universal quantification (``for all''),
\end{definition}
\index{quantification!vacuous truth}
\begin{remark}[Vacuous Truth]
\begin{remark}[Vacuous Truth]\remlabel{vacuous-truth}
By \dfnref{exists}, $\Exists{x\in\emptyset}P \equiv \false$
and by \dfnref{forall}, $\Forall{x\in\emptyset}P \equiv \true$.
And so we also have the tautologies $\infer \neg\Exists{x\in\emptyset}P$
@ -322,7 +322,7 @@ Given that, we have $f\bicomp{[]} = f\bicomp{[A]}$ for functions returning
noting that $\bicompi{[]}$ is \emph{not} a sensible construction.
\subsection{Monoids and Sequences}
\subsection{Monoids and Sequences}\seclabel{monoids}
\index{abstract algebra!monoid}
\index{monoid|see abstract algebra, monoid}
\begin{definition}[Monoid]\dfnlabel{monoid}

18
design/tpl/tpl.sty
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@ -35,6 +35,9 @@
% Creative Commons icons
\usepackage{ccicons}
% Dangerous Bend, from TeXbook
\usepackage{manfnt}
% Colors from Tango Icon Theme
% https://en.wikipedia.org/wiki/Tango_Desktop_Project
\definecolor{href}{HTML}{204a87}
@ -80,8 +83,10 @@
\newcommand\Matrices{\ensuremath{\Vectors^{\Vectors^\Real}}}
\let\union\cup
\let\Union\bigcup
\let\intersect\cap
\let\infer\vdash
\let\emptystr\epsilon
\newcommand\len[1]{\ensuremath{\left|#1\right|}}
@ -144,3 +149,16 @@
\newcommand\mremark[1]{\marginnote{\textsl{#1}}}
\newcommand\Monoid[3]{\left({#1},{#2},{#3}\right)}
% A really obnoxious notice making clear to the reader that this portion of
% the work is unfinished, to the point where it's probably even
% incorrect. Uses dangerous bend symbol from manfnt, which is admittedly a
% misuse given that it's often used to represent difficult problems. Though
% I suppose an unfinished work is a difficult problem.
\newcommand\INCOMPLETE[1]{%
\bigskip
\par
\todo{This is incomplete!}
\hfill\textdbend\textbf{\textsl{#1}}\textdbend\hfill
\bigskip
}