320 lines
11 KiB
TeX
320 lines
11 KiB
TeX
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\section{Notational Conventions}
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This section provides a fairly terse overview of the foundational
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mathematical concepts used in this paper.
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While we try to reason about \tame{} in terms of algebra,
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first-order logic;
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and set theory;
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notation varies even within those branches.
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To avoid ambiguity,
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especially while introducing our own notation,
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core operators and concepts are explicitly defined below.
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This section begins its numbering at~0.
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This is not only a hint that \tame{} (and this paper) use 0-indexing,
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but also because equations; definitions; theorems; corollaries; and the
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like are all numbered relative to their section.
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When you see any of these prefixed with ``0.'',
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this sets those references aside as foundational mathematical concepts
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that are not part of the theory and operation of \tame{} itself.
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\subsection{Propositional Logic}
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\index{boolean!false@\false{}}%
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\index{boolean!true@\true{}}%
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\index{boolean!FALSE@\tamefalse{}}%
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\index{boolean!TRUE@\tametrue{}}%
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\index{integer (\Int)}%
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We reproduce here certain axioms and corollaries of propositional logic for
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convenience and to clarify our interpretation of certain concepts.
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The use of the symbols $\logand$, $\logor$, and~$\neg$ are standard.
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The symbol $\vdash$ means ``infer''.
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We use $\implies$ in place of $\rightarrow$ for implication,
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since the latter is used to denote the mapping of a domain to a codomain
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in reference to functions.
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We further use $\equiv$ in place of $\leftrightarrow$ to represent material
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equivalence.
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\begin{definition}[Logical Conjunction]
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$p,q \vdash (p\logand q)$.
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\end{definition}
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\begin{definition}[Logical Disjunction]
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$p \vdash (p\logor q)$ and $q \vdash (p\logor q)$.
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\end{definition}
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\begin{definition}[Law of Excluded Middle]
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$\vdash (p \logor \neg p)$.
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\end{definition}
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\begin{definition}[Law of Non-Contradiction]
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$\vdash \neg(p \logand \neg p)$.
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\end{definition}
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\begin{definition}[De Morgan's Theorem]
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$\neg(p \logand q) \vdash (\neg p \logor \neg q)$
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and $\neg(p \logor q) \vdash (\neg p \logand \neg q)$.
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\end{definition}
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\index{equivalence!material (\ensuremath{\equiv})}
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\begin{definition}[Material Equivalence]
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$p\equiv q \vdash \big((p \logand q) \logor (\neg p \logand \neg q)\big)$.
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\end{definition}
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$\equiv$ denotes a logical identity.
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Consequently,
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it'll often be used as a definition operator.
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\begin{definition}[Logical Implication]
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$p\implies q \vdash (\neg p \logor q)$.
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\end{definition}
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\begin{definition}[Truth Values]\dfnlabel{truth-values}
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$\vdash\true$ and $\vdash\neg\false$.
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\end{definition}
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\subsection{First-Order Logic and Set Theory}
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The symbol $\emptyset$ represents the empty set---%
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the set of zero elements.
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We assume that the axioms of ZFC~set theory hold,
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but define $\in$ here for clarity.
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\index{set!membership@membership (\ensuremath{\in})}
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\begin{definition}[Set Membership]
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$x \in S \equiv \Set{x} \cap S \not= \emptyset.$
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\end{definition}
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\index{quantification|see {fist-order logic}}
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\index{first-order logic!quantification (\ensuremath{\forall, \exists})}
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$\forall$ denotes first-order universal quantification (``for all''),
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and $\exists$ first-order existential quantification (``there exists''),
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over some \gls{domain}.
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\index{disjunction|see {first-order logic}}
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\index{first-order logic!disjunction (\ensuremath{\logor})}
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\begin{definition}[Existential Quantification]\dfnlabel{exists}
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$\Exists{x\in X}{P(x)} \equiv
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\true \in \Set{P(x) \mid x\in X}$.
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\end{definition}
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\index{conjunction|see {first-order logic}}
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\index{first-order logic!conjunction (\ensuremath{\logand})}
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\begin{definition}[Universal Quantification]\dfnlabel{forall}
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$\Forall{x\in X}{P(x)} \equiv \neg\Exists{x\in X}{\neg P(x)}$.
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\end{definition}
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\index{set!empty (\ensuremath{\emptyset, \{\}})}
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\begin{remark}[Vacuous Truth]
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By Definition~7, $\Exists{x\in\emptyset}P \equiv \false$
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and by \dfnref{forall}, $\Forall{x\in\emptyset}P \equiv \true$.
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And so we also have the tautologies $\vdash \neg\Exists{x\in\emptyset}P$
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and $\vdash \Forall{x\in\emptyset}P$.
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\end{remark}
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\begin{definition}[Boolean/Integer Equivalency]\dfnlabel{bool-int}
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$\Set{0,1}\in\Int, \false \equiv 0$ and $\true \equiv 1$.
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\end{definition}
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\tamefalse{} and~\tametrue{} are constants in \tame{} mapping to the
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\gls{integer} values $\{0,1\}\in\Int$.
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\dfnref{bool-int} relates these constants to their
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\gls{boolean} counterparts so that they may be used in numeric contexts
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and vice-versa.
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\subsection{Functions}
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The notation $f = x \mapsto x' : A\rightarrow B$ represents a function~$f$
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that maps from~$x$ to~$x'$,
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where $x\in A$ (the domain of~$f$) and $x'\in B$ (the co-domain of~$f$).
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A function $A\rightarrow B$ can be represented as the Cartesian
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product of its domain and codomain, $A\times B$.
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For example,
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$x\mapsto x^2 : \Int\rightarrow\Int$ is represented by the set of ordered
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pairs $\Set{(x,x^2) \mid x\in\Int}$, which looks something like
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\begin{equation*}
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\Set{\ldots,\,(0,0),\,(1,1),\,(2,4),\,(3,9),\,\ldots}.
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\end{equation*}
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The set of values over which some function~$f$ ranges is its \emph{image},
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which is a subset of its codomain.
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In the example above,
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both the domain and codomain are the set of integers~$\Int$,
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but the image is $\Set{x^2 \mid x\in\Int}$,
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which is clearly a subset of~$\Int$.
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We therefore have
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\begin{align}
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A \rightarrow B &\subset A\times B, \\
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f : A \rightarrow B &\vdash f \subset A\times B, \\
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f = \alpha \mapsto \alpha' : A \rightarrow B
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&= \Set{(\alpha,\alpha')
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\mid \alpha\in A \logand \alpha'\in B}, \\
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f[D\subseteq A] &= \Set{f(\alpha) \mid \alpha\in D} \subset B, \\
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f[] &= f[A].
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\end{align}
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And ordered pair $(x,y)$ is also called a \emph{$2$-tuple}.
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Generally,
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an \emph{$n$-tuple} is used to represent an $n$-ary function,
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where by convention we have $(x)=x$.
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So $f(x,y) = f((x,y)) = x+y$.
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If we let $t=(x,y)$,
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then we also have $f(x,y) = ft$.
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Binary functions are often written using \emph{infix} notation;
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for example, we have $x+y$ rather than $+(x,y)$.
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\begin{equation}
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fx \in \Set{b \mid (x,b) \in f}
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\end{equation}
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\subsubsection{Binary Operations On Functions}
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Consider two unary functions $f$ and~$g$,
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and a binary relation~$R$.
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We introduce a notation~$\bicomp R$ to denote the composition of a binary
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function with two unary functions.\footnote{%
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The notation originates from~$\circ$ to denote ordinary function
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composition,
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as in $(f\circ g)(x) = f(g(x))$.}
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\begin{align}
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f &: A \rightarrow B \\
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g &: A \rightarrow D \\
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R &: B\times D \rightarrow F \\
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f \bicomp{R} g &= \alpha \mapsto f(\alpha)Rg(\alpha) : A \rightarrow F
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\end{align}
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Note that $f$ and~$g$ must share the same domain~$A$.
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In that sense,
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this is the mapping of the operation~$R$ over the domain~$A$.
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This is analogous to unary function composition~$f\circ g$.
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A scalar value~$x$ can be mapped onto some function~$f$ using a constant
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function.
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For example,
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consider adding some number~$x$ to each element in the image of~$f$:
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\begin{equation*}
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f \bicomp+ (\_\mapsto x) = \alpha \mapsto f(\alpha) + x.
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\end{equation*}
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The symbol~$\_$ is used to denote a variable that is never referenced.
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For convenience,
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we also define $\bicompi{R}$,
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which recursively handles combinations of function and scalar values.
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This notation is used to simplify definitions of the classification system
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(see \secpref{class})
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when dealing with vectors
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(see \secref{vec}).
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\begin{equation}\label{eq:bicompi}
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\alpha \bicompi{R} \beta =
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\begin{cases}
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\gamma \mapsto \alpha(\gamma) \bicompi{R} \beta(\gamma)
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&\text{if } (\alpha : A\rightarrow B) \logand (\beta : A\rightarrow D),\\
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\gamma \mapsto \alpha(\gamma) \bicompi{R} (\_ \mapsto \beta)
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&\text{if } (\alpha : A\rightarrow B) \logand (\beta \in\Real),\\
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\alpha R \beta &\text{otherwise}.
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\end{cases}
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\end{equation}
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Note that we consider the bracket notation for the image of a function
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$(f:A\rightarrow B)[A]$ to itself be a binary function.
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Given that, we have $f\bicomp{[]} = f\bicomp{[A]}$ for functions returning
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functions (such as vectors of vectors in \secref{vec}),
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noting that $\bicompi{[]}$ is \emph{not} a sensible construction.
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\subsection{Vectors and Index Sets}\seclabel{vec}
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\tame{} supports scalar, vector, and matrix values.
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Unfortunately,
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its implementation history leaves those concepts a bit tortured.
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A vector is a sequence of values, defined as a function of
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an~\gls{index set}.
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\begin{definition}[Vector]\dfnlabel{vec}
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Let $J\subset\Int$ represent an index set.
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A \emph{vector}~$v\in\Vectors^\Real$ is a totally ordered sequence of
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elements represented as a function of an element of its index set:
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\begin{equation}\label{vec}
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v = \Vector{v_0,\ldots,v_j}^{\Real}_{j\in J}
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= j \mapsto v_j : J \rightarrow \Real.
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\end{equation}
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\end{definition}
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This definition means that $v_j = v(j)$,
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making the subscript a notational convenience.
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We may omit the superscript such that $\Vectors^\Real=\Vectors$
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and $\Vector{\ldots}^\Real=\Vector{\ldots}$.
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\begin{definition}[Matrix]\dfnlabel{matrix}
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Let $J\subset\Int$ represent an index set.
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A \emph{matrix}~$M\in\Matrices$ is a totally ordered sequence of
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elements represented as a function of an element of its index set:
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\begin{equation}
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M = \Vector{M_0,\ldots,M_j}^{\Vectors^\Real}_{j\in J}
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= j \mapsto M_j : J \rightarrow \Vectors^\Real.
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\end{equation}
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\end{definition}
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The consequences of \dfnref{matrix}---%
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defining a matrix as a vector of independent vectors---%
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are important.
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This defines a matrix to be more like a multidimensional array,
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with no requirement that the lengths of the vectors be equal.
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\begin{corollary}[Matrix Row Length Variance]\corlabel{matrix-row-len}
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$\vdash \Exists{M\in\Matrices}{\neg\Forall*{j}{\Forall{k}{\len{M_j} = \len{M_k}}}}$.
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\end{corollary}
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\corref{matrix-row-len} can be read ``there exists some matrix~$M$ such that
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not all row lengths of~$M$ are equal''.
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In other words---%
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the inner vectors of a matrix can vary in length.
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Since a vector is a function,
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a vector or matrix can be converted into a set of unique elements like so:
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\begin{alignat*}{2}
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\bigcup\Vector{\Vector{0,1},\Vector{2,2},\Vector{2,0}}\!\bicomp{[]}
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&\mapsto &&\bigcup\Vector{\Vector{0,1}\![],\Vector{2,2}\![],\Vector{2,0}[]}\![] \\
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&\mapsto &&\bigcup\Vector{\Set{0,1},\Set{2},\Set{2,0}}\![] \\
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&\mapsto &&\bigcup\Set{\Set{0,1},\Set{2},\Set{2,0}} \\
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&= &&\Set{0,1,2}.
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\end{alignat*}
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We can also add two vectors, and scale them:
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\begin{align*}
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1 \bicomp{+} \Vector{1,2,3} \bicomp{+} \Vector{4,5,6}
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&= \Vector{1+1,\, 2+1,\, 3+1} \bicomp{+} \Vector{4,5,6} \\
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&= \Vector{2,3,4} \bicomp{+} \Vector{4,5,6} \\
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&= \Vector{2+4,\, 3+5,\, 4+6} \\
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&= \Vector{6, 8, 10}.
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\end{align*}
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\subsection{XML Notation}
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\index{XML}
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The grammar of \tame{} is XML.
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Equivalence relations will be used to map source expressions to an
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underlying mathematical expression.
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For example,
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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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\noindent
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defines that pattern of \xmlnode{match} expression to be materially
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equivalent to~$x=y$---%
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anywhere an equality relation appears,
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you could equivalently replace it with that XML representation without
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changing the meaning of the mathematical expression.
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