2021-04-30 09:16:01 -04:00
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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}
|
2021-05-11 11:33:12 -04:00
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\index{logic!propositional}
|
2021-04-30 09:16:01 -04:00
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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.
|
2021-05-10 16:54:19 -04:00
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\indexsym\infer{infer}
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\index{infer (\ensuremath\infer)}
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The symbol $\infer$ means ``infer''.
|
2021-04-30 09:16:01 -04:00
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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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2021-05-11 09:59:01 -04:00
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\indexsym\logand{conjunction}
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\index{conjunction (\ensuremath{\logand})}
|
2021-04-30 09:16:01 -04:00
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\begin{definition}[Logical Conjunction]
|
2021-05-10 16:54:19 -04:00
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$p,q \infer (p\logand q)$.
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2021-04-30 09:16:01 -04:00
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\end{definition}
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2021-05-11 09:59:01 -04:00
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\indexsym\logor{disjunction}
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\index{disjunction (\ensuremath{\logor})}
|
2021-04-30 09:16:01 -04:00
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\begin{definition}[Logical Disjunction]
|
2021-05-10 16:54:19 -04:00
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$p \infer (p\logor q)$ and $q \infer (p\logor q)$.
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2021-04-30 09:16:01 -04:00
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\end{definition}
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2021-05-11 11:33:12 -04:00
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\indexsym\neg{negation}
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\index{negation (\ensuremath{\neg})}
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\index{law of excluded middle}
|
2021-04-30 09:16:01 -04:00
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\begin{definition}[Law of Excluded Middle]
|
2021-05-10 16:54:19 -04:00
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$\infer (p \logor \neg p)$.
|
2021-04-30 09:16:01 -04:00
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\end{definition}
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|
2021-05-11 11:33:12 -04:00
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\index{law of non-contradiction}
|
2021-04-30 09:16:01 -04:00
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\begin{definition}[Law of Non-Contradiction]
|
2021-05-10 16:54:19 -04:00
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$\infer \neg(p \logand \neg p)$.
|
2021-04-30 09:16:01 -04:00
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\end{definition}
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|
2021-05-11 11:33:12 -04:00
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\index{De Morgan's theorem}
|
2021-04-30 09:16:01 -04:00
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\begin{definition}[De Morgan's Theorem]
|
2021-05-10 16:54:19 -04:00
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$\neg(p \logand q) \infer (\neg p \logor \neg q)$
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and $\neg(p \logor q) \infer (\neg p \logand \neg q)$.
|
2021-04-30 09:16:01 -04:00
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\end{definition}
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|
2021-05-10 16:50:30 -04:00
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\indexsym\equiv{equivalence}
|
2021-04-30 09:16:01 -04:00
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\index{equivalence!material (\ensuremath{\equiv})}
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\begin{definition}[Material Equivalence]
|
2021-05-10 16:54:19 -04:00
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$p\equiv q \infer \big((p \logand q) \logor (\neg p \logand \neg q)\big)$.
|
2021-04-30 09:16:01 -04:00
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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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|
2021-05-11 11:33:12 -04:00
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\indexsym{\!\!\implies\!\!}{implication}
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\index{implication (\ensuremath{\implies})}
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\begin{definition}[Implication]
|
2021-05-10 16:54:19 -04:00
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$p\implies q \infer (\neg p \logor q)$.
|
2021-04-30 09:16:01 -04:00
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\end{definition}
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|
2021-05-11 11:33:12 -04:00
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\indexsym{\true}{boolean, true}
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\indexsym{\false}{boolean, false}
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\index{boolean!FALSE@\tamefalse{} (\false)}%
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\index{boolean!TRUE@\tametrue{} (\true)}%
|
2021-04-30 09:16:01 -04:00
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\begin{definition}[Truth Values]\dfnlabel{truth-values}
|
2021-05-10 16:54:19 -04:00
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$\infer\true$ and $\infer\neg\false$.
|
2021-04-30 09:16:01 -04:00
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\end{definition}
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\subsection{First-Order Logic and Set Theory}
|
2021-05-11 11:33:12 -04:00
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\index{logic!first-order}
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\indexsym\emptyset{set empty}
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\indexsym{\Set{}}{set}
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\index{set!empty (\ensuremath{\emptyset, \{\}})}
|
2021-04-30 09:16:01 -04:00
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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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|
2021-05-11 11:33:12 -04:00
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% TODO: set-builder notation, union, intersection
|
2021-05-10 16:50:30 -04:00
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\indexsym\in{set membership}
|
2021-05-11 11:33:12 -04:00
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\indexsym\union{set, union}
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\indexsym\intersect{set, intersection}
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\index{set!membership@membership (\ensuremath\in)}
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\index{set!union (\ensuremath\union)}
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\index{set!intersection (\ensuremath\intersect)}
|
2021-04-30 09:16:01 -04:00
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\begin{definition}[Set Membership]
|
2021-05-11 11:33:12 -04:00
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$x \in S \equiv \Set{x} \intersect S \not= \emptyset.$
|
2021-04-30 09:16:01 -04:00
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\end{definition}
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$\forall$ denotes first-order universal quantification (``for all''),
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and $\exists$ first-order existential quantification (``there exists''),
|
2021-05-10 14:28:37 -04:00
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over some domain.
|
2021-04-30 09:16:01 -04:00
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|
2021-05-11 11:33:12 -04:00
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\indexsym\exists{quantification, existential}
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\index{quantification!existential (\ensuremath\exists)}
|
2021-04-30 09:16:01 -04:00
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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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|
2021-05-11 11:33:12 -04:00
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\indexsym\forall{quantification, universal}
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\index{quantification!universal (\ensuremath\forall)}
|
2021-04-30 09:16:01 -04:00
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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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|
2021-05-11 11:33:12 -04:00
|
|
|
\index{quantification!vacuous truth}
|
2021-04-30 09:16:01 -04:00
|
|
|
\begin{remark}[Vacuous Truth]
|
2021-05-11 11:33:12 -04:00
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|
By \dfnref{exists}, $\Exists{x\in\emptyset}P \equiv \false$
|
2021-04-30 09:16:01 -04:00
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|
and by \dfnref{forall}, $\Forall{x\in\emptyset}P \equiv \true$.
|
2021-05-10 16:54:19 -04:00
|
|
|
And so we also have the tautologies $\infer \neg\Exists{x\in\emptyset}P$
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|
and $\infer \Forall{x\in\emptyset}P$.
|
2021-04-30 09:16:01 -04:00
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|
|
\end{remark}
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|
2021-05-11 11:33:12 -04:00
|
|
|
\indexsym\Int{integer}
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|
\index{integer (\Int)}%
|
2021-04-30 09:16:01 -04:00
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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
|
2021-05-10 14:28:37 -04:00
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|
integer values $\{0,1\}\in\Int$.
|
2021-04-30 09:16:01 -04:00
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|
\dfnref{bool-int} relates these constants to their
|
2021-05-10 14:28:37 -04:00
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|
boolean counterparts so that they may be used in numeric contexts
|
2021-04-30 09:16:01 -04:00
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and vice-versa.
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\subsection{Functions}
|
2021-05-11 11:33:12 -04:00
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\indexsym{f, g}{function}
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\indexsym\mapsto{function, map}
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|
\indexsym\rightarrow{function, domain map}
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\index{function}
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\index{function!map (\ensuremath\mapsto)}
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|
\index{map|see {function}}
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|
\index{function!domain}
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|
\index{function!codomain}
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|
|
\index{domain|see {function, domain}}
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|
\index{function!domain map (\ensuremath\rightarrow)}
|
2021-04-30 09:16:01 -04:00
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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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|
2021-05-11 11:33:12 -04:00
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|
\indexsym\times{set, Cartesian product}
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\index{set!Cartesian product (\ensuremath\times)}
|
2021-04-30 09:16:01 -04:00
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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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|
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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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|
2021-05-11 11:33:12 -04:00
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|
\indexsym{[\,]}{function, image}
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|
\index{function!image (\ensuremath{[\,]})}
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|
|
\index{function!as a set}
|
2021-04-30 09:16:01 -04:00
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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, \\
|
2021-05-10 16:54:19 -04:00
|
|
|
f : A \rightarrow B &\infer f \subset A\times B, \\
|
2021-04-30 09:16:01 -04:00
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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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|
2021-05-11 11:33:12 -04:00
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|
|
\indexsym{()}{tuple}
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|
\index{tuple (\ensuremath{()})}
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|
\index{relation|see {function}}
|
2021-05-11 13:00:08 -04:00
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|
|
An ordered pair $(x,y)$ is also called a \emph{$2$-tuple}.
|
2021-04-30 09:16:01 -04:00
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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 $t=(x,y)$,
|
2021-05-10 14:08:11 -04:00
|
|
|
then we also have $f(x,y) = ft$,
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|
which we'll sometimes write as a subscript~$f_t$ where disambiguation is
|
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|
|
necessary and where parenthesis may add too much noise;
|
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|
|
this notation is especially well-suited to indexes,
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|
as in $f_1$.
|
2021-04-30 09:16:01 -04:00
|
|
|
Binary functions are often written using \emph{infix} notation;
|
|
|
|
for example, we have $x+y$ rather than $+(x,y)$.
|
|
|
|
|
|
|
|
\begin{equation}
|
2021-05-10 14:08:11 -04:00
|
|
|
f_x = f(x) \in \Set{b \mid (x,b) \in f}
|
2021-04-30 09:16:01 -04:00
|
|
|
\end{equation}
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|
|
\subsubsection{Binary Operations On Functions}
|
2021-05-11 11:33:12 -04:00
|
|
|
\indexsym{R}{relation}
|
2021-04-30 09:16:01 -04:00
|
|
|
Consider two unary functions $f$ and~$g$,
|
|
|
|
and a binary relation~$R$.
|
2021-05-11 11:33:12 -04:00
|
|
|
\indexsym{\bicomp{R}}{function, binary composition}
|
|
|
|
\index{function!binary composition (\ensuremath{\bicomp{R}})}
|
2021-04-30 09:16:01 -04:00
|
|
|
We introduce a notation~$\bicomp R$ to denote the composition of a binary
|
2021-05-11 11:33:12 -04:00
|
|
|
function with two unary functions.
|
2021-04-30 09:16:01 -04:00
|
|
|
|
|
|
|
\begin{align}
|
|
|
|
f &: A \rightarrow B \\
|
|
|
|
g &: A \rightarrow D \\
|
|
|
|
R &: B\times D \rightarrow F \\
|
2021-05-10 14:08:11 -04:00
|
|
|
f \bicomp{R} g &= \alpha \mapsto f_\alpha R g_\alpha : A \rightarrow F
|
2021-04-30 09:16:01 -04:00
|
|
|
\end{align}
|
|
|
|
|
2021-05-11 11:33:12 -04:00
|
|
|
\indexsym\circ{function, composition}
|
|
|
|
\index{function!composition (\ensuremath\circ)}
|
2021-04-30 09:16:01 -04:00
|
|
|
Note that $f$ and~$g$ must share the same domain~$A$.
|
|
|
|
In that sense,
|
|
|
|
this is the mapping of the operation~$R$ over the domain~$A$.
|
|
|
|
This is analogous to unary function composition~$f\circ g$.
|
|
|
|
|
2021-05-11 11:33:12 -04:00
|
|
|
\index{function!constant}
|
2021-04-30 09:16:01 -04:00
|
|
|
A scalar value~$x$ can be mapped onto some function~$f$ using a constant
|
|
|
|
function.
|
|
|
|
For example,
|
|
|
|
consider adding some number~$x$ to each element in the image of~$f$:
|
|
|
|
|
|
|
|
\begin{equation*}
|
2021-05-10 14:08:11 -04:00
|
|
|
f \bicomp+ (\_\mapsto x) = \alpha \mapsto f_\alpha + x.
|
2021-04-30 09:16:01 -04:00
|
|
|
\end{equation*}
|
|
|
|
|
2021-05-11 11:33:12 -04:00
|
|
|
\indexsym{\_}{variable, wildcard}
|
|
|
|
\index{variable!wildcard/hole (\ensuremath{\_})}
|
2021-05-11 12:47:28 -04:00
|
|
|
The symbol~$\_$ is used to denote a variable that matches anything but is
|
|
|
|
never referenced,
|
|
|
|
and is often referred to as a ``wildcard'' (since it matches anything)
|
|
|
|
or a ``hole'' (since its value goes nowhere).
|
2021-04-30 09:16:01 -04:00
|
|
|
|
2021-05-11 11:33:12 -04:00
|
|
|
\indexsym{\bicompi{R}}{function, binary composition, recursive}
|
|
|
|
\index{function!binary composition (\ensuremath{\bicomp{R}})!recursive (\ensuremath{\bicompi{R}})}
|
2021-04-30 09:16:01 -04:00
|
|
|
For convenience,
|
|
|
|
we also define $\bicompi{R}$,
|
|
|
|
which recursively handles combinations of function and scalar values.
|
|
|
|
This notation is used to simplify definitions of the classification system
|
|
|
|
(see \secpref{class})
|
|
|
|
when dealing with vectors
|
|
|
|
(see \secref{vec}).
|
|
|
|
|
|
|
|
\begin{equation}\label{eq:bicompi}
|
|
|
|
\alpha \bicompi{R} \beta =
|
|
|
|
\begin{cases}
|
2021-05-10 14:08:11 -04:00
|
|
|
\gamma \mapsto \alpha_\gamma \bicompi{R} \beta_\gamma
|
2021-04-30 09:16:01 -04:00
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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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2021-05-11 11:39:23 -04:00
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\goodbreak% Fits well on its own page, if we're near a page boundary
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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 index~set.
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2021-05-11 11:33:12 -04:00
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% TODO: font changes in index, making langle unavailable
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%\indexsym{\Vector{}}{vector}
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\index{vector!definition (\ensuremath{\Vector{}})}
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\index{sequence|see vector}
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\indexsym\Vectors{vector}
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\index{real number (\ensuremath\Real)}
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\indexsym\Real{real number}
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\indexsym{\Fam{a}jJ}{index set}
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\index{family|see {index set}}
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\index{index set (\ensuremath{\Fam{a}jJ})}
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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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2021-05-11 11:33:12 -04:00
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\index{vector!matrix}
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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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$\infer \Exists{M\in\Matrices}{\neg\Forall*{j}{\Forall{k}{\len{M_j} = \len{M_k}}}}$.
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2021-04-30 09:16:01 -04:00
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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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2021-05-11 13:49:18 -04:00
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1 \bicompi{+} \Vector{1,2,3} \bicompi{+} \Vector{4,5,6}
|
2021-04-30 09:16:01 -04:00
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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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2021-05-11 11:33:12 -04:00
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\indexsym{\xml{<>}}{XML}
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|
\index{XML!notation (\xml{<>})}
|
2021-04-30 09:16:01 -04:00
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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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|
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|
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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.
|