390 lines
14 KiB
TeX
390 lines
14 KiB
TeX
|
|
\section{Notational Conventions}
|
|
This section provides a fairly terse overview of the foundational
|
|
mathematical concepts used in this paper.
|
|
While we try to reason about \tame{} in terms of algebra,
|
|
first-order logic;
|
|
and set theory;
|
|
notation varies even within those branches.
|
|
To avoid ambiguity,
|
|
especially while introducing our own notation,
|
|
core operators and concepts are explicitly defined below.
|
|
|
|
This section begins its numbering at~0.
|
|
This is not only a hint that \tame{} (and this paper) use 0-indexing,
|
|
but also because equations; definitions; theorems; corollaries; and the
|
|
like are all numbered relative to their section.
|
|
When you see any of these prefixed with ``0.'',
|
|
this sets those references aside as foundational mathematical concepts
|
|
that are not part of the theory and operation of \tame{} itself.
|
|
|
|
|
|
\subsection{Propositional Logic}
|
|
\index{logic!propositional}
|
|
We reproduce here certain axioms and corollaries of propositional logic for
|
|
convenience and to clarify our interpretation of certain concepts.
|
|
The use of the symbols $\logand$, $\logor$, and~$\neg$ are standard.
|
|
\indexsym\infer{infer}
|
|
\index{infer (\ensuremath\infer)}
|
|
The symbol $\infer$ means ``infer''.
|
|
We use $\implies$ in place of $\rightarrow$ for implication,
|
|
since the latter is used to denote the mapping of a domain to a codomain
|
|
in reference to functions.
|
|
We further use $\equiv$ in place of $\leftrightarrow$ to represent material
|
|
equivalence.
|
|
|
|
\indexsym\logand{conjunction}
|
|
\index{conjunction (\ensuremath{\logand})}
|
|
\begin{definition}[Logical Conjunction]
|
|
$p,q \infer (p\logand q)$.
|
|
\end{definition}
|
|
|
|
\indexsym\logor{disjunction}
|
|
\index{disjunction (\ensuremath{\logor})}
|
|
\begin{definition}[Logical Disjunction]
|
|
$p \infer (p\logor q)$ and $q \infer (p\logor q)$.
|
|
\end{definition}
|
|
|
|
\indexsym\neg{negation}
|
|
\index{negation (\ensuremath{\neg})}
|
|
\index{law of excluded middle}
|
|
\begin{definition}[Law of Excluded Middle]
|
|
$\infer (p \logor \neg p)$.
|
|
\end{definition}
|
|
|
|
\index{law of non-contradiction}
|
|
\begin{definition}[Law of Non-Contradiction]
|
|
$\infer \neg(p \logand \neg p)$.
|
|
\end{definition}
|
|
|
|
\index{De Morgan's theorem}
|
|
\begin{definition}[De Morgan's Theorem]
|
|
$\neg(p \logand q) \infer (\neg p \logor \neg q)$
|
|
and $\neg(p \logor q) \infer (\neg p \logand \neg q)$.
|
|
\end{definition}
|
|
|
|
\indexsym\equiv{equivalence}
|
|
\index{equivalence!material (\ensuremath{\equiv})}
|
|
\begin{definition}[Material Equivalence]
|
|
$p\equiv q \infer \big((p \logand q) \logor (\neg p \logand \neg q)\big)$.
|
|
\end{definition}
|
|
|
|
$\equiv$ denotes a logical identity.
|
|
Consequently,
|
|
it'll often be used as a definition operator.
|
|
|
|
\indexsym{\!\!\implies\!\!}{implication}
|
|
\index{implication (\ensuremath{\implies})}
|
|
\begin{definition}[Implication]
|
|
$p\implies q \infer (\neg p \logor q)$.
|
|
\end{definition}
|
|
|
|
\indexsym{\true}{boolean, true}
|
|
\indexsym{\false}{boolean, false}
|
|
\index{boolean!FALSE@\tamefalse{} (\false)}%
|
|
\index{boolean!TRUE@\tametrue{} (\true)}%
|
|
\begin{definition}[Truth Values]\dfnlabel{truth-values}
|
|
$\infer\true$ and $\infer\neg\false$.
|
|
\end{definition}
|
|
|
|
|
|
\subsection{First-Order Logic and Set Theory}
|
|
\index{logic!first-order}
|
|
\indexsym\emptyset{set empty}
|
|
\indexsym{\Set{}}{set}
|
|
\index{set!empty (\ensuremath{\emptyset, \{\}})}
|
|
The symbol $\emptyset$ represents the empty set---%
|
|
the set of zero elements.
|
|
We assume that the axioms of ZFC~set theory hold,
|
|
but define $\in$ here for clarity.
|
|
|
|
\todo{Introduce set-builder notation, $\union$, $\intersect$.}
|
|
\indexsym\in{set membership}
|
|
\indexsym\union{set, union}
|
|
\indexsym\intersect{set, intersection}
|
|
\index{set!membership@membership (\ensuremath\in)}
|
|
\index{set!union (\ensuremath\union)}
|
|
\index{set!intersection (\ensuremath\intersect)}
|
|
\begin{definition}[Set Membership]
|
|
$x \in S \equiv \Set{x} \intersect S \not= \emptyset.$
|
|
\end{definition}
|
|
|
|
$\forall$ denotes first-order universal quantification (``for all''),
|
|
and $\exists$ first-order existential quantification (``there exists''),
|
|
over some domain.
|
|
|
|
\indexsym\exists{quantification, existential}
|
|
\index{quantification!existential (\ensuremath\exists)}
|
|
\begin{definition}[Existential Quantification]\dfnlabel{exists}
|
|
$\Exists{x\in X}{P(x)} \equiv
|
|
\true \in \Set{P(x) \mid x\in X}$.
|
|
\end{definition}
|
|
|
|
\indexsym\forall{quantification, universal}
|
|
\index{quantification!universal (\ensuremath\forall)}
|
|
\begin{definition}[Universal Quantification]\dfnlabel{forall}
|
|
$\Forall{x\in X}{P(x)} \equiv \neg\Exists{x\in X}{\neg P(x)}$.
|
|
\end{definition}
|
|
|
|
\index{quantification!vacuous truth}
|
|
\begin{remark}[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 $\infer \neg\Exists{x\in\emptyset}P$
|
|
and $\infer \Forall{x\in\emptyset}P$.
|
|
\end{remark}
|
|
|
|
\indexsym\Int{integer}
|
|
\index{integer (\Int)}%
|
|
\begin{definition}[Boolean/Integer Equivalency]\dfnlabel{bool-int}
|
|
$\Set{0,1}\in\Int, \false \equiv 0$ and $\true \equiv 1$.
|
|
\end{definition}
|
|
|
|
\tamefalse{} and~\tametrue{} are constants in \tame{} mapping to the
|
|
integer values $\{0,1\}\in\Int$.
|
|
\dfnref{bool-int} relates these constants to their
|
|
boolean counterparts so that they may be used in numeric contexts
|
|
and vice-versa.
|
|
|
|
|
|
\subsection{Functions}
|
|
\indexsym{f, g}{function}
|
|
\indexsym\mapsto{function, map}
|
|
\indexsym\rightarrow{function, domain map}
|
|
\index{function}
|
|
\index{function!map (\ensuremath\mapsto)}
|
|
\index{map|see {function}}
|
|
\index{function!domain}
|
|
\index{function!codomain}
|
|
\index{domain|see {function, domain}}
|
|
\index{function!domain map (\ensuremath\rightarrow)}
|
|
The notation $f = x \mapsto x' : A\rightarrow B$ represents a function~$f$
|
|
that maps from~$x$ to~$x'$,
|
|
where $x\in A$ (the domain of~$f$) and $x'\in B$ (the co-domain of~$f$).
|
|
|
|
\indexsym\times{set, Cartesian product}
|
|
\index{set!Cartesian product (\ensuremath\times)}
|
|
A function $A\rightarrow B$ can be represented as the Cartesian
|
|
product of its domain and codomain, $A\times B$.
|
|
For example,
|
|
$x\mapsto x^2 : \Int\rightarrow\Int$ is represented by the set of ordered
|
|
pairs $\Set{(x,x^2) \mid x\in\Int}$, which looks something like
|
|
|
|
\begin{equation*}
|
|
\Set{\ldots,\,(0,0),\,(1,1),\,(2,4),\,(3,9),\,\ldots}.
|
|
\end{equation*}
|
|
|
|
\indexsym{[\,]}{function, image}
|
|
\index{function!image (\ensuremath{[\,]})}
|
|
\index{function!as a set}
|
|
The set of values over which some function~$f$ ranges is its \emph{image},
|
|
which is a subset of its codomain.
|
|
In the example above,
|
|
both the domain and codomain are the set of integers~$\Int$,
|
|
but the image is $\Set{x^2 \mid x\in\Int}$,
|
|
which is clearly a subset of~$\Int$.
|
|
|
|
We therefore have
|
|
|
|
\begin{align}
|
|
A \rightarrow B &\subset A\times B, \\
|
|
f : A \rightarrow B &\infer f \subset A\times B, \\
|
|
f = \alpha \mapsto \alpha' : A \rightarrow B
|
|
&= \Set{(\alpha,\alpha')
|
|
\mid \alpha\in A \logand \alpha'\in B}, \\
|
|
f[D\subseteq A] &= \Set{f(\alpha) \mid \alpha\in D} \subset B, \\
|
|
f[] &= f[A].
|
|
\end{align}
|
|
|
|
\indexsym{()}{tuple}
|
|
\index{tuple (\ensuremath{()})}
|
|
\index{relation|see {function}}
|
|
An ordered pair $(x,y)$ is also called a \emph{$2$-tuple}.
|
|
Generally,
|
|
an \emph{$n$-tuple} is used to represent an $n$-ary function,
|
|
where by convention we have $(x)=x$.
|
|
So $f(x,y) = f((x,y)) = x+y$.
|
|
If we let $t=(x,y)$,
|
|
then we also have $f(x,y) = ft$,
|
|
which we'll sometimes write as a subscript~$f_t$ where disambiguation is
|
|
necessary and where parenthesis may add too much noise;
|
|
this notation is especially well-suited to indexes,
|
|
as in $f_1$.
|
|
Binary functions are often written using \emph{infix} notation;
|
|
for example, we have $x+y$ rather than $+(x,y)$.
|
|
|
|
\begin{equation}
|
|
f_x = f(x) \in \Set{b \mid (x,b) \in f}
|
|
\end{equation}
|
|
|
|
|
|
\subsubsection{Binary Operations On Functions}
|
|
\indexsym{R}{relation}
|
|
Consider two unary functions $f$ and~$g$,
|
|
and a binary relation~$R$.
|
|
\indexsym{\bicomp{R}}{function, binary composition}
|
|
\index{function!binary composition (\ensuremath{\bicomp{R}})}
|
|
We introduce a notation~$\bicomp R$ to denote the composition of a binary
|
|
function with two unary functions.
|
|
|
|
\begin{align}
|
|
f &: A \rightarrow B \\
|
|
g &: A \rightarrow D \\
|
|
R &: B\times D \rightarrow F \\
|
|
f \bicomp{R} g &= \alpha \mapsto f_\alpha R g_\alpha : A \rightarrow F
|
|
\end{align}
|
|
|
|
\indexsym\circ{function, composition}
|
|
\index{function!composition (\ensuremath\circ)}
|
|
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~$f\circ g$.
|
|
|
|
\index{function!constant}
|
|
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*}
|
|
f \bicomp+ (\_\mapsto x) = \alpha \mapsto f_\alpha + x.
|
|
\end{equation*}
|
|
|
|
\indexsym{\_}{variable, wildcard}
|
|
\index{variable!wildcard/hole (\ensuremath{\_})}
|
|
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).
|
|
|
|
\indexsym{\bicompi{R}}{function, binary composition, recursive}
|
|
\index{function!binary composition (\ensuremath{\bicomp{R}})!recursive (\ensuremath{\bicompi{R}})}
|
|
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}).
|
|
|
|
\mremark{$\bicompi{R}$ may be removed depending on how $\Classify$ is defined.}
|
|
\begin{equation}\label{eq:bicompi}
|
|
\alpha \bicompi{R} \beta =
|
|
\begin{cases}
|
|
\gamma \mapsto \alpha_\gamma \bicompi{R} \beta_\gamma
|
|
&\text{if } (\alpha : A\rightarrow B) \logand (\beta : A\rightarrow D),\\
|
|
\gamma \mapsto \alpha_\gamma \bicompi{R} (\_ \mapsto \beta)
|
|
&\text{if } (\alpha : A\rightarrow B) \logand (\beta \in\Real),\\
|
|
\alpha R \beta &\text{otherwise}.
|
|
\end{cases}
|
|
\end{equation}
|
|
|
|
Note that we consider the bracket notation for the image of a function
|
|
$(f:A\rightarrow B)[A]$ to itself be a binary function.
|
|
Given that, we have $f\bicomp{[]} = f\bicomp{[A]}$ for functions returning
|
|
functions (such as vectors of vectors in \secref{vec}),
|
|
noting that $\bicompi{[]}$ is \emph{not} a sensible construction.
|
|
|
|
|
|
\goodbreak% Fits well on its own page, if we're near a page boundary
|
|
\subsection{Vectors and Index Sets}\seclabel{vec}
|
|
\tame{} supports scalar, vector, and matrix values.
|
|
Unfortunately,
|
|
its implementation history leaves those concepts a bit tortured.
|
|
|
|
A vector is a sequence of values, defined as a function of
|
|
an index~set.
|
|
|
|
% TODO: font changes in index, making langle unavailable
|
|
%\indexsym{\Vector{}}{vector}
|
|
\index{vector!definition (\ensuremath{\Vector{}})}
|
|
\index{sequence|see vector}
|
|
\indexsym\Vectors{vector}
|
|
\index{real number (\ensuremath\Real)}
|
|
\indexsym\Real{real number}
|
|
\indexsym{\Fam{a}jJ}{index set}
|
|
\index{family|see {index set}}
|
|
\index{index set (\ensuremath{\Fam{a}jJ})}
|
|
\begin{definition}[Vector]\dfnlabel{vec}
|
|
Let $J\subset\Int$ represent an index set.
|
|
A \emph{vector}~$v\in\Vectors^\Real$ is a totally ordered sequence of
|
|
elements represented as a function of an element of its index set:
|
|
\begin{equation}\label{vec}
|
|
v = \Vector{v_0,\ldots,v_j}^{\Real}_{j\in J}
|
|
= j \mapsto v_j : J \rightarrow \Real.
|
|
\end{equation}
|
|
\end{definition}
|
|
|
|
This definition means that $v_j = v(j)$,
|
|
making the subscript a notational convenience.
|
|
We may omit the superscript such that $\Vectors^\Real=\Vectors$
|
|
and $\Vector{\ldots}^\Real=\Vector{\ldots}$.
|
|
|
|
\index{vector!matrix}
|
|
\begin{definition}[Matrix]\dfnlabel{matrix}
|
|
Let $J\subset\Int$ represent an index set.
|
|
A \emph{matrix}~$M\in\Matrices$ is a totally ordered sequence of
|
|
elements represented as a function of an element of its index set:
|
|
\begin{equation}
|
|
M = \Vector{M_0,\ldots,M_j}^{\Vectors^\Real}_{j\in J}
|
|
= j \mapsto M_j : J \rightarrow \Vectors^\Real.
|
|
\end{equation}
|
|
\end{definition}
|
|
|
|
The consequences of \dfnref{matrix}---%
|
|
defining a matrix as a vector of independent vectors---%
|
|
are important.
|
|
This defines a matrix to be more like a multidimensional array,
|
|
with no requirement that the lengths of the vectors be equal.
|
|
|
|
\begin{corollary}[Matrix Row Length Variance]\corlabel{matrix-row-len}
|
|
$\infer \Exists{M\in\Matrices}{\neg\Forall*{j}{\Forall{k}{\len{M_j} = \len{M_k}}}}$.
|
|
\end{corollary}
|
|
|
|
\corref{matrix-row-len} can be read ``there exists some matrix~$M$ such that
|
|
not all row lengths of~$M$ are equal''.
|
|
In other words---%
|
|
the inner vectors of a matrix can vary in length.
|
|
|
|
Since a vector is a function,
|
|
a vector or matrix can be converted into a set of unique elements like so:
|
|
|
|
\begin{alignat*}{2}
|
|
\bigcup\Vector{\Vector{0,1},\Vector{2,2},\Vector{2,0}}\!\bicomp{[]}
|
|
&\mapsto &&\bigcup\Vector{\Vector{0,1}\![],\Vector{2,2}\![],\Vector{2,0}[]}\![] \\
|
|
&\mapsto &&\bigcup\Vector{\Set{0,1},\Set{2},\Set{2,0}}\![] \\
|
|
&\mapsto &&\bigcup\Set{\Set{0,1},\Set{2},\Set{2,0}} \\
|
|
&= &&\Set{0,1,2}.
|
|
\end{alignat*}
|
|
|
|
We can also add two vectors, and scale them:
|
|
|
|
\begin{align*}
|
|
1 \bicompi{+} \Vector{1,2,3} \bicompi{+} \Vector{4,5,6}
|
|
&= \Vector{1+1,\, 2+1,\, 3+1} \bicomp{+} \Vector{4,5,6} \\
|
|
&= \Vector{2,3,4} \bicomp{+} \Vector{4,5,6} \\
|
|
&= \Vector{2+4,\, 3+5,\, 4+6} \\
|
|
&= \Vector{6, 8, 10}.
|
|
\end{align*}
|
|
|
|
|
|
\subsection{XML Notation}
|
|
\indexsym{\xml{<>}}{XML}
|
|
\index{XML!notation (\xml{<>})}
|
|
The grammar of \tame{} is XML.
|
|
Equivalence relations will be used to map source expressions to an
|
|
underlying mathematical expression.
|
|
For example,
|
|
|
|
\begin{equation*}
|
|
\xml{<match on="$x$" value="$y$" />} \equiv x = y
|
|
\end{equation*}
|
|
|
|
\noindent
|
|
defines that pattern of \xmlnode{match} expression to be materially
|
|
equivalent to~$x=y$---%
|
|
anywhere an equality relation appears,
|
|
you could equivalently replace it with that XML representation without
|
|
changing the meaning of the mathematical expression.
|