From: Nathaniel Wesley Filardo Date: Tue, 2 Feb 2016 22:39:10 +0000 (-0500) Subject: Introduce \alphabet X-Git-Url: https://hydra-www.ietfng.org/gitweb/?a=commitdiff_plain;h=1d4f6cf059934dbd3110c27d2f4df440af49a0c9;p=autzoo Introduce \alphabet --- diff --git a/main.tex b/main.tex index 1d7b3fe..810afb2 100644 --- a/main.tex +++ b/main.tex @@ -220,6 +220,7 @@ or \url{http://www.gnu.org/licenses/agpl-3.0.txt}) or any later version.}} \newcommand{\apriori}{\latin{a priori}\xspace} \newcommand{\perse}{\latin{per se}\xspace} +\newcommand{\alphabet}{{\ensuremath\mathcal{A}}} \newcommand{\defn}[1]{{\boldmath\bfseries #1}} \newcommand{\config}{{\ensuremath\mathcal{C}}} \newcommand{\alang}[1]{\mathcal{L}\paren{#1}} diff --git a/strlang.tex b/strlang.tex index f527c09..1b313f1 100644 --- a/strlang.tex +++ b/strlang.tex @@ -1,14 +1,14 @@ A string language $L$ is a set of finite strings over some alphabet -$\Sigma$.% +$\alphabet$.% % \footnote{More formally, it is a subset of the underlying set of the free -monoid of $\Sigma$. Following the literature, we tend to suppress writing -the forgetful functor, and simply refer to the set as $\Sigma^*$.} +monoid of $\alphabet$. Following the literature, we tend to suppress writing +the forgetful functor, and simply refer to the set as $\alphabet^*$.} % -For example, if $\Sigma = \set{a,b}$, then ``the string $a$'', ``the set of +For example, if $\alphabet = \set{a,b}$, then ``the string $a$'', ``the set of strings with an even number of $a$s and an odd number of $b$s (and no other characters)'', and ``all strings composed of $x$ many $a$s, $y$ many $b$s, -then $x*y$ more $a$s'' are all languages over $\Sigma$. +then $x*y$ more $a$s'' are all languages over $\alphabet$. We may define string languages in terms of other string languages by \begin{enumerate} @@ -20,9 +20,9 @@ We may define string languages in terms of other string languages by $L_1 \cup L_2 \defeq \set{ s \middle\vert s \in L_1 ~\vee~ s \in L_2 }$ \item Relative complement: $L_1 \setminus L_2 \defeq \set{s \middle\vert s \in L_1 ~\wedge~ s \not\in L_2 }$ - \item (General) complement, $L^C$: the set of strings over $\Sigma$ + \item (General) complement, $L^C$: the set of strings over $\alphabet$ which are not in $L$. We might write this as - $F_\text{mon}\Sigma \setminus L$ or $\Sigma^* \setminus L$ + $F_\text{mon}\alphabet \setminus L$ or $\alphabet^* \setminus L$ (see below). \end{enumerate} @@ -50,6 +50,6 @@ in the language'' ($\epsilon \in L$) and ``the language is empty'' ($L = \item Miscellaneous operations: \begin{enumerate} \item Concatenation: $L_1 \cdot L_2 \defeq \set{ s \cdot t \middle\vert s \in L_1 ~\wedge~ t \in L_2}$ - \item Reversal: $L^R \defeq \set{ c_1 \ldots c_n \middle\vert c_i \in \Sigma ~\wedge~ c_n \ldots c_1 \in L }$ + \item Reversal: $L^R \defeq \set{ c_1 \ldots c_n \middle\vert c_i \in \alphabet ~\wedge~ c_n \ldots c_1 \in L }$ \end{enumerate} \end{enumerate} diff --git a/treelang.tex b/treelang.tex index d8dfb20..c9bb8d3 100644 --- a/treelang.tex +++ b/treelang.tex @@ -5,20 +5,20 @@ a new set of operations. For an excellent and thourough introduction, see \cite[Preliminaries]{tata}. We limit ourselves here to a quick summary. A {\em ranked} tree language $L$ is a set of finite trees over some -\defn{signature} (also \defn{ranked alphabet}) $\Sigma$, with arity function -$\mbox{ar} : \Sigma \to \mathbb{N}$. Every node of a tree labeled with -$\sigma \in \Sigma$ has exactly $\mbox{ar}\paren{\sigma}$-many children.% +\defn{signature} (also \defn{ranked alphabet}) $\alphabet$, with arity function +$\mbox{ar} : \alphabet \to \mathbb{N}$. Every node of a tree labeled with +$\sigma \in \alphabet$ has exactly $\mbox{ar}\paren{\sigma}$-many children.% % \footnote{More formally, a ranked tree language is a subset of the carrier -of the free algebra over $\paren{\Sigma,\mbox{ar}}$.} We use -$\mathcal{T}(\Sigma,\mbox{ar})$ for such a set of ranked trees; often -$\mbox{ar}$ will be implicit and we will just write $\mathcal{T}(\Sigma)$. -We use the notation $\mathcal{T}(\Sigma \sqcup X)$ to mean the set of trees -whose labels come either from $\Sigma$ (with the appropriate arity) or a +of the free algebra over $\paren{\alphabet,\mbox{ar}}$.} We use +$\mathcal{T}(\alphabet,\mbox{ar})$ for such a set of ranked trees; often +$\mbox{ar}$ will be implicit and we will just write $\mathcal{T}(\alphabet)$. +We use the notation $\mathcal{T}(\alphabet \sqcup X)$ to mean the set of trees +whose labels come either from $\alphabet$ (with the appropriate arity) or a (disjoint) set of ``variables'' $X$, with $\mbox{ar}(x \in X) \defeq 0$. -The notation $\mathcal{T}(\Sigma \times \mathcal{Q})$ will be used for trees -whose labels are pairs of elements from $\Sigma$ and $\mathcal{Q}$; we -define $\mbox{ar}(\sigma \times q \in \Sigma \times \mathcal{Q}) \defeq +The notation $\mathcal{T}(\alphabet \times \mathcal{Q})$ will be used for trees +whose labels are pairs of elements from $\alphabet$ and $\mathcal{Q}$; we +define $\mbox{ar}(\sigma \times q \in \alphabet \times \mathcal{Q}) \defeq \mbox{ar}(\sigma)$. The set-theoretic operations carry over as might be expected.