Introduce \alphabet

diff --git a/main.tex b/main.tex
index 1d7b3fe2b315c9dd2e77e037f73bda2b15ebbf19..810afb2d2f12d0dee7c10c85f12159fd6db2af80 100644 (file)
--- 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 f527c092f00f49f3ffac6e370fd337e926f3a7d0..1b313f1f556d0eca61ccfd133f76494b05a4933e 100644 (file)
--- 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 d8dfb20be3b644a6e81e2aded417ee9aee6ac3d4..c9bb8d30a26d8af825cb0c5798de74572370dd4f 100644 (file)
--- 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.