# HG changeset patch
# User Eugen Sawin <sawine@me73.com>
# Date 1310753254 -7200
# Node ID 82c3364b42b59b1614d5b62bd165d10eb82a7d2e
# Parent  8e48b5f173ac151d3f663636b28e20108aa339b2
Automaton example.

diff -r 8e48b5f173ac -r 82c3364b42b5 paper/src/paper.tex
--- a/paper/src/paper.tex	Fri Jul 15 19:21:26 2011 +0200
+++ b/paper/src/paper.tex	Fri Jul 15 20:07:34 2011 +0200
@@ -8,6 +8,8 @@
 \usepackage{multicol}
 \usepackage{tikz}
 \usetikzlibrary{automata}
+\usepackage{sidecap}
+\usepackage{wrapfig}
 %\usepackage{fullpage}
 %\usepackage{a4wide}
 \usepackage[left=3.9cm, right=3.9cm]{geometry}
@@ -184,8 +186,13 @@
 Given all legal computations of an automaton, we have defined the acceptance condition. A computation is accepting, if it passes through an accepting state infinitely times. Since the set of states $S$ is finite, there must be a state $s \in S$, which occurs infinitely often within an infinite run; but it is not necessary, that $s$ is an accepting state; notice that $F$ can be an empty set.
 
 \subsection{Example}
-Let $\A_1 = (\Sigma, S, S_0, \Delta, F)$ be an automaton with $\Sigma = \{a, b\}$, $S = \{q_0, q_1, q_2\},$ $S_0 = \{q_0\}$ and $\Delta = \{(q_0, a, q_1)\}, F = \{q_2\}$.
-
+\label{ex:automaton}
+%\begin{figure}[h]
+\begin{wrapfigure}{r}{0.5\textwidth}
+\label{fig:automaton}
+\vspace{-22pt}
+%\centering
+\begin{center}
 \begin{tikzpicture}[shorten >=0pt, node distance=2cm, auto, semithick, >=stealth
     %every state/.style={fill, draw=none, gray, text=white},
     ,accepting/.style={fill, gray!50!black, text=white}
@@ -194,19 +201,36 @@
 %\draw[help lines] (0,0) grid (3,2);
 \node[state,initial, initial text=] (q_0) {$q_0$};
 \node[state] (q_1) [above right of= q_0] {$q_1$};
-%\node[state] (q_2) [below right of= q_0] {$q_2$};
 \node[state,accepting](q_2) [below right of= q_1] {$q_2$};
 \path[->] 
 (q_0) edge node {$a$} (q_1)
   edge [loop above] node {$b$} ()
 (q_1) edge node {$b$} (q_2)
   edge [loop above] node {$a$} ()
-%(q_2) edge node [swap] {$a$} (q_1)
-%  edge [loop below] node {$b$} ()
 (q_2) %edge node {$a$} (q_1)
   edge node {$b$} (q_0);
 \draw[->] (q_2) .. controls +(up:1cm) and +(right:1cm) .. node[above] {$a$} (q_1);
 \end{tikzpicture}
+\end{center}
+\caption{B\"uchi automaton $\A_1$ (\ref{ex:automaton} Example)}
+\vspace{-42pt}
+%\end{figure}
+\end{wrapfigure}
+
+Let $\A_1 = (\Sigma, S, S_0, \Delta, F)$ be an automaton with:
+\begin{itemize}
+\item $\Sigma = \{a, b\}$
+\item $S = \{q_0, q_1, q_2\}$
+\item $S_0 = \{q_0\}$
+\item $\Delta = \{(q_0, a, q_0), (q_0, b, q_0),$\\
+\hspace*{0.9cm}$(q_1, a, q_1), (q_1, b, q_2),$\\
+\hspace*{0.9cm}$(q_2, a, q_1), (q_2, b, q_0)\}$
+\item $F = \{q_2\}$
+\end{itemize}
+%
+Figure \ref{fig:automaton} shows $\A_1$, the initial state is marked with an unlabelled incoming arrow and the only accepting state $q_2$ is a filled circle.
+
+By some further investigation of the automaton, we identify the recognised language $L_\omega(\A_1)$ of the automaton, it is the set of all infinite words with infinitely many occurrences of the sequence $ab$.
 
 \subsection{Generalised B\"uchi automata}
 In the following two sections, we present the modal logic, which is used for the specification of system properties and automata construction on such specifications. To provide a more convenient link between linear temporal logic and B\"uchi automata, we introduce a slightly different formalisation of automata with an extended acceptance condition.