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+++ b/slides/slides02.tex Sat Jun 15 09:23:18 2013 -0400
@@ -0,0 +1,494 @@
+\documentclass[dvipsnames,14pt,t]{beamer}
+\usepackage{beamerthemeplainculight}
+\usepackage[T1]{fontenc}
+\usepackage[latin1]{inputenc}
+\usepackage{mathpartir}
+\usepackage[absolute,overlay]{textpos}
+\usepackage{ifthen}
+\usepackage{tikz}
+\usepackage{pgf}
+\usepackage{calc}
+\usepackage{ulem}
+\usepackage{courier}
+\usepackage{listings}
+\renewcommand{\uline}[1]{#1}
+\usetikzlibrary{arrows}
+\usetikzlibrary{automata}
+\usetikzlibrary{shapes}
+\usetikzlibrary{shadows}
+\usetikzlibrary{positioning}
+\usetikzlibrary{calc}
+\usepackage{graphicx}
+
+\definecolor{javared}{rgb}{0.6,0,0} % for strings
+\definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments
+\definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords
+\definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc
+
+\lstset{language=Java,
+ basicstyle=\ttfamily,
+ keywordstyle=\color{javapurple}\bfseries,
+ stringstyle=\color{javagreen},
+ commentstyle=\color{javagreen},
+ morecomment=[s][\color{javadocblue}]{/**}{*/},
+ numbers=left,
+ numberstyle=\tiny\color{black},
+ stepnumber=1,
+ numbersep=10pt,
+ tabsize=2,
+ showspaces=false,
+ showstringspaces=false}
+
+\lstdefinelanguage{scala}{
+ morekeywords={abstract,case,catch,class,def,%
+ do,else,extends,false,final,finally,%
+ for,if,implicit,import,match,mixin,%
+ new,null,object,override,package,%
+ private,protected,requires,return,sealed,%
+ super,this,throw,trait,true,try,%
+ type,val,var,while,with,yield},
+ otherkeywords={=>,<-,<\%,<:,>:,\#,@},
+ sensitive=true,
+ morecomment=[l]{//},
+ morecomment=[n]{/*}{*/},
+ morestring=[b]",
+ morestring=[b]',
+ morestring=[b]"""
+}
+
+\lstset{language=Scala,
+ basicstyle=\ttfamily,
+ keywordstyle=\color{javapurple}\bfseries,
+ stringstyle=\color{javagreen},
+ commentstyle=\color{javagreen},
+ morecomment=[s][\color{javadocblue}]{/**}{*/},
+ numbers=left,
+ numberstyle=\tiny\color{black},
+ stepnumber=1,
+ numbersep=10pt,
+ tabsize=2,
+ showspaces=false,
+ showstringspaces=false}
+
+% beamer stuff
+\renewcommand{\slidecaption}{AFL 02, King's College London, 3.~October 2012}
+\newcommand{\bl}[1]{\textcolor{blue}{#1}}
+\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
+
+\begin{document}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}<1>[t]
+\frametitle{%
+ \begin{tabular}{@ {}c@ {}}
+ \\[-3mm]
+ \LARGE Automata and \\[-2mm]
+ \LARGE Formal Languages (2)\\[3mm]
+ \end{tabular}}
+
+ %\begin{center}
+ %\includegraphics[scale=0.3]{pics/ante1.jpg}\hspace{5mm}
+ %\includegraphics[scale=0.31]{pics/ante2.jpg}\\
+ %\footnotesize\textcolor{gray}{Antikythera automaton, 100 BC (Archimedes?)}
+ %\end{center}
+
+\normalsize
+ \begin{center}
+ \begin{tabular}{ll}
+ Email: & christian.urban at kcl.ac.uk\\
+ Of$\!$fice: & S1.27 (1st floor Strand Building)\\
+ Slides: & KEATS
+ \end{tabular}
+ \end{center}
+
+
+\end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Languages\end{tabular}}
+
+A \alert{language} is a set of strings.\bigskip
+
+A \alert{regular expression} specifies a set of strings or language.
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}
+
+Their inductive definition:
+
+
+\begin{textblock}{6}(2,5)
+ \begin{tabular}{@ {}rrl@ {\hspace{13mm}}l}
+ \bl{r} & \bl{$::=$} & \bl{$\varnothing$} & null\\
+ & \bl{$\mid$} & \bl{$\epsilon$} & empty string / "" / []\\
+ & \bl{$\mid$} & \bl{c} & character\\
+ & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$} & sequence\\
+ & \bl{$\mid$} & \bl{r$_1$ + r$_2$} & alternative / choice\\
+ & \bl{$\mid$} & \bl{r$^*$} & star (zero or more)\\
+ \end{tabular}
+ \end{textblock}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}
+
+Their implementation in Scala:
+
+{\lstset{language=Scala}\fontsize{8}{10}\selectfont
+\texttt{\lstinputlisting{app51.scala}}}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}The Meaning of a\\[-2mm] Regular Expression\end{tabular}}
+
+\begin{textblock}{15}(1,4)
+ \begin{tabular}{@ {}rcl}
+ \bl{$L$($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$}\\
+ \bl{$L$($\epsilon$)} & \bl{$\dn$} & \bl{$\{$""$\}$}\\
+ \bl{$L$(c)} & \bl{$\dn$} & \bl{$\{$"c"$\}$}\\
+ \bl{$L$(r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{$L$(r$_1$) $\cup$ $L$(r$_2$)}\\
+ \bl{$L$(r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{$L$(r$_1$) @ $L$(r$_2$)}\\
+ \bl{$L$(r$^*$)} & \bl{$\dn$} & \bl{$\bigcup_{n \ge 0}$ $L$(r)$^n$}\\
+ \end{tabular}\bigskip
+
+\hspace{5mm}\textcolor{gray}{$L$(r)$^0$ $\;\dn\;$ $\{$""$\}$}\\
+\textcolor{gray}{$L$(r)$^{n+1}$ $\;\dn\;$ $L$(r) @ $L$(r)$^n$}
+\end{textblock}
+
+\only<2->{
+\begin{textblock}{5}(11,5)
+\textcolor{gray}{\small
+A @ B\\
+\ldots you take out every string from A and
+concatenate it with every string in B
+}
+\end{textblock}}
+
+\only<3->{
+\begin{textblock}{6}(9,12)\small
+\bl{$L$} is a function from regular expressions to sets of strings\\
+\bl{$L$ : Rexp $\Rightarrow$ Set[String]}
+\end{textblock}}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\large
+\begin{center}
+What is \bl{$L$(a$^*$)}?
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+\newcommand{\YES}{\textcolor{gray}{yes}}
+\newcommand{\NO}{\textcolor{gray}{no}}
+\newcommand{\FORALLR}{\textcolor{gray}{$\forall$ r.}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Reg Exp Equivalences\end{tabular}}
+
+\begin{center}
+\begin{tabular}{l@ {\hspace{7mm}}rcl@ {\hspace{7mm}}l}
+&\bl{(a + b) + c} & \bl{$\equiv^?$} & \bl{a + (b + c)} & \onslide<2->{\YES}\\
+&\bl{a + a} & \bl{$\equiv^?$} & \bl{a} & \onslide<3->{\YES}\\
+&\bl{(a $\cdot$ b) $\cdot$ c} & \bl{$\equiv^?$} & \bl{a $\cdot$ (b $\cdot$ c)} & \onslide<4->{\YES}\\
+&\bl{a $\cdot$ a} & \bl{$\equiv^?$} & \bl{a} & \onslide<5->{\NO}\\
+&\bl{$\epsilon^*$} & \bl{$\equiv^?$} & \bl{$\epsilon$} & \onslide<6->{\YES}\\
+&\bl{$\varnothing^*$} & \bl{$\equiv^?$} & \bl{$\varnothing$} & \onslide<7->{\NO}\\
+\FORALLR &\bl{r $\cdot$ $\epsilon$} & \bl{$\equiv^?$} & \bl{r} & \onslide<8->{\YES}\\
+\FORALLR &\bl{r + $\epsilon$} & \bl{$\equiv^?$} & \bl{r} & \onslide<9->{\NO}\\
+\FORALLR &\bl{r + $\varnothing$} & \bl{$\equiv^?$} & \bl{r} & \onslide<10->{\YES}\\
+\FORALLR &\bl{r $\cdot$ $\varnothing$} & \bl{$\equiv^?$} & \bl{r} & \onslide<11->{\NO}\\
+&\bl{c $\cdot$ (a + b)} & \bl{$\equiv^?$} & \bl{(c $\cdot$ a) + (c $\cdot$ b)} & \onslide<12->{\YES}\\
+&\bl{a$^*$} & \bl{$\equiv^?$} & \bl{$\epsilon$ + (a $\cdot$ a$^*$)} & \onslide<13->{\YES}
+\end{tabular}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}The Meaning of Matching\end{tabular}}
+
+\large
+a regular expression \bl{r} matches a string \bl{s} is defined as
+
+\begin{center}
+\bl{s $\in$ $L$(r)}\\
+\end{center}\bigskip\bigskip\pause
+
+\small
+if \bl{r$_1$ $\equiv$ r$_2$}, then \bl{$s$ $\in$ $L$(r$_1$)} iff \bl{$s$ $\in$ $L$(r$_2$)}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+\frametitle{\begin{tabular}{c}A Matching Algorithm\end{tabular}}
+
+\begin{itemize}
+\item given a regular expression \bl{r} and a string \bl{s}, say yes or no for whether
+\begin{center}
+\bl{s $\in$ $L$(r)}
+\end{center}
+or not.\bigskip\bigskip\pause
+\end{itemize}\pause
+
+\small
+\begin{itemize}
+\item Identifiers (strings of letters or digits, starting with a letter)
+\item Integers (a non-empty sequence of digits)
+\item Keywords (else, if, while, \ldots)
+\item White space (a non-empty sequence of blanks, newlines and tabs)
+\end{itemize}
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}A Matching Algorithm\end{tabular}}
+
+\small
+whether a regular expression matches the empty string:\medskip
+
+
+{\lstset{language=Scala}\fontsize{8}{10}\selectfont
+\texttt{\lstinputlisting{app5.scala}}}
+
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}The Derivative of a Rexp\end{tabular}}
+
+\large
+If \bl{r} matches the string \bl{c::s}, what is a regular expression that matches \bl{s}?\bigskip\bigskip\bigskip\bigskip
+
+\small
+\bl{der c r} gives the answer
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}The Derivative of a Rexp (2)\end{tabular}}
+
+\begin{center}
+\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
+ \bl{der c ($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$} & \\
+ \bl{der c ($\epsilon$)} & \bl{$\dn$} & \bl{$\varnothing$} & \\
+ \bl{der c (d)} & \bl{$\dn$} & \bl{if c $=$ d then $\epsilon$ else $\varnothing$} & \\
+ \bl{der c (r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\
+ \bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{if nullable r$_1$}\\
+ & & \bl{then ((der c r$_1$) $\cdot$ r$_2$) + (der c r$_2$)}\\
+ & & \bl{else (der c r$_1$) $\cdot$ r$_2$}\\
+ \bl{der c (r$^*$)} & \bl{$\dn$} & \bl{(der c r) $\cdot$ (r$^*$)} &\smallskip\\\pause
+
+ \bl{ders [] r} & \bl{$\dn$} & \bl{r} & \\
+ \bl{ders (c::s) r} & \bl{$\dn$} & \bl{ders s (der c r)} & \\
+ \end{tabular}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}The Derivative\end{tabular}}
+
+
+{\lstset{language=Scala}\fontsize{8}{10}\selectfont
+\texttt{\lstinputlisting{app6.scala}}}
+
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}The Rexp Matcher\end{tabular}}
+
+
+{\lstset{language=Scala}\fontsize{8}{10}\selectfont
+\texttt{\lstinputlisting{app7.scala}}}
+
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+\frametitle{\begin{tabular}{c}Proofs about Rexp\end{tabular}}
+
+Remember their inductive definition:\\[5cm]
+
+\begin{textblock}{6}(5,5)
+ \begin{tabular}{@ {}rrl}
+ \bl{r} & \bl{$::=$} & \bl{$\varnothing$}\\
+ & \bl{$\mid$} & \bl{$\epsilon$} \\
+ & \bl{$\mid$} & \bl{c} \\
+ & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$}\\
+ & \bl{$\mid$} & \bl{r$_1$ + r$_2$} \\
+ & \bl{$\mid$} & \bl{r$^*$} \\
+ \end{tabular}
+ \end{textblock}
+
+If we want to prove something, say a property \bl{$P$(r)}, for all regular expressions \bl{r} then \ldots
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Proofs about Rexp (2)\end{tabular}}
+
+\begin{itemize}
+\item \bl{$P$} holds for \bl{$\varnothing$}, \bl{$\epsilon$} and \bl{c}\bigskip
+\item \bl{$P$} holds for \bl{r$_1$ + r$_2$} under the assumption that \bl{$P$} already
+holds for \bl{r$_1$} and \bl{r$_2$}.\bigskip
+\item \bl{$P$} holds for \bl{r$_1$ $\cdot$ r$_2$} under the assumption that \bl{$P$} already
+holds for \bl{r$_1$} and \bl{r$_2$}.
+\item \bl{$P$} holds for \bl{r$^*$} under the assumption that \bl{$P$} already
+holds for \bl{r}.
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Proofs about Rexp (3)\end{tabular}}
+
+Assume \bl{$P(r)$} is the property:
+
+\begin{center}
+\bl{nullable(r)} if and only if \bl{"" $\in$ $L$(r)}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Proofs about Strings\end{tabular}}
+
+If we want to prove something, say a property \bl{$P$(s)}, for all strings \bl{s} then \ldots\bigskip
+
+\begin{itemize}
+\item \bl{$P$} holds for the empty string, and\medskip
+\item \bl{$P$} holds for the string \bl{c::s} under the assumption that \bl{$P$}
+already holds for \bl{s}
+\end{itemize}
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Proofs about Strings (2)\end{tabular}}
+
+Let \bl{Der c A} be the set defined as
+
+\begin{center}
+\bl{Der c A $\dn$ $\{$ s $|$ c::s $\in$ A$\}$ }
+\end{center}
+
+Assume that \bl{$L$(der c r) = Der c ($L$(r))}. Prove that
+
+\begin{center}
+\bl{matcher(r, s) if and only if s $\in$ $L$(r)}
+\end{center}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Regular Languages\end{tabular}}
+
+A language (set of strings) is \alert{regular} iff there exists
+a regular expression that recognises all its strings.
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Automata\end{tabular}}
+
+A deterministic finite automaton consists of:
+
+\begin{itemize}
+\item a set of states
+\item one of these states is the start state
+\item some states are accepting states, and
+\item there is transition function\medskip
+
+\small
+which takes a state as argument and a character and produces a new state\smallskip\\
+this function might not always be defined
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
+