
diff -r e85600529ca5 -r 4794759139ea slides02.tex
--- a/slides02.tex	Sat Jun 15 09:11:11 2013 -0400
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,494 +0,0 @@
-\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: 
-




