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+\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 04, King's College London, 17.~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 (4)\\[3mm]
+ \end{tabular}}
+
+ \normalsize
+ \begin{center}
+ \begin{tabular}{ll}
+ Email: & christian.urban at kcl.ac.uk\\
+ Of$\!$fice: & S1.27 (1st floor Strand Building)\\
+ Slides: & KEATS (also home work is there)\\
+ \end{tabular}
+ \end{center}
+
+
+\end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Last Week\end{tabular}}
+
+Last week I showed you
+
+\begin{itemize}
+\item tokenizer
+
+\item tokenization identifies lexeme in an input stream of characters (or string)
+and categorizes them into tokens
+
+\item maximal munch rule
+\end{itemize}
+
+\url{http://www.technologyreview.com/tr10/?year=2011}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}The Derivative of a Rexp\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$^*$)}\\
+ \end{tabular}
+\end{center}
+
+``the regular expression after \bl{c} has been recognised''
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+For this we defined the set \bl{Der c A} as
+
+\begin{center}
+\bl{Der c A $\dn$ $\{$ s $|$ c::s $\in$ A$\}$ }
+\end{center}
+
+which is called the semantic derivative of a set
+and proved
+
+\begin{center}
+\bl{$L$(der c r) $=$ Der c ($L$(r))}
+\end{center}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}The Idea of the Algorithm\end{tabular}}
+
+If we want to recognise the string \bl{abc} with regular expression \bl{r}
+then\medskip
+
+\begin{enumerate}
+\item \bl{Der a ($L$(r))}\pause
+\item \bl{Der b (Der a ($L$(r)))}
+\item \bl{Der c (Der b (Der a ($L$(r))))}\pause
+\item finally we test whether the empty string is in set\pause\medskip
+\end{enumerate}
+
+The matching algorithm works similarly, just over regular expression than sets.
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+Input: string \bl{abc} and regular expression \bl{r}
+
+\begin{enumerate}
+\item \bl{der a r}
+\item \bl{der b (der a r)}
+\item \bl{der c (der b (der a r))}\pause
+\item finally check whether the latter regular expression can match the empty string
+\end{enumerate}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+We need to prove
+
+\begin{center}
+\bl{$L$(der c r) $=$ Der c ($L$(r))}
+\end{center}
+
+by induction on the regular expression.
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Proofs about Rexp\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 Natural Numbers\\ and Strings\end{tabular}}
+
+\begin{itemize}
+\item \bl{$P$} holds for \bl{$0$} and
+\item \bl{$P$} holds for \bl{$n + 1$} under the assumption that \bl{$P$} already
+holds for \bl{$n$}
+\end{itemize}\bigskip
+
+\begin{itemize}
+\item \bl{$P$} holds for \bl{\texttt{""}} and
+\item \bl{$P$} holds for \bl{$c\!::\!s$} under the assumption that \bl{$P$} already
+holds for \bl{$s$}
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}
+
+\begin{center}
+ \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}\bigskip\pause
+ \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.\bigskip
+
+A language is \alert{regular} iff there exists
+a regular expression that recognises all its strings.\bigskip\bigskip\pause
+
+\textcolor{gray}{not all languages are regular, e.g.~\bl{a$^n$b$^n$}.}
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}
+
+\begin{center}
+ \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}\bigskip
+ \end{center}
+
+How about ranges \bl{[a-z]}, \bl{r$^\text{+}$} and \bl{!r}?
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Negation of Regular Expr's\end{tabular}}
+
+\begin{itemize}
+\item \bl{!r} \hspace{6mm} (everything that \bl{r} cannot recognise)\medskip
+\item \bl{$L$(!r) $\dn$ UNIV - $L$(r)}\medskip
+\item \bl{nullable (!r) $\dn$ not (nullable(r))}\medskip
+\item \bl{der\,c\,(!r) $\dn$ !(der\,c\,r)}
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Regular Exp's for Lexing\end{tabular}}
+
+Lexing separates strings into ``words'' / components.
+
+\begin{itemize}
+\item Identifiers (non-empty strings of letters or digits, starting with a letter)
+\item Numbers (non-empty sequences of digits omitting leading zeros)
+\item Keywords (else, if, while, \ldots)
+\item White space (a non-empty sequence of blanks, newlines and tabs)
+\item Comments
+\end{itemize}
+
+\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:
+