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+++ b/slides05.tex Wed Oct 24 03:40:50 2012 +0100
@@ -0,0 +1,420 @@
+\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 05, King's College London, 24.~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 (5)\\[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}[t]
+\frametitle{\begin{tabular}{c}Deterministic Finite Automata\end{tabular}}
+
+A DFA \bl{$A(Q, q_0, F, \delta)$} consists of:
+
+\begin{itemize}
+\item a finite set of states \bl{$Q$}
+\item one of these states is the start state \bl{$q_0$}
+\item some states are accepting states \bl{$F$}
+\item there is transition function \bl{$\delta$}
+\end{itemize}\pause
+
+\onslide<2->{
+\begin{center}
+\begin{tabular}{l}
+\bl{$\hat{\delta}(q, \texttt{""}) = q$}\\
+\bl{$\hat{\delta}(q, c\!::\!s) = \hat{\delta}(\delta(q, c), s)$}
+\end{tabular}
+\end{center}}
+
+\only<3,4>{
+\begin{center}
+\begin{tikzpicture}[scale=2, line width=0.5mm]
+ \node[state, initial] (q02) at ( 0,1) {$q_{0}$};
+ \node[state] (q13) at ( 1,1) {$q_{1}$};
+ \node[state, accepting] (q4) at ( 2,1) {$q_2$};
+ \path[->] (q02) edge[bend left] node[above] {$a$} (q13)
+ (q13) edge[bend left] node[below] {$b$} (q02)
+ (q13) edge node[above] {$a$} (q4)
+ (q02) edge [loop below] node {$b$} ()
+ (q4) edge [loop right] node {$a, b$} ()
+ ;
+\end{tikzpicture}
+\end{center}}%
+%
+\only<5>{
+\begin{center}
+\bl{$L(A) \dn \{ s \;|\; \hat{\delta}(q_0, s) \in F\}$}
+\end{center}}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+\frametitle{\begin{tabular}{c}Non-Deterministic\\[-1mm] Finite Automata\end{tabular}}
+
+An NFA \bl{$A(Q, q_0, F, \delta)$} consists again of:
+
+\begin{itemize}
+\item a finite set of states
+\item one of these states is the start state
+\item some states are accepting states
+\item there is transition \alert{relation}\medskip
+\end{itemize}
+
+
+\begin{center}
+\begin{tabular}{c}
+\bl{(q$_1$, a) $\rightarrow$ q$_2$}\\
+\bl{(q$_1$, a) $\rightarrow$ q$_3$}\\
+\end{tabular}
+\hspace{10mm}
+\begin{tabular}{c}
+\bl{(q$_1$, $\epsilon$) $\rightarrow$ q$_2$}\\
+\end{tabular}
+\end{center}\pause\medskip
+
+A string \bl{s} is accepted by an NFA, if there is a ``lucky'' sequence to an accepting state.
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Last Week\end{tabular}}
+
+Last week I showed you\bigskip
+
+\begin{itemize}
+\item an algorithm for automata minimisation
+
+\item an algorithm for transforming a regular expression into an NFA
+
+\item an algorithm for transforming an NFA into a DFA (subset construction)
+
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}This Week\end{tabular}}
+
+Go over the algorithms again, but with two new things and \ldots\medskip
+
+\begin{itemize}
+\item with the example: what is the regular expression that accepts every string, except those ending
+in \bl{aa}?\medskip
+
+\item Go over the proof for \bl{$L(rev(r)) = Rev(L(r))$}.\medskip
+
+\item Anything else so far.
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}Proofs By Induction\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}
+
+\begin{center}
+\bl{$P(r):\;\;L(rev(r)) = Rev(L(r))$}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+
+What is the regular expression that accepts every string, except those ending
+in \bl{aa}?\pause\bigskip
+
+\begin{center}
+\begin{tabular}{l}
+\bl{(a + b)$^*$ba}\\
+\bl{(a + b)$^*$ab}\\
+\bl{(a + b)$^*$bb}\\\pause
+\bl{a}\\
+\bl{\texttt{""}}
+\end{tabular}
+\end{center}\pause
+
+What are the strings to be avoided?\pause\medskip
+
+\begin{center}
+\bl{(a + b)$^*$aa}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+
+An NFA for \bl{(a + b)$^*$aa}
+
+\begin{center}
+\begin{tikzpicture}[scale=2, line width=0.5mm]
+ \node[state, initial] (q0) at ( 0,1) {$q_0$};
+ \node[state] (q1) at ( 1,1) {$q_1$};
+ \node[state, accepting] (q2) at ( 2,1) {$q_2$};
+ \path[->] (q0) edge node[above] {$a$} (q1)
+ (q1) edge node[above] {$a$} (q2)
+ (q0) edge [loop below] node {$a$} ()
+ (q0) edge [loop above] node {$b$} ()
+ ;
+\end{tikzpicture}
+\end{center}\pause
+
+Minimisation for DFAs\\
+Subset Construction for NFAs
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{\begin{tabular}{c}DFA Minimisation\end{tabular}}
+
+
+\begin{enumerate}
+\item Take all pairs \bl{(q, p)} with \bl{q $\not=$ p}
+\item Mark all pairs that accepting and non-accepting states
+\item For all unmarked pairs \bl{(q, p)} and all characters \bl{c} tests wether
+\begin{center}
+\bl{($\delta$(q,c), $\delta$(p,c))}
+\end{center}
+are marked. If yes, then also mark \bl{(q, p)}.
+\item Repeat last step until nothing changed.
+\item All unmarked pairs can be merged.
+\end{enumerate}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+Minimal DFA \only<1>{\bl{(a + b)$^*$aa}}\only<2->{\alert{not} \bl{(a + b)$^*$aa}}
+
+\begin{center}
+\begin{tikzpicture}[scale=2, line width=0.5mm]
+ \only<1>{\node[state, initial] (q0) at ( 0,1) {$q_0$};}
+ \only<2->{\node[state, initial,accepting] (q0) at ( 0,1) {$q_0$};}
+ \only<1>{\node[state] (q1) at ( 1,1) {$q_1$};}
+ \only<2->{\node[state,accepting] (q1) at ( 1,1) {$q_1$};}
+ \only<1>{\node[state, accepting] (q2) at ( 2,1) {$q_2$};}
+ \only<2->{\node[state] (q2) at ( 2,1) {$q_2$};}
+ \path[->] (q0) edge[bend left] node[above] {$a$} (q1)
+ (q1) edge[bend left] node[above] {$b$} (q0)
+ (q2) edge[bend left=50] node[below] {$b$} (q0)
+ (q1) edge node[above] {$a$} (q2)
+ (q2) edge [loop right] node {$a$} ()
+ (q0) edge [loop below] node {$b$} ()
+ ;
+\end{tikzpicture}
+\end{center}
+
+\onslide<3>{How to get from a DFA to a regular expression?}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\begin{center}
+\begin{tikzpicture}[scale=2, line width=0.5mm]
+ \only<1->{\node[state, initial] (q0) at ( 0,1) {$q_0$};}
+ \only<1->{\node[state] (q1) at ( 1,1) {$q_1$};}
+ \only<1->{\node[state] (q2) at ( 2,1) {$q_2$};}
+ \path[->] (q0) edge[bend left] node[above] {$a$} (q1)
+ (q1) edge[bend left] node[above] {$b$} (q0)
+ (q2) edge[bend left=50] node[below] {$b$} (q0)
+ (q1) edge node[above] {$a$} (q2)
+ (q2) edge [loop right] node {$a$} ()
+ (q0) edge [loop below] node {$b$} ()
+ ;
+\end{tikzpicture}
+\end{center}\pause\bigskip
+
+\onslide<2->{
+\begin{center}
+\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
+\bl{$q_0$} & \bl{$=$} & \bl{$2\, q_0 + 3 \,q_1 + 4\, q_2$}\\
+\bl{$q_1$} & \bl{$=$} & \bl{$2 \,q_0 + 3\, q_1 + 1\, q_2$}\\
+\bl{$q_2$} & \bl{$=$} & \bl{$1\, q_0 + 5\, q_1 + 2\, q_2$}\\
+
+\end{tabular}
+\end{center}
+}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\begin{center}
+\begin{tikzpicture}[scale=2, line width=0.5mm]
+ \only<1->{\node[state, initial] (q0) at ( 0,1) {$q_0$};}
+ \only<1->{\node[state] (q1) at ( 1,1) {$q_1$};}
+ \only<1->{\node[state] (q2) at ( 2,1) {$q_2$};}
+ \path[->] (q0) edge[bend left] node[above] {$a$} (q1)
+ (q1) edge[bend left] node[above] {$b$} (q0)
+ (q2) edge[bend left=50] node[below] {$b$} (q0)
+ (q1) edge node[above] {$a$} (q2)
+ (q2) edge [loop right] node {$a$} ()
+ (q0) edge [loop below] node {$b$} ()
+ ;
+\end{tikzpicture}
+\end{center}\bigskip
+
+\onslide<2->{
+\begin{center}
+\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
+\bl{$q_0$} & \bl{$=$} & \bl{$\epsilon + q_0\,b + q_1\,b + q_2\,b$}\\
+\bl{$q_1$} & \bl{$=$} & \bl{$q_0\,a$}\\
+\bl{$q_2$} & \bl{$=$} & \bl{$q_1\,a + q_2\,a$}\\
+
+\end{tabular}
+\end{center}
+}
+
+\onslide<3->{
+Arden's Lemma:
+\begin{center}
+If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
+\end{center}
+}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
+