
diff -r e85600529ca5 -r 4794759139ea slides05.tex
--- a/slides05.tex	Sat Jun 15 09:11:11 2013 -0400
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,504 +0,0 @@
-\documentclass[dvipsnames,14pt,t]{beamer}
-\usepackage{beamerthemeplainculight}
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-\usepackage{mathpartir}
-\usepackage[absolute,overlay]{textpos}
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-\usepackage{pgf}
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-
-% 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 a 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 a 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}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}Algorithms on Automata\end{tabular}}
-
-
-\begin{itemize}
-\item Reg $\rightarrow$ NFA: Thompson-McNaughton-Yamada method\medskip
-\item NFA $\rightarrow$ DFA: Subset Construction\medskip
-\item DFA $\rightarrow$ Reg: Brzozowski's Algebraic Method\medskip
-\item DFA minimisation: Hopcrofts Algorithm\medskip
-\item complement DFA
-\end{itemize}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-\newcommand{\qq}{\mbox{\texttt{"}}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}Grammars\end{tabular}}
-
-\begin{center}
-\bl{\begin{tabular}{lcl}
-$E$ & $\rightarrow$ &  $F + (F \cdot \qq*\qq \cdot F) + (F \cdot \qq\backslash\qq \cdot F)$\\
-$F$ & $\rightarrow$ & $T + (T \cdot \qq\texttt{+}\qq \cdot T) + (T \cdot \qq\texttt{-}\qq \cdot T)$\\
-$T$ & $\rightarrow$ & $num + (\qq\texttt{(}\qq \cdot E \cdot \qq\texttt{)}\qq)$\\
-\end{tabular}}
-\end{center}
-
-\bl{$E$}, \bl{$F$} and \bl{$T$} are non-terminals\\
-\bl{$E$} is start symbol\\
-\bl{$num$}, \bl{(}, \bl{)}, \bl{+} \ldots are terminals\bigskip\\
-
-
-\bl{\texttt{(2*3)+(3+4)}}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-\begin{center}
-\bl{\begin{tabular}{lcl}
-$E$ & $\rightarrow$ &  $F + (F \cdot \qq*\qq \cdot F) + (F \cdot \qq\backslash\qq \cdot F)$\\
-$F$ & $\rightarrow$ & $T + (T \cdot \qq\texttt{+}\qq \cdot T) + (T \cdot \qq\texttt{-}\qq \cdot T)$\\
-$T$ & $\rightarrow$ & $num + (\qq\texttt{(}\qq \cdot E \cdot \qq\texttt{)}\qq)$\\
-\end{tabular}}
-\end{center}
-
-\begin{center}
-\begin{tikzpicture}[level distance=8mm, blue]
-  \node {E}
-    child {node {F} 
-     child {node {T} 
-                 child {node {\qq(\qq\,E\,\qq)\qq}
-                            child {node{F \qq*\qq{} F}
-                                  child {node {T} child {node {2}}}
-                                  child {node {T} child {node {3}}} 
-                               }
-                          }
-              }
-     child {node {\qq+\qq}}
-     child {node {T}
-       child {node {\qq(\qq\,E\,\qq)\qq} 
-       child {node {F}
-       child {node {T \qq+\qq{} T}
-                    child {node {3}}
-                    child {node {4}} 
-                 }
-                 }}
-    }};
-\end{tikzpicture}
-\end{center}
-
-\begin{textblock}{5}(1, 5)
-\bl{\texttt{(2*3)+(3+4)}}
-\end{textblock}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-\end{document}
-
-%%% Local Variables:  
-%%% mode: latex
-%%% TeX-master: t
-%%% End: 
-




