
diff -r 9da175d5eb63 -r 4725bba8ef26 slides/slides05.tex
--- a/slides/slides05.tex	Mon Oct 21 15:02:54 2013 +0100
+++ b/slides/slides05.tex	Tue Oct 22 23:17:53 2013 +0100
@@ -1,7 +1,7 @@
 \documentclass[dvipsnames,14pt,t]{beamer}
-\usepackage{beamerthemeplainculight}
-\usepackage[T1]{fontenc}
-\usepackage[latin1]{inputenc}
+\usepackage{beamerthemeplaincu}
+%\usepackage[T1]{fontenc}
+%\usepackage[latin1]{inputenc}
 \usepackage{mathpartir}
 \usepackage[absolute,overlay]{textpos}
 \usepackage{ifthen}
@@ -25,8 +25,13 @@
 \definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords
 \definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc
 
+\makeatletter
+\lst@CCPutMacro\lst@ProcessOther {"2D}{\lst@ttfamily{-{}}{-{}}}
+\@empty\z@\@empty
+\makeatother
+
 \lstset{language=Java,
-	basicstyle=\ttfamily,
+	basicstyle=\consolas,
 	keywordstyle=\color{javapurple}\bfseries,
 	stringstyle=\color{javagreen},
 	commentstyle=\color{javagreen},
@@ -46,8 +51,8 @@
     new,null,object,override,package,%
     private,protected,requires,return,sealed,%
     super,this,throw,trait,true,try,%
-    type,val,var,while,with,yield},
-  otherkeywords={=>,<-,<\%,<:,>:,\#,@},
+    type,val,var,while,with,yield, then},
+  otherkeywords={=>,<-,<\%,<:,>:,\#,@,->},
   sensitive=true,
   morecomment=[l]{//},
   morecomment=[n]{/*}{*/},
@@ -57,7 +62,7 @@
 }
 
 \lstset{language=Scala,
-	basicstyle=\ttfamily,
+	basicstyle=\consolas,
 	keywordstyle=\color{javapurple}\bfseries,
 	stringstyle=\color{javagreen},
 	commentstyle=\color{javagreen},
@@ -69,9 +74,10 @@
 	tabsize=2,
 	showspaces=false,
 	showstringspaces=false}
+	
 
 % beamer stuff 
-\renewcommand{\slidecaption}{AFL 05, King's College London, 24.~October 2012}
+\renewcommand{\slidecaption}{AFL 05, King's College London, 23.~October 2013}
 \newcommand{\bl}[1]{\textcolor{blue}{#1}}       
 \newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
 
@@ -91,7 +97,7 @@
   \begin{center}
   \begin{tabular}{ll}
   Email:  & christian.urban at kcl.ac.uk\\
-  Of$\!$fice: & S1.27 (1st floor Strand Building)\\
+  Office: & S1.27 (1st floor Strand Building)\\
   Slides: & KEATS (also home work is there)\\
   \end{tabular}
   \end{center}
@@ -102,93 +108,209 @@
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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{frame}[c]
+\frametitle{DFA Minimisation}
 
-\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
+\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 no chance.
+\item All unmarked pairs can be merged.
+\end{enumerate}
 
-\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}}
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}<1-2>[c]
 
-\only<3,4>{
+\begin{center}
+\begin{tikzpicture}[>=stealth',very thick,auto,
+                             every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
+\node[state,initial]  (q_0)  {$q_0$};
+\node[state] (q_1) [right=of q_0] {$q_1$};
+\node[state] (q_2) [below right=of q_0] {$q_2$};
+\node[state] (q_3) [right=of q_2] {$q_3$};
+\node[state, accepting] (q_4) [right=of q_1] {$q_4$};
+\path[->] (q_0) edge node [above]  {\alert{$a$}} (q_1);
+\path[->] (q_1) edge node [above]  {\alert{$a$}} (q_4);
+\path[->] (q_4) edge [loop right] node  {\alert{$a, b$}} ();
+\path[->] (q_3) edge node [right]  {\alert{$a$}} (q_4);
+\path[->] (q_2) edge node [above]  {\alert{$a$}} (q_3);
+\path[->] (q_1) edge node [right]  {\alert{$b$}} (q_2);
+\path[->] (q_0) edge node [above]  {\alert{$b$}} (q_2);
+\path[->] (q_2) edge [loop left] node  {\alert{$b$}} ();
+\path[->] (q_3) edge [bend left=95, looseness=1.3] node [below]  {\alert{$b$}} (q_0);
+\end{tikzpicture}
+\end{center}
+
+\mbox{}\\[-20mm]\mbox{}
+
 \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}}
+\begin{tikzpicture}[scale=0.8,line width=0.8mm]
+\draw (0,0) -- (4,0);
+\draw (0,1) -- (4,1);
+\draw (0,2) -- (3,2);
+\draw (0,3) -- (2,3);
+\draw (0,4) -- (1,4);
+
+\draw (0,0) -- (0, 4);
+\draw (1,0) -- (1, 4);
+\draw (2,0) -- (2, 3);
+\draw (3,0) -- (3, 2);
+\draw (4,0) -- (4, 1);
+
+\draw (0.5,-0.5) node {$q_0$}; 
+\draw (1.5,-0.5) node {$q_1$}; 
+\draw (2.5,-0.5) node {$q_2$}; 
+\draw (3.5,-0.5) node {$q_3$};
+ 
+\draw (-0.5, 3.5) node {$q_1$}; 
+\draw (-0.5, 2.5) node {$q_2$}; 
+\draw (-0.5, 1.5) node {$q_3$}; 
+\draw (-0.5, 0.5) node {$q_4$}; 
+
+\draw (0.5,0.5) node {\large$\star$}; 
+\draw (1.5,0.5) node {\large$\star$}; 
+\draw (2.5,0.5) node {\large$\star$}; 
+\draw (3.5,0.5) node {\large$\star$};
+\end{tikzpicture}\\
+\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{frame}<1-2>[c]
 
-\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}{@{\hspace{-8mm}}cc@{}}
+\begin{tikzpicture}[>=stealth',very thick,auto,
+                             every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
+\node[state,initial]  (q_0)  {$q_0$};
+\node[state] (q_1) [right=of q_0] {$q_1$};
+\node[state] (q_2) [below right=of q_0] {$q_2$};
+\node[state] (q_3) [right=of q_2] {$q_3$};
+\node[state, accepting] (q_4) [right=of q_1] {$q_4$};
+\path[->] (q_0) edge node [above]  {\alert{$a$}} (q_1);
+\path[->] (q_1) edge node [above]  {\alert{$a$}} (q_4);
+\path[->] (q_4) edge [loop right] node  {\alert{$a, b$}} ();
+\path[->] (q_3) edge node [right]  {\alert{$a$}} (q_4);
+\path[->] (q_2) edge node [above]  {\alert{$a$}} (q_3);
+\path[->] (q_1) edge node [right]  {\alert{$b$}} (q_2);
+\path[->] (q_0) edge node [above]  {\alert{$b$}} (q_2);
+\path[->] (q_2) edge [loop left] node  {\alert{$b$}} ();
+\path[->] (q_3) edge [bend left=95, looseness=1.3] node [below]  {\alert{$b$}} (q_0);
+\end{tikzpicture}
+&
+\raisebox{9mm}{\begin{tikzpicture}[scale=0.6,line width=0.8mm]
+\draw (0,0) -- (4,0);
+\draw (0,1) -- (4,1);
+\draw (0,2) -- (3,2);
+\draw (0,3) -- (2,3);
+\draw (0,4) -- (1,4);
+
+\draw (0,0) -- (0, 4);
+\draw (1,0) -- (1, 4);
+\draw (2,0) -- (2, 3);
+\draw (3,0) -- (3, 2);
+\draw (4,0) -- (4, 1);
+
+\draw (0.5,-0.5) node {$q_0$}; 
+\draw (1.5,-0.5) node {$q_1$}; 
+\draw (2.5,-0.5) node {$q_2$}; 
+\draw (3.5,-0.5) node {$q_3$};
+ 
+\draw (-0.5, 3.5) node {$q_1$}; 
+\draw (-0.5, 2.5) node {$q_2$}; 
+\draw (-0.5, 1.5) node {$q_3$}; 
+\draw (-0.5, 0.5) node {$q_4$}; 
+
+\draw (0.5,0.5) node {\large$\star$}; 
+\draw (1.5,0.5) node {\large$\star$}; 
+\draw (2.5,0.5) node {\large$\star$}; 
+\draw (3.5,0.5) node {\large$\star$};
+\draw (0.5,1.5) node {\large$\star$}; 
+\draw (2.5,1.5) node {\large$\star$}; 
+\draw (0.5,3.5) node {\large$\star$}; 
+\draw (1.5,2.5) node {\large$\star$}; 
+\end{tikzpicture}}
+\end{tabular}
+\end{center}
 
 
+\mbox{}\\[-20mm]\mbox{}
+
 \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.
+\begin{tikzpicture}[>=stealth',very thick,auto,
+                             every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
+\node[state,initial]  (q_02)  {$q_{0, 2}$};
+\node[state] (q_13) [right=of q_02] {$q_{1, 3}$};
+\node[state, accepting] (q_4) [right=of q_13] {$q_{4\phantom{,0}}$};
+\path[->] (q_02) edge [bend left] node [above]  {\alert{$a$}} (q_13);
+\path[->] (q_13) edge [bend left] node [below]  {\alert{$b$}} (q_02);
+\path[->] (q_02) edge [loop below] node  {\alert{$b$}} ();
+\path[->] (q_13) edge node [above]  {\alert{$a$}} (q_4);
+\path[->] (q_4) edge [loop above] node  {\alert{$a, b$}} ();
+\end{tikzpicture}\\
+minimal automaton
+\end{center}
 
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
+
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
-\frametitle{\begin{tabular}{c}Last Week\end{tabular}}
 
-Last week I showed you\bigskip
+\begin{center}
+\begin{tikzpicture}[>=stealth',very thick,auto,
+                             every state/.style={minimum size=0pt,inner sep=2pt,draw=blue!50,very thick,fill=blue!20},]
+\only<1>{\node[state,initial]  (q_0)  {$q_0$};}
+\only<2->{\node[state,accepting]  (q_0)  {$q_0$};}
+\node[state] (q_1) [right=of q_0] {$q_1$};
+\node[state] (q_2) [below right=of q_0] {$q_2$};
+\node[state] (q_3) [right=of q_2] {$q_3$};
+\only<1>{\node[state, accepting] (q_4) [right=of q_1] {$q_4$};}
+\only<2->{\node[state, initial right] (q_4) [right=of q_1] {$q_4$};}
+\only<1-2>{
+\path[->] (q_0) edge node [above]  {\alert{$a$}} (q_1);
+\path[->] (q_1) edge node [above]  {\alert{$a$}} (q_4);
+\path[->] (q_4) edge [loop above] node  {\alert{$a, b$}} ();
+\path[->] (q_3) edge node [right]  {\alert{$a$}} (q_4);
+\path[->] (q_2) edge node [above]  {\alert{$a$}} (q_3);
+\path[->] (q_1) edge node [right]  {\alert{$b$}} (q_2);
+\path[->] (q_0) edge node [above]  {\alert{$b$}} (q_2);
+\path[->] (q_2) edge [loop left] node  {\alert{$b$}} ();
+\path[->] (q_3) edge [bend left=95, looseness=1.3] node [below]  {\alert{$b$}} (q_0);}
+\only<3->{
+\path[<-] (q_0) edge node [above]  {\alert{$a$}} (q_1);
+\path[<-] (q_1) edge node [above]  {\alert{$a$}} (q_4);
+\path[<-] (q_4) edge [loop above] node  {\alert{$a, b$}} ();
+\path[<-] (q_3) edge node [right]  {\alert{$a$}} (q_4);
+\path[<-] (q_2) edge node [above]  {\alert{$a$}} (q_3);
+\path[<-] (q_1) edge node [right]  {\alert{$b$}} (q_2);
+\path[<-] (q_0) edge node [above]  {\alert{$b$}} (q_2);
+\path[<-] (q_2) edge [loop left] node  {\alert{$b$}} ();
+\path[<-] (q_3) edge [bend left=95, looseness=1.3] node [below]  {\alert{$b$}} (q_0);}
+\end{tikzpicture}
+\end{center}
 
 \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)
+\item<2-> exchange initial / accepting states
+\item<3-> reverse all edges
+\item<4-> subset construction $\Rightarrow$ DFA
+\item<5-> repeat once more \onslide<6->{$\Rightarrow$ minimal DFA}
 
 \end{itemize}
 
@@ -198,112 +320,8 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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}
+\texttt{\consolas\lstinputlisting{../progs/fib.while}}
 
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
@@ -312,120 +330,14 @@
 \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?}
+\texttt{\consolas\lstinputlisting{../progs/collatz.while}}
 
 \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>{




