\documentclass[dvipsnames,14pt,t]{beamer}
\usepackage{../slides}
\usepackage{../graphics}
\usepackage{../langs}
\usepackage{../data}

\hfuzz=220pt 

\pgfplotsset{compat=1.11}

\newcommand{\bl}[1]{\textcolor{blue}{#1}}  

\renewcommand{\slidecaption}{AFL 04, King's College London}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[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\\
  Office: & S1.27 (1st floor Strand Building)\\
  Slides: & KEATS (also home work is there)\\
  \end{tabular}
  \end{center}
\end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     


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\mode<presentation>{
\begin{frame}[c]
\frametitle{Regexps and Automata}

\begin{center}
\begin{tikzpicture}
\node (rexp)  {\bl{\bf Regexps}};
\node (nfa) [right=of rexp] {\bl{\bf NFAs}};
\node (dfa) [right=of nfa] {\bl{\bf DFAs}};
\onslide<3->{\node (mdfa) [right=of dfa] {\bl{\bf \begin{tabular}{c}minimal\\ DFAs\end{tabular}}};}
\path[->, red, line width=2mm] (rexp) edge node [above=4mm, black] {\begin{tabular}{c@{\hspace{9mm}}}Thompson's\\[-1mm] construction\end{tabular}} (nfa);
\path[->, red, line width=2mm] (nfa) edge node [above=4mm, black] {\begin{tabular}{c}subset\\[-1mm] construction\end{tabular}}(dfa);
\onslide<3->{\path[->, red, line width=2mm] (dfa) edge node [below=9mm, black] {minimisation} (mdfa);}
\onslide<2->{\path[->, red, line width=2mm] (dfa) edge [bend left=45] (rexp);}
\end{tikzpicture}\\
\end{center}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   


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\begin{frame}[t]
\frametitle{\bl{$(a?\{n\}) \cdot a\{n\}$}}

\mbox{}\\[-13mm]

\begin{tikzpicture}[y=.2cm, x=.09cm]
 	%axis
	\draw (0,0) -- coordinate (x axis mid) (100,0);
    	\draw (0,0) -- coordinate (y axis mid) (0,30);
    	%ticks
    	\foreach \x in {0,10,...,100}
     		\draw (\x,1pt) -- (\x,-3pt)
			node[anchor=north] {\x};
    	\foreach \y in {0,5,...,30}
     		\draw (1pt,\y) -- (-3pt,\y) 
     			node[anchor=east] {\y}; 
	%labels      
	\node[below=0.6cm] at (x axis mid) {\bl{a}s};
	\node[rotate=90, left=1.2cm] at (y axis mid) {secs};

	%plots
	\draw[color=blue] plot[mark=*, mark options={fill=white}] 
		file {re-python.data};
	\draw[color=red] plot[mark=triangle*, mark options={fill=white} ] 
		file {nfa.data};	  
	\draw[color=brown] plot[mark=pentagon*, mark options={fill=white} ] 
		file {re-ruby.data};
		
    
	%legend
	\begin{scope}[shift={(4,20)}] 
	\draw[color=blue] (0,0) -- 
		plot[mark=*, mark options={fill=white}] (0.25,0) -- (0.5,0) 
		node[right]{\small Python};
	\draw[yshift=-\baselineskip, color=brown] (0,0) -- 
		plot[mark=pentagon*, mark options={fill=white}] (0.25,0) -- (0.5,0)
		node[right]{\small Ruby};
	\draw[yshift=\baselineskip, color=red] (0,0) -- 
		plot[mark=triangle*, mark options={fill=white}] (0.25,0) -- (0.5,0)
		node[right]{\small NFA 1};		
	\end{scope}
\end{tikzpicture}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}\bl{$(a?\{n\}) \cdot a\{n\}$}\end{tabular}}

\mbox{}\\[-13mm]

\begin{tikzpicture}[y=.2cm, x=.3cm]
 	%axis
	\draw (0,0) -- coordinate (x axis mid) (30,0);
    	\draw (0,0) -- coordinate (y axis mid) (0,30);
    	%ticks
    	\foreach \x in {0,5,...,30}
     		\draw (\x,1pt) -- (\x,-3pt)
			node[anchor=north] {\x};
    	\foreach \y in {0,5,...,30}
     		\draw (1pt,\y) -- (-3pt,\y) 
     			node[anchor=east] {\y}; 
	%labels      
	\node[below=0.6cm] at (x axis mid) {\bl{a}s};
	\node[rotate=90, left=1.2cm] at (y axis mid) {secs};

	%plots
	\draw[color=blue] plot[mark=*, mark options={fill=white}] 
		file {re-python.data};
	\draw[color=red] plot[mark=triangle*, mark options={fill=white} ] 
		file {nfasearch.data};	  
	\draw[color=brown] plot[mark=pentagon*, mark options={fill=white} ] 
		file {re-ruby.data};
    
	%legend
	\begin{scope}[shift={(4,20)}] 
	\draw[color=blue] (0,0) -- 
		plot[mark=*, mark options={fill=white}] (0.25,0) -- (0.5,0) 
		node[right]{\small Python};
	\draw[yshift=-\baselineskip, color=brown] (0,0) -- 
		plot[mark=pentagon*, mark options={fill=white}] (0.25,0) -- (0.5,0)
		node[right]{\small Ruby};
	\draw[yshift=\baselineskip, color=red] (0,0) -- 
		plot[mark=triangle*, mark options={fill=white}] (0.25,0) -- (0.5,0)
		node[right]{\small NFA 2};		
	\end{scope}
\end{tikzpicture}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<2>[c]
\frametitle{DFA to Rexp}

\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{DFA Minimisation}

\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{$c$} test whether
\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 change.
\item All unmarked pairs can be merged.
\end{enumerate}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]

\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=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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]

\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}[>=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}<1-2>[c]

\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}\pause

\mbox{}\\[-20mm]\mbox{}

\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_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]

\begin{itemize}
\item Assuming you have the alphabet \bl{$\{a, b, c\}$}\bigskip
\item Give a regular expression that can recognise all strings that have at least one \bl{$b$}.
\end{itemize}

\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   


\end{document}

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