\documentclass[dvipsnames,14pt,t]{beamer}

\usepackage{../langs}

\usepackage{../data}

\begin{document}

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\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}

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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]

\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}

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\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}}

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\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}}

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\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}}

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\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}}

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\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

\begin{center}

\bl{$(\delta(q, c), \delta(p,c))$}

\end{center} 

are marked. If yes, then also mark \bl{$(q, p)$}.

\item All unmarked pairs can be merged.

\end{enumerate}

\end{frame}}

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\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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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}}

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\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}}

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\end{document}

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