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\makeatletter

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\@empty\z@\@empty

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\renewcommand{\slidecaption}{AFL 04, King's College London, 16.~October 2013}

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

\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions

\begin{document}

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\mode<presentation>{

\begin{frame}<1>[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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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}DFAs\end{tabular}}

A deterministic finite automaton 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 there is transition function, \bl{$\delta$}, and

\item some states are accepting states, \bl{$F$}

\medskip 

\end{itemize}

\begin{center}

\bl{$A(Q, q_0, \delta, F)$}

\end{center}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}State Nodes\end{tabular}}

\hspace{5mm}\mbox{{\lstset{language=Scala}\consolas\fontsize{9}{10}\selectfont

\lstinputlisting{../progs/appA.scala}}}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}DFAs\;\;\;\end{tabular}}

\mbox{}\\[7mm]

\mbox{{\lstset{language=Scala}\consolas\fontsize{9}{10}\selectfont

\lstinputlisting{../progs/appB.scala}}}

\only<2->{

\begin{textblock}{9}(7.5,0.5)

\begin{tikzpicture}

\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

{\normalsize\color{darkgray}

\begin{minipage}{6cm}

\begin{tabular}{l}

\bl{$\hat{\delta}(q, \texttt{""}) \dn q$}\\

\bl{$\hat{\delta}(q, c::s) \dn \hat{\delta}(\delta(q, c), s)$}\\[4mm]

\bl{$\hat{\delta}(q_0, s) \in F$}

\end{tabular}

\end{minipage}};

\end{tikzpicture}

\end{textblock}}

\end{frame}}

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\mode<presentation>{

\begin{frame}[t]

\mbox{}\hspace{-10mm}

\begin{tikzpicture}[scale=0.6,>=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}

\only<1->{

\begin{textblock}{9}(7.4,3.5)

\begin{tikzpicture}

\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

{\normalsize\color{darkgray}

\begin{minipage}{6.6cm}

\hspace{5mm}\mbox{{\lstset{language=Scala}\consolas\fontsize{8}{10}\selectfont

\lstinputlisting{../progs/appC.scala}}}

\end{minipage}};

\end{tikzpicture}

\end{textblock}}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}NFAs\end{tabular}}

A non-deterministic finite automaton \bl{$A(Q, q_0, \delta, F)$} consists of:\medskip

\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 \alert{relation}, \bl{$\delta$}, and

\item there are \alert{silent} transitions\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}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\hspace{5mm}\mbox{{\lstset{language=Scala}\consolas\fontsize{9}{10}\selectfont

\lstinputlisting{../progs/appD.scala}}}

\end{frame}}

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\mode<presentation>{

\begin{frame}[t]

\mbox{}\hspace{-10mm}

\begin{tikzpicture}[scale=0.6,>=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) [above=of q_0] {$q_1$};

\node[state, accepting] (q_2) [below=of q_0] {$q_2$};

\path[->] (q_0) edge node [left]  {\alert{$\epsilon$}} (q_1);

\path[->] (q_0) edge node [left]  {\alert{$\epsilon$}} (q_2);

\path[->] (q_0) edge [loop right] node  {\alert{$a$}} ();

\path[->] (q_1) edge [loop above] node  {\alert{$a$}} ();

\path[->] (q_2) edge [loop below] node  {\alert{$b$}} ();

\end{tikzpicture}

\only<1->{

\begin{textblock}{9}(6,1.5)

\begin{tikzpicture}

\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 

{\normalsize\color{darkgray}

\begin{minipage}{7cm}

\hspace{5mm}\mbox{{\lstset{language=Scala}\consolas\fontsize{8}{10}\selectfont

\lstinputlisting{../progs/appE.scala}}}

\end{minipage}};

\end{tikzpicture}

\end{textblock}}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{Rexp to NFA}

Thompson's construction of a NFA from a regular expression:

\begin{center}

\begin{tabular}[t]{l@{\hspace{10mm}}l}

\raisebox{1mm}{\bl{$\varnothing$}} & 

\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]

\node[state, initial]  (q_0)  {$\mbox{}$};

\end{tikzpicture}\\\\

\raisebox{1mm}{\bl{$\epsilon$}} & 

\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]

\node[state, initial, accepting]  (q_0)  {$\mbox{}$};

\end{tikzpicture}\\\\

\raisebox{2mm}{\bl{$c$}} & 

\begin{tikzpicture}[scale=0.7,>=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]

\node[state, initial]  (q_0)  {$\mbox{}$};

\node[state, accepting]  (q_1)  [right=of q_0] {$\mbox{}$};

\path[->] (q_0) edge node [below]  {\alert{$c$}} (q_1);

\end{tikzpicture}\\\\

\end{tabular}

\end{center}

\end{frame}}

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\mode<presentation>{

\begin{frame}[t]

\frametitle{Case $r_1\cdot r_2$}

\mbox{}\bigskip

\onslide<1>{By recursion we are given two automata:\bigskip}

{\centering\begin{tikzpicture}[node distance=3mm,

                             >=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]

\node[state, initial]  (q_0)  {$\mbox{}$};

\node (r_1)  [right=of q_0] {$\ldots$};

\only<1>{

\node[state, accepting]  (t_1)  [right=of r_1] {$\mbox{}$};

\node[state, accepting]  (t_2)  [above=of t_1] {$\mbox{}$};

\node[state, accepting]  (t_3)  [below=of t_1] {$\mbox{}$};}

\only<2>{

\node[state]  (t_1)  [right=of r_1] {$\mbox{}$};

\node[state]  (t_2)  [above=of t_1] {$\mbox{}$};

\node[state]  (t_3)  [below=of t_1] {$\mbox{}$};}

\only<1>{\node[state, initial]  (a_0)  [right=2.5cm of t_1] {$\mbox{}$};}

\only<2>{\node[state]  (a_0)  [right=2.5cm of t_1] {$\mbox{}$};}

\node (b_1)  [right=of a_0] {$\ldots$};

\node[state, accepting]  (c_1)  [right=of b_1] {$\mbox{}$};

\node[state, accepting]  (c_2)  [above=of c_1] {$\mbox{}$};

\node[state, accepting]  (c_3)  [below=of c_1] {$\mbox{}$};

\only<2>{

\path[->] (t_1) edge node [above, pos=0.3]  {\alert{$\epsilon$}} (a_0);

\path[->] (t_2) edge node [above]  {\alert{$\epsilon$}} (a_0);

\path[->] (t_3) edge node [below]  {\alert{$\epsilon$}} (a_0);

}

\begin{pgfonlayer}{background}

\only<1>{\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (q_0) (r_1) (t_1) (t_2) (t_3)] {};}

\only<1>{\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (a_0) (b_1) (c_1) (c_2) (c_3)] {};}

\only<2>{\node (3) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (q_0) (c_1) (c_2) (c_3)] {};}

\only<1>{\node [yshift=2mm] at (1.north) {\bl{$r_1$}};}

\only<1>{\node [yshift=2mm] at (2.north) {\bl{$r_2$}};}

\only<2>{\node [yshift=2mm] at (3.north) {\bl{$r_1\cdot r_2$}};}

\end{pgfonlayer}

\end{tikzpicture}}\bigskip\bigskip

\small

We need to (1) change the accepting nodes of the first automaton into non-accepting nodes, and (2) connect them

via $\epsilon$-transitions to the starting state of the second automaton. 

\end{frame}}

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

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\mode<presentation>{

\begin{frame}[t]

\frametitle{Case $r_1+ r_2$}

\onslide<1>{By recursion we are given two automata:\smallskip}

\hspace{2cm}\begin{tikzpicture}[node distance=3mm,

                             >=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]

\onslide<1>{\node at (0,0)  (1)  {$\mbox{}$};}

\onslide<2>{\node at (0,0) [state, initial]  (1)  {$\mbox{}$};}

\only<1>{

\node[state, initial]  (2)  [above right=16mm of 1] {$\mbox{}$};

\node[state, initial]  (3)  [below right=16mm of 1] {$\mbox{}$};}

\only<2>{

\node[state]  (2)  [above right=16mm of 1] {$\mbox{}$};

\node[state]  (3)  [below right=16mm of 1] {$\mbox{}$};}

\node (a)  [right=of 2] {$\ldots$};

\node[state, accepting]  (a1)  [right=of a] {$\mbox{}$};

\node[state, accepting]  (a2)  [above=of a1] {$\mbox{}$};

\node[state, accepting]  (a3)  [below=of a1] {$\mbox{}$};

\node (b)  [right=of 3] {$\ldots$};

\node[state, accepting]  (b1)  [right=of b] {$\mbox{}$};

\node[state, accepting]  (b2)  [above=of b1] {$\mbox{}$};

\node[state, accepting]  (b3)  [below=of b1] {$\mbox{}$};

\only<2>{

\path[->] (1) edge node [above]  {\alert{$\epsilon$}} (2);

\path[->] (1) edge node [below]  {\alert{$\epsilon$}} (3);

}

\begin{pgfonlayer}{background}

\only<1>{\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (2) (a1) (a2) (a3)] {};}

\only<1>{\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (3) (b1) (b2) (b3)] {};}

\only<2>{\node (3) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (1) (a2) (a3) (b2) (b3)] {};}

\only<1>{\node [yshift=3mm] at (1.north) {\bl{$r_1$}};}

\only<1>{\node [yshift=3mm] at (2.north) {\bl{$r_2$}};}

\only<2>{\node [yshift=3mm] at (3.north) {\bl{$r_1+ r_2$}};}

\end{pgfonlayer}

\end{tikzpicture}

\small

We (1) need to introduce a new starting state and (2) connect it to the original two starting states.

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{Case $r^*$}

\onslide<1>{By recursion we are given an automaton for $r$:\smallskip}

\hspace{2cm}\begin{tikzpicture}[node distance=3mm,

                             >=stealth',very thick, every state/.style={minimum size=3pt,draw=blue!50,very thick,fill=blue!20},]

\onslide<1>{\node at (0,0)  (1)  {$\mbox{}$};}

\onslide<2->{\node at (0,0) [state, initial,accepting]  (1)  {$\mbox{}$};}

\only<1>{\node[state, initial]  (2)  [right=16mm of 1] {$\mbox{}$};}

\only<2->{\node[state]  (2)  [right=16mm of 1] {$\mbox{}$};}

\node (a)  [right=of 2] {$\ldots$};

\only<1>{

\node[state, accepting]  (a1)  [right=of a] {$\mbox{}$};

\node[state, accepting]  (a2)  [above=of a1] {$\mbox{}$};

\node[state, accepting]  (a3)  [below=of a1] {$\mbox{}$};}

\only<2->{

\node[state]  (a1)  [right=of a] {$\mbox{}$};

\node[state]  (a2)  [above=of a1] {$\mbox{}$};

\node[state]  (a3)  [below=of a1] {$\mbox{}$};}

\only<2->{

\path[->] (1) edge node [above]  {\alert{$\epsilon$}} (2);

\path[->] (a1) edge [bend left=45] node [above]  {\alert{$\epsilon$}} (1);

\path[->] (a2) edge [bend right] node [below]  {\alert{$\epsilon$}} (1);

\path[->] (a3) edge [bend left=45] node [below]  {\alert{$\epsilon$}} (1);

}

\begin{pgfonlayer}{background}

\only<1>{\node (1) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (2) (a1) (a2) (a3)] {};}

\only<2->{\node (2) [rounded corners, inner sep=1mm, thick, draw=black!60, fill=black!20, fit= (1) (a2) (a3)] {};}

\only<1>{\node [yshift=3mm] at (1.north) {\bl{$r$}};}

\only<2->{\node [yshift=3mm] at (2.north) {\bl{$r^*$}};}

\end{pgfonlayer}

\end{tikzpicture}\bigskip

\onslide<3->{

Why can't we just have an epsilon transition from the accepting states to the starting state?}

\end{frame}}

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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=.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}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{Greedy Depth-First}

\hspace{5mm}\mbox{{\lstset{language=Scala}\consolas\fontsize{9}{10}\selectfont

\lstinputlisting{../progs/appG.scala}}}

\end{frame}}

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

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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