\usepackage{beamerthemeplaincu}

%\usepackage[T1]{fontenc}

%\usepackage[latin1]{inputenc}

\makeatletter

\lst@CCPutMacro\lst@ProcessOther {"2D}{\lst@ttfamily{-{}}{-{}}}

\@empty\z@\@empty

\makeatother

	basicstyle=\consolas,

    type,val,var,while,with,yield, then},

  otherkeywords={=>,<-,<\%,<:,>:,\#,@,->},

	basicstyle=\consolas,

\renewcommand{\slidecaption}{AFL 05, King's College London, 23.~October 2013}

  Office: & S1.27 (1st floor Strand Building)\\

\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{$c$} 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}

\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}

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

\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}

\begin{frame}<1-2>[c]

\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{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}

\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}

\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}

\texttt{\consolas\lstinputlisting{../progs/fib.while}}

\texttt{\consolas\lstinputlisting{../progs/collatz.while}}