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% beamer stuff 
\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

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1>[t]
\frametitle{%
  \begin{tabular}{@ {}c@ {}}
  \\[-3mm]
  \LARGE Automata and \\[-2mm] 
  \LARGE Formal Languages (5)\\[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}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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{$c$} 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{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}<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{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{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<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}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]

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

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]

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

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




\newcommand{\qq}{\mbox{\texttt{"}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Grammars\end{tabular}}

\begin{center}
\bl{\begin{tabular}{lcl}
$E$ & $\rightarrow$ &  $F + (F \cdot \qq*\qq \cdot F) + (F \cdot \qq\backslash\qq \cdot F)$\\
$F$ & $\rightarrow$ & $T + (T \cdot \qq\texttt{+}\qq \cdot T) + (T \cdot \qq\texttt{-}\qq \cdot T)$\\
$T$ & $\rightarrow$ & $num + (\qq\texttt{(}\qq \cdot E \cdot \qq\texttt{)}\qq)$\\
\end{tabular}}
\end{center}

\bl{$E$}, \bl{$F$} and \bl{$T$} are non-terminals\\
\bl{$E$} is start symbol\\
\bl{$num$}, \bl{(}, \bl{)}, \bl{+} \ldots are terminals\bigskip\\


\bl{\texttt{(2*3)+(3+4)}}

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]

\begin{center}
\bl{\begin{tabular}{lcl}
$E$ & $\rightarrow$ &  $F + (F \cdot \qq*\qq \cdot F) + (F \cdot \qq\backslash\qq \cdot F)$\\
$F$ & $\rightarrow$ & $T + (T \cdot \qq\texttt{+}\qq \cdot T) + (T \cdot \qq\texttt{-}\qq \cdot T)$\\
$T$ & $\rightarrow$ & $num + (\qq\texttt{(}\qq \cdot E \cdot \qq\texttt{)}\qq)$\\
\end{tabular}}
\end{center}

\begin{center}
\begin{tikzpicture}[level distance=8mm, blue]
  \node {E}
    child {node {F} 
     child {node {T} 
                 child {node {\qq(\qq\,E\,\qq)\qq}
                            child {node{F \qq*\qq{} F}
                                  child {node {T} child {node {2}}}
                                  child {node {T} child {node {3}}} 
                               }
                          }
              }
     child {node {\qq+\qq}}
     child {node {T}
       child {node {\qq(\qq\,E\,\qq)\qq} 
       child {node {F}
       child {node {T \qq+\qq{} T}
                    child {node {3}}
                    child {node {4}} 
                 }
                 }}
    }};
\end{tikzpicture}
\end{center}

\begin{textblock}{5}(1, 5)
\bl{\texttt{(2*3)+(3+4)}}
\end{textblock}

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

\end{document}

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