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

\usepackage{../langs}

\usepackage{../data}

% beamer stuff 

\begin{document}

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\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}

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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

\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 All unmarked pairs can be merged.

\end{enumerate}

\end{frame}}

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\mode<presentation>{

\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}}

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\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}}

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\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}}

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\mode<presentation>{

\begin{frame}[c]

\mbox{\lstinputlisting[language=while]{../progs/fib.while}}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\mbox{\lstinputlisting[language=while]{../progs/collatz.while}}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\mbox{\lstinputlisting[language=while]{../progs/collatz.while}}

\begin{textblock}{6}(10,2)

\begin{tikzpicture}[scale=0.46]

\begin{axis}[axis x line=bottom, axis y line=left, ylabel=secs,

    xlabel=n,

    enlargelimits=0.05,

    ybar interval=0.7, legend style=small]

\addplot file {interpreted2.data};

\addplot file {compiled2.data};

%\legend{interpreted, compiled}

\end{axis}

\end{tikzpicture}

\end{textblock}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\begin{tikzpicture}[scale=1]

  \draw[line width=1mm] (-.3, 0) rectangle (1.5,2);

  \draw (4.2,1) node {Code Gen};

  \draw (0.6,1.7) node {\footnotesize Parser};

  \draw (-2.7,1.7) node {\footnotesize Lexer};

  \draw[line width=1mm] (-1.8, 0) rectangle (-3.6,2);

  \draw[white] (1.7,1) node (X) {};

  \draw[white] (3.2,1) node (Y) {};

  \draw[red, ->, line width = 2mm] (X) -- (Y);

  \draw[red, <-, line width = 2mm] (-0.6,1) -- (-1.6,1);

  \draw[red, <-, line width = 2mm] (-3.8,1) -- (-4.8,1);

\end{tikzpicture}

\end{frame}}

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\mode<presentation>{

\begin{frame}[t]

\consolas

\begin{center}

"if true then then 42 else +"

\end{center}

\begin{tabular}{@{}l}

KEYWORD: \\

\hspace{5mm}{if}, {then}, {else},\\ 

WHITESPACE:\\

\hspace{5mm}{" "}, {$\backslash$n},\\ 

IDENT:\\

\hspace{5mm}LETTER $\cdot$ (LETTER + DIGIT + {\_})$^*$\\ 

NUM:\\

\hspace{5mm}(NONZERODIGIT $\cdot$ DIGIT$^*$) + {0}\\

OP:\\

\hspace{5mm}{+}\\

COMMENT:\\

\hspace{5mm}{$\slash$*} $\cdot$ (ALL$^*$ $\cdot$ {*$\slash$} $\cdot$ ALL$^*$) $\cdot$ {*$\slash$}

\end{tabular}

\end{frame}}

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\mode<presentation>{

\begin{frame}[t]

\consolas

\begin{center}

"if true then then 42 else +"

\end{center}

\only<1>{

\small\begin{tabular}{l}

KEYWORD(if),\\ 

WHITESPACE,\\ 

IDENT(true),\\ 

WHITESPACE,\\ 

KEYWORD(then),\\ 

WHITESPACE,\\ 

KEYWORD(then),\\ 

WHITESPACE,\\ 

NUM(42),\\ 

WHITESPACE,\\ 

KEYWORD(else),\\ 

WHITESPACE,\\ 

OP(+)

\end{tabular}}

\only<2>{

\small\begin{tabular}{l}

KEYWORD(if),\\ 

IDENT(true),\\ 

KEYWORD(then),\\ 

KEYWORD(then),\\ 

NUM(42),\\ 

KEYWORD(else),\\ 

OP(+)

\end{tabular}}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

There is one small problem with the tokenizer. How should we 

tokenize:

\begin{center}

{\consolas "x - 3"}

\end{center}

\consolas

\begin{tabular}{@{}l}

OP:\\

\hspace{5mm}\texttt{"+"}, \texttt{"-"}\\

NUM:\\

\hspace{5mm}(NONZERODIGIT $\cdot$ DIGIT$^*$) + {''0''}\\

NUMBER:\\

\hspace{5mm}NUM +  (\texttt{"-"} $\cdot$ NUM)\\

\end{tabular}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\frametitle{\begin{tabular}{c}Two Rules\end{tabular}}

\begin{itemize}

\item Longest match rule (``maximal munch rule''): The 

longest initial substring matched by any regular expression is taken

as next token.\bigskip

\item Rule priority:

For a particular longest initial substring, the first regular

expression that can match determines the token.

\end{itemize}

%\url{http://www.technologyreview.com/tr10/?year=2011}

%finite deterministic automata/ nondeterministic automaton

%\item problem with infix operations, for example i-12

\end{frame}}

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\mode<presentation>{

\begin{frame}[t]

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

\small

\ldots{}whether a regular expression can match the empty string:

\begin{center}

\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}

\bl{$nullable(\varnothing)$}      & \bl{$\dn$} & \bl{$f\!\/alse$}\\

\bl{$nullable(\epsilon)$}           & \bl{$\dn$} &  \bl{$true$}\\

\bl{$nullable (c)$}                    & \bl{$\dn$} &  \bl{$f\!alse$}\\

\bl{$nullable (r_1 + r_2)$}       & \bl{$\dn$} &  \bl{$nullable(r_1) \vee nullable(r_2)$} \\ 

\bl{$nullable (r_1 \cdot r_2)$} & \bl{$\dn$} &  \bl{$nullable(r_1) \wedge nullable(r_2)$} \\

\bl{$nullable (r^*)$}                 & \bl{$\dn$} & \bl{$true$} \\

\end{tabular}

\end{center}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

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

\small

\ldots{}whether a regular expression can match nothing:

\begin{center}

\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}

\bl{$zeroable(\varnothing)$}      & \bl{$\dn$} & \bl{$true$}\\

\bl{$zeroable(\epsilon)$}           & \bl{$\dn$} &  \bl{$f\!alse$}\\

\bl{$zeroable (c)$}                    & \bl{$\dn$} &  \bl{$f\!alse$}\\

\bl{$zeroable (r_1 + r_2)$}       & \bl{$\dn$} &  \bl{$zeroable(r_1) \wedge zeroable(r_2)$} \\ 

\bl{$zeroable (r_1 \cdot r_2)$} & \bl{$\dn$} &  \bl{$zeroable(r_1) \vee zeroable(r_2)$} \\

\bl{$zeroable (r^*)$}                 & \bl{$\dn$} & \bl{$f\!alse$} \\

\end{tabular}

\end{center}\bigskip\pause

\begin{center}

\bl{$zeroable(r) \Leftrightarrow L(r) = \varnothing$}

\end{center}

\end{frame}}

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\mode<presentation>{

\begin{frame}[c]

\begin{itemize}

\item The star-case in our proof about the matcher needs the following lemma

\begin{center}

\bl{$Der\,c\,A^* = (Der\,c\,A)\,@\, A^*$}

\end{center}

\end{itemize}\bigskip\bigskip

\begin{itemize}

\item \bl{$A^* = \{""\} \cup A\,@\,A^*$}

\item If \bl{\texttt{""} $\in A$}, then\\ \bl{$Der\,c\,(A @ B) = (Der\,c\,A) @ B \cup (Der\,c\,B)$}\medskip

\item If \bl{\texttt{""} $\not\in A$}, then\\ \bl{$Der\,c\,(A @ B) = (Der\,c\,A) @ B$}