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% beamer stuff 
\renewcommand{\slidecaption}{AFL 05, King's College London, 24.~October 2012}
\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\\
  Of$\!$fice: & S1.27 (1st floor Strand Building)\\
  Slides: & KEATS (also home work is there)\\
  \end{tabular}
  \end{center}


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}Deterministic Finite Automata\end{tabular}}

A DFA \bl{$A(Q, q_0, F, \delta)$} 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 some states are accepting states \bl{$F$}
\item a transition function \bl{$\delta$}
\end{itemize}\pause

\onslide<2->{
\begin{center}
\begin{tabular}{l}
\bl{$\hat{\delta}(q, \texttt{""}) = q$}\\
\bl{$\hat{\delta}(q, c\!::\!s) = \hat{\delta}(\delta(q, c), s)$}
\end{tabular}
\end{center}}

\only<3,4>{
\begin{center}
\begin{tikzpicture}[scale=2, line width=0.5mm]
  \node[state, initial]        (q02) at ( 0,1) {$q_{0}$};
  \node[state]                    (q13) at ( 1,1) {$q_{1}$};
  \node[state, accepting] (q4) at ( 2,1) {$q_2$};
  \path[->] (q02) edge[bend left] node[above] {$a$} (q13)
                  (q13) edge[bend left] node[below] {$b$} (q02)
                  (q13) edge node[above] {$a$} (q4)
                  (q02) edge [loop below] node {$b$} ()
                  (q4) edge [loop right] node {$a, b$} ()
            ;
\end{tikzpicture}
\end{center}}%
%
\only<5>{
\begin{center}
\bl{$L(A) \dn \{ s \;|\; \hat{\delta}(q_0, s) \in F\}$}
\end{center}}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[t]
\frametitle{\begin{tabular}{c}Non-Deterministic\\[-1mm] Finite Automata\end{tabular}}

An NFA \bl{$A(Q, q_0, F, \delta)$} consists again of:

\begin{itemize}
\item a finite set of states
\item one of these states is the start state
\item some states are accepting states
\item a transition \alert{relation}\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}\pause\medskip

A string \bl{s} is accepted by an NFA, if there is a ``lucky'' sequence to an accepting state.

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Last Week\end{tabular}}

Last week I showed you\bigskip

\begin{itemize}
\item an algorithm for automata minimisation

\item an algorithm for transforming a regular expression into an NFA

\item an algorithm for transforming an NFA into a DFA (subset construction)

\end{itemize}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}This Week\end{tabular}}

Go over the algorithms again, but with two new things and \ldots\medskip

\begin{itemize}
\item with the example: what is the regular expression that accepts every string, except those ending 
in \bl{aa}?\medskip

\item Go over the proof for \bl{$L(rev(r)) = Rev(L(r))$}.\medskip

\item Anything else so far.
\end{itemize}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Proofs By Induction\end{tabular}}

\begin{itemize}
\item \bl{$P$} holds for \bl{$\varnothing$}, \bl{$\epsilon$} and \bl{c}\bigskip
\item \bl{$P$} holds for \bl{r$_1$ + r$_2$} under the assumption that \bl{$P$} already
holds for \bl{r$_1$} and \bl{r$_2$}.\bigskip
\item \bl{$P$} holds for \bl{r$_1$ $\cdot$ r$_2$} under the assumption that \bl{$P$} already
holds for \bl{r$_1$} and \bl{r$_2$}.
\item \bl{$P$} holds for \bl{r$^*$} under the assumption that \bl{$P$} already
holds for \bl{r}.
\end{itemize}

\begin{center}
\bl{$P(r):\;\;L(rev(r)) = Rev(L(r))$}
\end{center}

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

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

What is the regular expression that accepts every string, except those ending 
in \bl{aa}?\pause\bigskip

\begin{center}
\begin{tabular}{l}
\bl{(a + b)$^*$ba}\\
\bl{(a + b)$^*$ab}\\
\bl{(a + b)$^*$bb}\\\pause
\bl{a}\\
\bl{\texttt{""}}
\end{tabular}
\end{center}\pause

What are the strings to be avoided?\pause\medskip

\begin{center}
\bl{(a + b)$^*$aa}
\end{center}

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

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

An NFA for \bl{(a + b)$^*$aa}

\begin{center}
\begin{tikzpicture}[scale=2, line width=0.5mm]
  \node[state, initial]        (q0) at ( 0,1) {$q_0$};
  \node[state]                    (q1) at ( 1,1) {$q_1$};
  \node[state, accepting] (q2) at ( 2,1) {$q_2$};
  \path[->] (q0) edge node[above] {$a$} (q1)
                  (q1) edge node[above] {$a$} (q2)
                  (q0) edge [loop below] node {$a$} ()
                  (q0) edge [loop above] node {$b$} ()
            ;
\end{tikzpicture}
\end{center}\pause

Minimisation for DFAs\\
Subset Construction for NFAs

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}DFA Minimisation\end{tabular}}


\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 nothing changed.
\item All unmarked pairs can be merged.
\end{enumerate}

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

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

Minimal DFA \only<1>{\bl{(a + b)$^*$aa}}\only<2->{\alert{not} \bl{(a + b)$^*$aa}}

\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}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}[c]
\frametitle{\begin{tabular}{c}Algorithms on Automata\end{tabular}}


\begin{itemize}
\item Reg $\rightarrow$ NFA: Thompson-McNaughton-Yamada method\medskip
\item NFA $\rightarrow$ DFA: Subset Construction\medskip
\item DFA $\rightarrow$ Reg: Brzozowski's Algebraic Method\medskip
\item DFA minimisation: Hopcrofts Algorithm\medskip
\item complement DFA
\end{itemize}

\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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