# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1381496555 -3600
# Node ID 69cec773736b8b406cf8cf1fb6ad1ffb2d7303d8
# Parent  cc19e6cbcb6884ce8eeadeae4884e64ba21ba29e
added pictures

diff -r cc19e6cbcb68 -r 69cec773736b slides/slides03.pdf
Binary file slides/slides03.pdf has changed
diff -r cc19e6cbcb68 -r 69cec773736b slides/slides03.tex
--- a/slides/slides03.tex	Wed Oct 09 20:54:52 2013 +0100
+++ b/slides/slides03.tex	Fri Oct 11 14:02:35 2013 +0100
@@ -18,6 +18,8 @@
 \usetikzlibrary{shadows}
 \usetikzlibrary{positioning}
 \usetikzlibrary{calc}
+\usetikzlibrary{fit}
+\usetikzlibrary{backgrounds}
 \usepackage{graphicx} 
 
 \definecolor{javared}{rgb}{0.6,0,0} % for strings
@@ -436,9 +438,25 @@
 \begin{frame}[c]
 
 \begin{center}
-\includegraphics[scale=0.7]{pics/ch3.jpg}
+\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}\pause
 
+
 \begin{itemize}
 \item start can be an accepting state
 \item it is possible that there is no accepting state
@@ -453,15 +471,30 @@
 \begin{frame}[c]
 
 \begin{center}
-\includegraphics[scale=0.7]{pics/ch3.jpg}
+\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}
 
 for this automaton \bl{$\delta$} is the function\\
 
 \begin{center}
 \begin{tabular}{lll}
-\bl{(q$_0$, a) $\rightarrow$ q$_1$} & \bl{(q$_1$, a) $\rightarrow$ q$_4$} & \bl{(q$_4$, a) $\rightarrow$ q$_4$}\\
-\bl{(q$_0$, b) $\rightarrow$ q$_2$} & \bl{(q$_1$, b) $\rightarrow$ q$_2$} & \bl{(q$_4$, b) $\rightarrow$ q$_4$}\\
+\bl{$(q_0, a) \rightarrow q_1$} & \bl{$(q_1, a) \rightarrow q_4$} & \bl{$(q_4, a) \rightarrow q_4$}\\
+\bl{$(q_0, b) \rightarrow q_2$} & \bl{$(q_1, b) \rightarrow q_2$} & \bl{$(q_4, b) \rightarrow q_4$}\\
 \end{tabular}\ldots
 \end{center}
 
@@ -515,12 +548,12 @@
 
 \begin{center}
 \begin{tabular}{c}
-\bl{(q$_1$, a) $\rightarrow$ q$_2$}\\
-\bl{(q$_1$, a) $\rightarrow$ q$_3$}\\
+\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$_2$}\\
+\bl{$(q_1, \epsilon) \rightarrow q_2$}\\
 \end{tabular}
 \end{center}
 
@@ -530,7 +563,7 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
-\frametitle{An NFA}
+\frametitle{An NFA Example}
 
 \begin{center}
 \begin{tikzpicture}[scale=0.7,>=stealth',very thick,
@@ -549,12 +582,24 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
+\frametitle{Rexp to NFA}
 
 \begin{center}
-\begin{tabular}[b]{ll}
-\bl{$\varnothing$} & \includegraphics[scale=0.7]{pics/NULL.jpg}\\\\
-\bl{$\epsilon$} & \includegraphics[scale=0.7]{pics/epsilon.jpg}\\\\
-\bl{c} & \includegraphics[scale=0.7]{pics/char.jpg}\\
+\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}
 
@@ -563,33 +608,102 @@
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
-\begin{frame}[c]
+\begin{frame}[t]
+\frametitle{Case $r_1\cdot r_2$}
+
+\mbox{}\bigskip
+\onslide<1>{By recursion we are given two automata:\bigskip}
+
+\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. 
 
-\begin{center}
-\begin{tabular}[t]{ll}
-\bl{r$_1$ $\cdot$ r$_2$} & \includegraphics[scale=0.6]{pics/seq.jpg}\\\\
-\end{tabular}
-\end{center}
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+\frametitle{Case $r_1+ r_2$}
+
+\onslide<1>{By recursion we are given two automata:\smallskip}
+
+\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}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
-
-\begin{center}
-\begin{tabular}[t]{ll}
-\bl{r$_1$ + r$_2$} & \includegraphics[scale=0.7]{pics/alt.jpg}\\\\
-\end{tabular}
-\end{center}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
+\frametitle{Rexp to NFA}
 
 \begin{center}
 \begin{tabular}[b]{ll}