--- a/Slides/Slides1.thy	Wed Aug 24 08:03:42 2011 +0000
+++ b/Slides/Slides1.thy	Wed Aug 24 10:01:24 2011 +0000
@@ -314,18 +314,45 @@
   \begin{frame}[c]
   \frametitle{\LARGE The Myhill-Nerode Theorem}
 
-  \mbox{}\\[5cm]
-
+  \begin{center}
+  \only<1>{%
+  \begin{tikzpicture}[scale=3]
+  \draw[very thick] (0.5,0.5) circle (.6cm);
+  \end{tikzpicture}}%
+  \only<2->{%
+  \begin{tikzpicture}[scale=3]
+  \draw[very thick] (0.5,0.5) circle (.6cm);
+  \clip[draw] (0.5,0.5) circle (.6cm);
+  \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
+  \end{tikzpicture}}
+  \end{center}
+  
   \begin{itemize}
   \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
   \end{itemize}
 
-  \only<2->{
+  \begin{textblock}{5}(2.1,5.3)
+  \begin{tikzpicture}
+  \node at (0,0) [single arrow, fill=red,text=white, minimum height=2cm]
+  {$U\!N\!IV$};
+  \draw (-0.3,-1.1) node {\begin{tabular}{l}set of all\\[-1mm] strings\end{tabular}};
+  \end{tikzpicture}
+  \end{textblock}
+
+  \only<2->{%
+  \begin{textblock}{5}(9.4,7.2)
+  \begin{tikzpicture}
+  \node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]
+  {@{text "\<lbrakk>x\<rbrakk>"}$_{\approx_{A}}$};
+  \draw (0.9,-1.1) node {\begin{tabular}{l}equivalence class\end{tabular}};
+  \end{tikzpicture}
+  \end{textblock}}
+
+  \only<3->{
   \begin{textblock}{11.9}(1.7,3)
   \begin{block}{}
   \begin{minipage}{11.4cm}\raggedright
   Two directions:\medskip\\
-
   \begin{tabular}{@ {}ll}
   1.)\;finite $\Rightarrow$ regular\\
   \;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_A) \Rightarrow \exists r.\;A = {\cal L}(r)}\\[3mm]
@@ -347,17 +374,47 @@
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
   \begin{frame}[c]
-  \frametitle{\LARGE Final States}
+  \frametitle{\LARGE Initial and Final {\sout{\textcolor{gray}{States}}}}
+
+  \begin{textblock}{8}(10, 2)
+  \textcolor{black}{Equivalence Classes}
+  \end{textblock}
+
 
-  \mbox{}\\[3cm]
+  \begin{center}
+  \begin{tikzpicture}[scale=3]
+  \draw[very thick] (0.5,0.5) circle (.6cm);
+  \clip[draw] (0.5,0.5) circle (.6cm);
+  \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
+  \only<2->{\draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);}
+  \only<3->{\draw[red, fill] (0.2, 0.2) rectangle (0.4, 0.4);
+  \draw[red, fill] (0.4, 0.8) rectangle (0.6, 1.0);
+  \draw[red, fill] (0.6, 0.0) rectangle (0.8, 0.2);
+  \draw[red, fill] (0.8, 0.4) rectangle (1.0, 0.6);}
+  \end{tikzpicture}
+  \end{center}
 
   \begin{itemize}
-  \item \smath{\text{final}_A\,X \dn}\\
-  \smath{\hspace{6mm}X \in (U\!N\!IV /\!/\approx_A) \;\wedge\; \forall s \in X.\; s \in A}
+  \item \smath{\text{final}_A\,X \dn \{[\!|x|\!]_{\approx_{A}}\;|\;x \in A\}}
   \smallskip
   \item we can prove: \smath{A = \bigcup \{X.\;\text{final}_A\,X\}}
+  \end{itemize}
 
-  \end{itemize}
+  \only<2->{%
+  \begin{textblock}{5}(2.1,4.6)
+  \begin{tikzpicture}
+  \node at (0,0) [single arrow, fill=blue,text=white, minimum height=2cm]
+  {$[] \in X$};
+  \end{tikzpicture}
+  \end{textblock}}
+
+  \only<3->{%
+  \begin{textblock}{5}(10,7.4)
+  \begin{tikzpicture}
+  \node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]
+  {a final};
+  \end{tikzpicture}
+  \end{textblock}}
 
   \end{frame}}
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
@@ -367,40 +424,33 @@
 text_raw {*
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{\LARGE Transitions between Eq-Classes}
+  \begin{frame}<-1>[c]
+  \frametitle{\begin{tabular}{@ {}l}\LARGE% 
+  Transitions between Eq-Classes\end{tabular}}
 
-  \smath{L = \{[c]\}}
+  \begin{center}
+  \begin{tikzpicture}[scale=3]
+  \draw[very thick] (0.5,0.5) circle (.6cm);
+  \clip[draw] (0.5,0.5) circle (.6cm);
+  \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
+  \draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);
+  \draw[blue, fill] (0.8, 0.4) rectangle (1.0, 0.6);
+  \draw[white] (0.1,0.7) node {$X$};
+  \draw[white] (0.9,0.5) node {$Y$};
+  \end{tikzpicture}
+  \end{center}
 
-  \begin{tabular}{@ {\hspace{-7mm}}cc}
+  \begin{center}
+  \smath{X \stackrel{c}{\longrightarrow} Y \;\dn\; X ; c \subseteq Y}
+  \end{center}
+
+  \onslide<8>{
   \begin{tabular}{c}
   \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
   \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
-
-  %\draw[help lines] (0,0) grid (3,2);
-
-  \node[state,initial]   (q_0)                        {$R_1$};
-  \node[state,accepting] (q_1) [above right of=q_0]   {$R_2$};
-  \node[state]           (q_2) [below right of=q_0]   {$R_3$};
-
-  \path[->] (q_0) edge                node        {c} (q_1)
-                  edge                node [swap] {$\Sigma-{c}$} (q_2)
-            (q_2) edge [loop below]   node        {$\Sigma$} ()
-            (q_1) edge                node        {$\Sigma$} (q_2);
+  \node[state,initial] (q_0) {$R_1$};
   \end{tikzpicture}
-  \end{tabular}
-  &
-  \begin{tabular}[t]{ll}
-  \\[-20mm]
-  \multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm]
-
-  \smath{R_1}: & \smath{\{[]\}}\\
-  \smath{R_2}: & \smath{\{[c]\}}\\
-  \smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm]
-  \multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ; [c] \subseteq Y}}}
-  \end{tabular}
-
-  \end{tabular}
+  \end{tabular}}
 
   \end{frame}}
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%