317   \mbox{}\\[5cm]

   317   \begin{center}

   318 

   318   \only<1>{%

   319   \begin{tikzpicture}[scale=3]

   320   \draw[very thick] (0.5,0.5) circle (.6cm);

   321   \end{tikzpicture}}%

   322   \only<2->{%

   323   \begin{tikzpicture}[scale=3]

   324   \draw[very thick] (0.5,0.5) circle (.6cm);

   325   \clip[draw] (0.5,0.5) circle (.6cm);

   326   \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);

   327   \end{tikzpicture}}

   328   \end{center}

   329   

   323   \only<2->{

   334   \begin{textblock}{5}(2.1,5.3)

   335   \begin{tikzpicture}

   336   \node at (0,0) [single arrow, fill=red,text=white, minimum height=2cm]

   337   {$U\!N\!IV$};

   338   \draw (-0.3,-1.1) node {\begin{tabular}{l}set of all\\[-1mm] strings\end{tabular}};

   339   \end{tikzpicture}

   340   \end{textblock}

   341 

   342   \only<2->{%

   343   \begin{textblock}{5}(9.4,7.2)

   344   \begin{tikzpicture}

   345   \node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]

   346   {@{text "\<lbrakk>x\<rbrakk>"}$_{\approx_{A}}$};

   347   \draw (0.9,-1.1) node {\begin{tabular}{l}equivalence class\end{tabular}};

   348   \end{tikzpicture}

   349   \end{textblock}}

   350 

   351   \only<3->{

   350   \frametitle{\LARGE Final States}

   377   \frametitle{\LARGE Initial and Final {\sout{\textcolor{gray}{States}}}}

   351 

   378 

   352   \mbox{}\\[3cm]

   379   \begin{textblock}{8}(10, 2)

   353 

   380   \textcolor{black}{Equivalence Classes}

   354   \begin{itemize}

   381   \end{textblock}

   355   \item \smath{\text{final}_A\,X \dn}\\

   382 

   356   \smath{\hspace{6mm}X \in (U\!N\!IV /\!/\approx_A) \;\wedge\; \forall s \in X.\; s \in A}

   383 

   384   \begin{center}

   385   \begin{tikzpicture}[scale=3]

   386   \draw[very thick] (0.5,0.5) circle (.6cm);

   387   \clip[draw] (0.5,0.5) circle (.6cm);

   388   \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);

   389   \only<2->{\draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);}

   390   \only<3->{\draw[red, fill] (0.2, 0.2) rectangle (0.4, 0.4);

   391   \draw[red, fill] (0.4, 0.8) rectangle (0.6, 1.0);

   392   \draw[red, fill] (0.6, 0.0) rectangle (0.8, 0.2);

   393   \draw[red, fill] (0.8, 0.4) rectangle (1.0, 0.6);}

   394   \end{tikzpicture}

   395   \end{center}

   396 

   397   \begin{itemize}

   398   \item \smath{\text{final}_A\,X \dn \{[\!|x|\!]_{\approx_{A}}\;|\;x \in A\}}

   359 

   401   \end{itemize}

   360   \end{itemize}

   402 

   361 

   403   \only<2->{%

   362   \end{frame}}

   404   \begin{textblock}{5}(2.1,4.6)

   363   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

   405   \begin{tikzpicture}

   364 *}

   406   \node at (0,0) [single arrow, fill=blue,text=white, minimum height=2cm]

   365 

   407   {$[] \in X$};

   366 

   408   \end{tikzpicture}

   367 text_raw {*

   409   \end{textblock}}

   368   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   410 

   369   \mode<presentation>{

   411   \only<3->{%

   370   \begin{frame}[c]

   412   \begin{textblock}{5}(10,7.4)

   371   \frametitle{\LARGE Transitions between Eq-Classes}

   413   \begin{tikzpicture}

   372 

   414   \node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]

   373   \smath{L = \{[c]\}}

   415   {a final};

   374 

   416   \end{tikzpicture}

   375   \begin{tabular}{@ {\hspace{-7mm}}cc}

   417   \end{textblock}}

   418 

   419   \end{frame}}

   420   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

   421 *}

   422 

   423 

   424 text_raw {*

   425   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   426   \mode<presentation>{

   427   \begin{frame}<-1>[c]

   428   \frametitle{\begin{tabular}{@ {}l}\LARGE% 

   429   Transitions between Eq-Classes\end{tabular}}

   430 

   431   \begin{center}

   432   \begin{tikzpicture}[scale=3]

   433   \draw[very thick] (0.5,0.5) circle (.6cm);

   434   \clip[draw] (0.5,0.5) circle (.6cm);

   435   \draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);

   436   \draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);

   437   \draw[blue, fill] (0.8, 0.4) rectangle (1.0, 0.6);

   438   \draw[white] (0.1,0.7) node {$X$};

   439   \draw[white] (0.9,0.5) node {$Y$};

   440   \end{tikzpicture}

   441   \end{center}

   442 

   443   \begin{center}

   444   \smath{X \stackrel{c}{\longrightarrow} Y \;\dn\; X ; c \subseteq Y}

   445   \end{center}

   446 

   447   \onslide<8>{

   379 

   451   \node[state,initial] (q_0) {$R_1$};

   380   %\draw[help lines] (0,0) grid (3,2);

   452   \end{tikzpicture}

   381 

   453   \end{tabular}}

   382   \node[state,initial]   (q_0)                        {$R_1$};

   383   \node[state,accepting] (q_1) [above right of=q_0]   {$R_2$};

   384   \node[state]           (q_2) [below right of=q_0]   {$R_3$};

   385 

   386   \path[->] (q_0) edge                node        {c} (q_1)

   387                   edge                node [swap] {$\Sigma-{c}$} (q_2)

   388             (q_2) edge [loop below]   node        {$\Sigma$} ()

   389             (q_1) edge                node        {$\Sigma$} (q_2);

   390   \end{tikzpicture}

   391   \end{tabular}

   392   &

   393   \begin{tabular}[t]{ll}

   394   \\[-20mm]

   395   \multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm]

   396 

   397   \smath{R_1}: & \smath{\{[]\}}\\

   398   \smath{R_2}: & \smath{\{[c]\}}\\

   399   \smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm]

   400   \multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ; [c] \subseteq Y}}}

   401   \end{tabular}

   402 

   403   \end{tabular}