# HG changeset patch
# User Christian Urban <christian.urban@kcl.ac.uk>
# Date 1603113438 -3600
# Node ID 06cbaaad3ba886f5aa1d8298832792c27e7e3b3b
# Parent  a26a20acd1c2a301f7e397992187f12999408015
updated

diff -r a26a20acd1c2 -r 06cbaaad3ba8 slides/slides03.pdf
Binary file slides/slides03.pdf has changed
diff -r a26a20acd1c2 -r 06cbaaad3ba8 slides/slides03.tex
--- a/slides/slides03.tex	Sat Oct 17 13:14:19 2020 +0100
+++ b/slides/slides03.tex	Mon Oct 19 14:17:18 2020 +0100
@@ -1269,8 +1269,8 @@
 \begin{center}
 \begin{tikzpicture}[scale=2,>=stealth',very thick,
                              every state/.style={minimum size=0pt,draw=blue!50,very thick,fill=blue!20},]
-  \only<1->{\node[state, initial]        (q0) at ( 0,1) {$\mbox{Q}_0$};}
-  \only<1->{\node[state]                    (q1) at ( 1,1) {$\mbox{Q}_1$};}
+  \only<1->{\node[state, initial,accepting]        (q0) at ( 0,1) {$\mbox{Q}_0$};}
+  \only<1->{\node[state,accepting]                    (q1) at ( 1,1) {$\mbox{Q}_1$};}
   \only<1->{\node[state] (q2) at ( 2,1) {$\mbox{Q}_2$};}
   \path[->] (q0) edge[bend left] node[above] {\alert{$a$}} (q1)
                   (q1) edge[bend left] node[above] {\alert{$b$}} (q0)
@@ -1327,12 +1327,96 @@
 \end{center}
 }
 
-\onslide<3->{
+
+\only<3-9>{\small
+\begin{textblock}{6}(1,0.8)
+\begin{bubble}[6.7cm]
+\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
+\multicolumn{3}{@{}l}{substitute \bl{$\mbox{Q}_1$} into \bl{$\mbox{Q}_0$} \& \bl{$\mbox{Q}_2$}:}\\    
+\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,b + \mbox{Q}_0\,a\,b +  \mbox{Q}_2\,b + \ONE$}\\
+\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a + \mbox{Q}_2\,a$}
+\end{tabular}
+\end{bubble}
+\end{textblock}}
+
+\only<4-9>{\small
+\begin{textblock}{6}(2,4.15)
+\begin{bubble}[6.7cm]
+\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
+\multicolumn{3}{@{}l}{simplifying \bl{$\mbox{Q}_0$}:}\\    
+\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,(b + a\,b) + \mbox{Q}_2\,b + \ONE$}\\
+\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a + \mbox{Q}_2\,a$}
+\end{tabular}
+\end{bubble}
+\end{textblock}}
+
+\only<6-9>{\small
+\begin{textblock}{6}(3,7.55)
+\begin{bubble}[6.7cm]
+\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
+  \multicolumn{3}{@{}l}{Arden for \bl{$\mbox{Q}_2$}:}\\    
+\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,(b + a\,b) + \mbox{Q}_2\,b + \ONE$}\\
+\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a\,(a^*)$}
+\end{tabular}
+\end{bubble}
+\end{textblock}}
+
+\only<7-9>{\small
+\begin{textblock}{6}(4,10.9)
+\begin{bubble}[7.5cm]
+\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
+  \multicolumn{3}{@{}l}{Substitute \bl{$\mbox{Q}_2$} and simplify:}\\    
+\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,(b + a\,b + a\,a\,(a^*)\,b) + \ONE$}\\
+\end{tabular}
+\end{bubble}
+\end{textblock}}
+
+\only<8-9>{\small
+\begin{textblock}{6}(5,13.4)
+\begin{bubble}[7.5cm]
+\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
+  \multicolumn{3}{@{}l}{Arden again for \bl{$\mbox{Q}_0$}:}\\    
+\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*$}\\
+\end{tabular}
+\end{bubble}
+\end{textblock}}
+
+
+\only<9-10>{\small
+\begin{textblock}{6}(6,11.5)
+\begin{bubble}[6.7cm]
+\begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
+\multicolumn{3}{@{}l}{Finally:}\\    
+\bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*$}\\
+\bl{$\mbox{Q}_1$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*\,a$}\\
+\bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*\,a\,a\,(a^*)$}\\
+\end{tabular}
+\end{bubble}
+\end{textblock}}
+
+
+
+
+
+\only<5-6>{
+\begin{textblock}{6}(0.7,11.9)
+\begin{bubble}[6.7cm]
 Arden's Lemma:
 \begin{center}
 If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
 \end{center}
-}
+\end{bubble}
+\end{textblock}}
+
+\only<8>{
+\begin{textblock}{6}(1.1,7.8)
+\begin{bubble}[6.7cm]
+Arden's Lemma:
+\begin{center}
+If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}
+\end{center}
+\end{bubble}
+\end{textblock}}
 
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
@@ -1376,35 +1460,35 @@
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}[c]
-
-Given the function 
-
-\begin{center}
-\bl{\begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}
-$rev(\ZERO)$   & $\dn$ & $\ZERO$\\
-$rev(\ONE)$         & $\dn$ & $\ONE$\\
-$rev(c)$                      & $\dn$ & $c$\\
-$rev(r_1 + r_2)$        & $\dn$ & $rev(r_1) + rev(r_2)$\\
-$rev(r_1 \cdot r_2)$  & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\
-$rev(r^*)$                   & $\dn$ & $rev(r)^*$\\
-\end{tabular}}
-\end{center}
-
-
-and the set
-
-\begin{center}
-\bl{$Rev\,A \dn \{s^{-1} \;|\; s \in A\}$}
-\end{center}
-
-prove whether
-
-\begin{center}
-\bl{$L(rev(r)) = Rev (L(r))$}
-\end{center}
-
-\end{frame}
+%\begin{frame}[c]
+%
+%Given the function 
+%
+%\begin{center}
+%\bl{\begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}
+%$rev(\ZERO)$   & $\dn$ & $\ZERO$\\
+%$rev(\ONE)$         & $\dn$ & $\ONE$\\
+%$rev(c)$                      & $\dn$ & $c$\\
+%$rev(r_1 + r_2)$        & $\dn$ & $rev(r_1) + rev(r_2)$\\
+%$rev(r_1 \cdot r_2)$  & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\
+%$rev(r^*)$                   & $\dn$ & $rev(r)^*$\\
+%\end{tabular}}
+%\end{center}
+%
+%
+%and the set
+%
+%\begin{center}
+%\bl{$Rev\,A \dn \{s^{-1} \;|\; s \in A\}$}
+%\end{center}
+%
+%prove whether
+%
+%\begin{center}
+%\bl{$L(rev(r)) = Rev (L(r))$}
+%\end{center}
+%
+%\end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%