1272   \only<1->{\node[state, initial]        (q0) at ( 0,1) {$\mbox{Q}_0$};}

  1272   \only<1->{\node[state, initial,accepting]        (q0) at ( 0,1) {$\mbox{Q}_0$};}

  1273   \only<1->{\node[state]                    (q1) at ( 1,1) {$\mbox{Q}_1$};}

  1273   \only<1->{\node[state,accepting]                    (q1) at ( 1,1) {$\mbox{Q}_1$};}

  1330 \onslide<3->{

  1330 

  1331 \only<3-9>{\small

  1332 \begin{textblock}{6}(1,0.8)

  1333 \begin{bubble}[6.7cm]

  1334 \begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}

  1335 \multicolumn{3}{@{}l}{substitute \bl{$\mbox{Q}_1$} into \bl{$\mbox{Q}_0$} \& \bl{$\mbox{Q}_2$}:}\\    

  1336 \bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,b + \mbox{Q}_0\,a\,b +  \mbox{Q}_2\,b + \ONE$}\\

  1337 \bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a + \mbox{Q}_2\,a$}

  1338 \end{tabular}

  1339 \end{bubble}

  1340 \end{textblock}}

  1341 

  1342 \only<4-9>{\small

  1343 \begin{textblock}{6}(2,4.15)

  1344 \begin{bubble}[6.7cm]

  1345 \begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}

  1346 \multicolumn{3}{@{}l}{simplifying \bl{$\mbox{Q}_0$}:}\\    

  1347 \bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,(b + a\,b) + \mbox{Q}_2\,b + \ONE$}\\

  1348 \bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a + \mbox{Q}_2\,a$}

  1349 \end{tabular}

  1350 \end{bubble}

  1351 \end{textblock}}

  1352 

  1353 \only<6-9>{\small

  1354 \begin{textblock}{6}(3,7.55)

  1355 \begin{bubble}[6.7cm]

  1356 \begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}

  1357   \multicolumn{3}{@{}l}{Arden for \bl{$\mbox{Q}_2$}:}\\    

  1358 \bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,(b + a\,b) + \mbox{Q}_2\,b + \ONE$}\\

  1359 \bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$\mbox{Q}_0\,a\,a\,(a^*)$}

  1360 \end{tabular}

  1361 \end{bubble}

  1362 \end{textblock}}

  1363 

  1364 \only<7-9>{\small

  1365 \begin{textblock}{6}(4,10.9)

  1366 \begin{bubble}[7.5cm]

  1367 \begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}

  1368   \multicolumn{3}{@{}l}{Substitute \bl{$\mbox{Q}_2$} and simplify:}\\    

  1369 \bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$\mbox{Q}_0\,(b + a\,b + a\,a\,(a^*)\,b) + \ONE$}\\

  1370 \end{tabular}

  1371 \end{bubble}

  1372 \end{textblock}}

  1373 

  1374 \only<8-9>{\small

  1375 \begin{textblock}{6}(5,13.4)

  1376 \begin{bubble}[7.5cm]

  1377 \begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}

  1378   \multicolumn{3}{@{}l}{Arden again for \bl{$\mbox{Q}_0$}:}\\    

  1379 \bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*$}\\

  1380 \end{tabular}

  1381 \end{bubble}

  1382 \end{textblock}}

  1383 

  1384 

  1385 \only<9-10>{\small

  1386 \begin{textblock}{6}(6,11.5)

  1387 \begin{bubble}[6.7cm]

  1388 \begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}

  1389 \multicolumn{3}{@{}l}{Finally:}\\    

  1390 \bl{$\mbox{Q}_0$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*$}\\

  1391 \bl{$\mbox{Q}_1$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*\,a$}\\

  1392 \bl{$\mbox{Q}_2$} & \bl{$=$} & \bl{$(b + a\,b + a\,a\,(a^*)\,b)^*\,a\,a\,(a^*)$}\\

  1393 \end{tabular}

  1394 \end{bubble}

  1395 \end{textblock}}

  1396 

  1397 

  1398 

  1399 

  1400 

  1401 \only<5-6>{

  1402 \begin{textblock}{6}(0.7,11.9)

  1403 \begin{bubble}[6.7cm]

  1335 }

  1408 \end{bubble}

  1409 \end{textblock}}

  1410 

  1411 \only<8>{

  1412 \begin{textblock}{6}(1.1,7.8)

  1413 \begin{bubble}[6.7cm]

  1414 Arden's Lemma:

  1415 \begin{center}

  1416 If \bl{$q = q\,r + s$}\; then\; \bl{$q = s\, r^*$}

  1417 \end{center}

  1418 \end{bubble}

  1419 \end{textblock}}

  1379 \begin{frame}[c]

  1463 %\begin{frame}[c]

  1380 

  1464 %

  1381 Given the function 

  1465 %Given the function 

  1382 

  1466 %

  1383 \begin{center}

  1467 %\begin{center}

  1384 \bl{\begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}

  1468 %\bl{\begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}

  1385 $rev(\ZERO)$   & $\dn$ & $\ZERO$\\

  1469 %$rev(\ZERO)$   & $\dn$ & $\ZERO$\\

  1386 $rev(\ONE)$         & $\dn$ & $\ONE$\\

  1470 %$rev(\ONE)$         & $\dn$ & $\ONE$\\

  1387 $rev(c)$                      & $\dn$ & $c$\\

  1471 %$rev(c)$                      & $\dn$ & $c$\\

  1388 $rev(r_1 + r_2)$        & $\dn$ & $rev(r_1) + rev(r_2)$\\

  1472 %$rev(r_1 + r_2)$        & $\dn$ & $rev(r_1) + rev(r_2)$\\

  1389 $rev(r_1 \cdot r_2)$  & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\

  1473 %$rev(r_1 \cdot r_2)$  & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\

  1390 $rev(r^*)$                   & $\dn$ & $rev(r)^*$\\

  1474 %$rev(r^*)$                   & $\dn$ & $rev(r)^*$\\

  1391 \end{tabular}}

  1475 %\end{tabular}}

  1392 \end{center}

  1476 %\end{center}

  1393 

  1477 %

  1394 

  1478 %

  1395 and the set

  1479 %and the set

  1396 

  1480 %

  1397 \begin{center}

  1481 %\begin{center}

  1398 \bl{$Rev\,A \dn \{s^{-1} \;|\; s \in A\}$}

  1482 %\bl{$Rev\,A \dn \{s^{-1} \;|\; s \in A\}$}

  1399 \end{center}

  1483 %\end{center}

  1400 

  1484 %

  1401 prove whether

  1485 %prove whether

  1402 

  1486 %

  1403 \begin{center}

  1487 %\begin{center}

  1404 \bl{$L(rev(r)) = Rev (L(r))$}

  1488 %\bl{$L(rev(r)) = Rev (L(r))$}

  1405 \end{center}

  1489 %\end{center}

  1406 

  1490 %

  1407 \end{frame}

  1491 %\end{frame}