1056 \end{frame}

  1056 \only<2>{

  1057 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

  1057 \begin{textblock}{4}(10.5,8)

  1058 \small

  1059 Let

  1060 

  1061 \bl{$A = \{foo, bar\}$} \bl{$B = \{a, b\}$}

  1062 \[

  1063 \bl{A \,@\, B = \{fooa, foob, bara, barb\}}

  1064 \]

  1065 \end{textblock}}  

  1066 

  1067 \end{frame}

  1068 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

  1069 

  1070 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1071 \begin{frame}[c]

  1072   \frametitle{Two Corner Cases}

  1073    

  1074   \Large

  1075   \begin{center}

  1076   \bl{$A \,@\, \{[]\} = \;?$}\bigskip\bigskip\pause\\

  1077   \bl{$A \,@\, \{\} = \;?$}

  1078   \end{center}  

  1079     

  1080   \end{frame}

  1081   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

  1082   

  1097  \bl{$L(r^*)$}           & \bl{$\dn$} & \onslide<4->{\bl{$\bigcup_{0 \le n} L(r)^n$}}\\

  1153  \bl{$L(r^*)$}           & \bl{$\dn$} & \onslide<2->{\bl{$\bigcup_{0 \le n} L(r)^n$}}\\

  1100 \onslide<2->{

  1156 %\onslide<2->{

  1101 \hspace{5mm}\bl{$L(r)^0 \;\dn\; \{[]\}$}\\

  1157 %\hspace{5mm}\bl{$L(r)^0 \;\dn\; \{[]\}$}\\

  1102 \bl{$L(r)^{n+1} \;\dn\; L(r) \,@\, L(r)^n$}\hspace{9mm}\onslide<3->{\small\textcolor{gray}{(append on sets)}\\

  1158 %\bl{$L(r)^{n+1} \;\dn\; L(r) \,@\, L(r)^n$}\hspace{9mm}\onslide<3->{\small\textcolor{gray}{(append on sets)}\\

  1103 \small\hspace{5cm}\textcolor{gray}{$\{ s_1 @ s_2 \;|\; s_1\in L(r) \wedge s_2 \in L(r)^n \}$}}

  1159 %\small\hspace{5cm}\textcolor{gray}{$\{ s_1 @ s_2 \;|\; s_1\in L(r) \wedge s_2 \in L(r)^n \}$}}

  1104 }  

  1160 %}  

  1105     \end{textblock}

  1161 \end{textblock}

  1106 

  1162 

  1107 \end{frame}

  1163 \end{frame}

  1108 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

  1164 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

  1109 

  1165 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1110 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1166 \begin{frame}[c]

  1111 \begin{frame}[c]

  1167   \frametitle{The Star Operation}

  1112 \frametitle{The Meaning of Matching}

  1113 

  1114 \begin{bubble}[10cm]

  1115 \large\bf 

  1116 A regular expression \bl{$r$} matches a string~\bl{$s$} 

  1117 provided

  1118 

  1119 \begin{center}

  1120 \bl{$s \in L(r)$}\\ 

  1121 \end{center}

  1122 \end{bubble}\bigskip\bigskip

  1123 

  1124 \ldots and the point of the next lecture is 

  1125 to decide this problem as fast as possible (unlike Python,

  1126 Ruby, Java)

  1127 

  1128 \end{frame}

  1129 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

  1130 

  1131 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1132 \begin{frame}[c]

  1133   \frametitle{The Power Operation}

  1136   \item The \alert{\textbf{\boldmath$n$th Power}} of a language:

  1170   \item The \alert{\bf Kleene Star} of a \underline{language}:

  1171   \bigskip

  1139   \begin{tabular}{lcl}

  1174   \begin{tabular}{c}

  1140   \bl{$A^0$}    & \bl{$\dn$} & \bl{$\{[]\}$}\\

  1175   \bl{$A\star \dn \bigcup_{0\le n} A^n$}

  1141   \bl{$A^{n+1}$} & \bl{$\dn$} & \bl{$A \,@\, A^n$}

  1145   \item[] For example

  1179   \item[] This expands to 

  1146   

  1180   

  1147   \begin{center}

  1181   \[

  1148   \begin{tabular}{lcl@{\hspace{10mm}}l}

  1182   \bl{A^0 \cup A^1 \cup A^2 \cup A^3 \cup A^4 \cup \ldots}

  1149   \bl{$A^4$} & \bl{$=$} & \bl{$A \,@\, A \,@\, A \,@\, A$} & \bl{$(@\,\{[]\})$}\\

  1183   \]

  1150   \bl{$A^1$} & \bl{$=$} & \bl{$A$} & \bl{$(@\,\{[]\})$}\\

  1184   

  1151   \bl{$A^0$} & \bl{$=$} & \bl{$\{[]\}$}\\

  1185   or

  1152   \end{tabular}

  1186   

  1153   \end{center}

  1187   \small

  1188   \[

  1189   \bl{\{[]\} \;\cup\; A \;\cup\; A\,@\,A \;\cup\; 

  1190     A\,@\,A\,@\,A \;\cup\; A\,@\,A\,@\,A\,@\,A \cup \ldots}

  1191   \]

  1158 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

  1196   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

  1159 

  1160 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1161 \begin{frame}[c]

  1162   \frametitle{Questions}

  1163   

  1164   \begin{itemize}

  1165   \item Say \bl{$A = \{[a],[b],[c],[d]\}$}.\bigskip

  1166   

  1167   \item[]

  1168   How many strings are in \bl{$A^4$}\,?

  1169   \bigskip\medskip\pause

  1170   

  1171   

  1172  \item[]

  1173   What if \bl{$A = \{[a],[b],[c],[]\}$};\\ 

  1174   how many strings are then in \bl{$A^4$}\,?

  1175   \end{itemize}  

  1176   

  1177 \end{frame}

  1178 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

  1179 

  1180 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1181 \begin{frame}[c]

  1182 \frametitle{Languages (Sets of Strings)}

  1183 

  1184 \begin{itemize}

  1185 

  1186 \item A \alert{\bf Language} is a set of strings, for example\medskip

  1187 \begin{center}

  1188 \bl{$\{[], hello, foobar, a, abc\}$}

  1189 \end{center}\bigskip

  1190 

  1191 \item \alert{\bf Concatenation} for strings and languages

  1192 

  1193 \begin{center}

  1194 \begin{tabular}{rcl}

  1195 \bl{$foo\;@\;bar$} & \bl{$=$} & \bl{$foobar$}\medskip\\

  1196 \bl{$A\;@\;B$}     & \bl{$\dn$} & \bl{$\{ s_1\,@\,s_2 \;\mid\; s_1 \in A \wedge s_2 \in B\}$}

  1197 \end{tabular}

  1198 \end{center}

  1199 \bigskip

  1200 

  1201 \small

  1202 \item [] For example \bl{$A = \{foo, bar\}$}, \bl{$B = \{a, b\}$}

  1203 

  1204 \[

  1205 \bl{A \,@\, B = \{fooa, foob, bara, barb\}}

  1206 \]

  1207 

  1208 \end{itemize}  

  1209 \end{frame}

  1210 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

  1211 

  1212 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1213 \begin{frame}[c]

  1214   \frametitle{Two Corner Cases}

  1215    

  1216   \Large

  1217   \begin{center}

  1218   \bl{$A \,@\, \{[]\} = \;?$}\bigskip\bigskip\pause\\

  1219   \bl{$A \,@\, \{\} = \;?$}

  1220   \end{center}  

  1221     

  1222   \end{frame}

  1223   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

  1224   

  1225 

  1231  ...all the strings a regular expression can match.   

  1202 \begin{textblock}{15}(1,4)

  1232 

  1233 \begin{center}

  1239  \bl{$L(r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{$L(r_1) \,@\, L(r_2)$}\\

  1208  \bl{$L(r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{$\{ s_1 \,@\, s_2 \;|\; s_1 \in L(r_1) \wedge s_2 \in L(r_2) \}$}\\

  1240  \bl{$L(r^*)$}           & \bl{$\dn$} & \\

  1209  \bl{$L(r^*)$}           & \bl{$\dn$} & \bl{$(L(r))\star$}\\

  1243 

  1229 \end{bubble}\bigskip\bigskip

  1244 \begin{textblock}{14}(1.5,13.5)\small

  1230 

  1245 \bl{$L$} is a function from regular expressions to 

  1231 \ldots and the point of the next lecture is 

  1246 sets of strings (languages):\smallskip\\

  1232 to decide this problem as fast as possible (unlike Python,

  1247 \bl{\quad$L$ : Rexp $\Rightarrow$ Set$[$String$]$}

  1233 Ruby, Java)

  1248 \end{textblock}

  1234 

  1249 

  1235 \end{frame}

  1250 \end{frame}

  1236 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      

  1251 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1237 

  1252 

  1238 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1253 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1239 \begin{frame}[c]

  1254 \begin{frame}[c]

  1240   \frametitle{Questions}

  1255 \frametitle{The Power Operation}

  1241   

  1256 

  1242   \begin{itemize}

  1257 \begin{itemize}

  1243   \item Say \bl{$A = \{[a],[b],[c],[d]\}$}.\bigskip

  1258 \item The \alert{\textbf{\boldmath$n$th Power}} of a language:

  1244   

  1259 

  1245   \item[]

  1260 \begin{center}

  1246   How many strings are in \bl{$A^4$}\,?

  1261 \begin{tabular}{lcl}

  1247   \bigskip\medskip\pause

  1262 \bl{$A^0$}    & \bl{$\dn$} & \bl{$\{[]\}$}\\

  1248   

  1263 \bl{$A^{n+1}$} & \bl{$\dn$} & \bl{$A \,@\, A^n$}

  1249   

  1264 \end{tabular}

  1250  \item[]

  1265 \end{center}\bigskip

  1251   What if \bl{$A = \{[a],[b],[c],[]\}$};\\ 

  1266 

  1252   how many strings are then in \bl{$A^4$}\,?

  1267 \item[] For example

  1253   \end{itemize}  

  1268 

  1254   

  1269 \begin{center}

  1255 \end{frame}

  1270 \begin{tabular}{lcl@{\hspace{10mm}}l}

  1256 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

  1271 \bl{$A^4$} & \bl{$=$} & \bl{$A \,@\, A \,@\, A \,@\, A$} & \bl{$(@\,\{[]\})$}\\

  1257 

  1272 \bl{$A^1$} & \bl{$=$} & \bl{$A$} & \bl{$(@\,\{[]\})$}\\

  1258 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1273 \bl{$A^0$} & \bl{$=$} & \bl{$\{[]\}$}\\

  1259 % \begin{frame}[c]

  1274 \end{tabular}

  1260 % \frametitle{Languages (Sets of Strings)}

  1275 \end{center}

  1261 

  1276 

  1262 % \begin{itemize}

  1277 \end{itemize}  

  1263 

  1278 

  1264 % \item A \alert{\bf Language} is a set of strings, for example\medskip

  1279 \end{frame}

  1265 % \begin{center}

  1266 % \bl{$\{[], hello, foobar, a, abc\}$}

  1267 % \end{center}\bigskip

  1268 

  1269 % \item \alert{\bf Concatenation} for strings and languages

  1270 

  1271 % \begin{center}

  1272 % \begin{tabular}{rcl}

  1273 % \bl{$foo\;@\;bar$} & \bl{$=$} & \bl{$foobar$}\medskip\\

  1274 % \bl{$A\;@\;B$}     & \bl{$\dn$} & \bl{$\{ s_1\,@\,s_2 \;\mid\; s_1 \in A \wedge s_2 \in B\}$}

  1275 % \end{tabular}

  1276 % \end{center}

  1277 % \bigskip

  1278 

  1279 % \small

  1280 % \item [] For example \bl{$A = \{foo, bar\}$}, \bl{$B = \{a, b\}$}

  1281 

  1282 % \[

  1283 % \bl{A \,@\, B = \{fooa, foob, bara, barb\}}

  1284 % \]

  1285 

  1286 

  1287 

  1288 

  1289 % \end{itemize}  

  1290 % \end{frame}

  1291 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

  1292 

  1293 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1294 % \begin{frame}[c]

  1295 %   \frametitle{Two Corner Cases}

  1296    

  1297 %   \Large

  1298 %   \begin{center}

  1299 %   \bl{$A \,@\, \{[]\} = \;?$}\bigskip\bigskip\pause\\

  1300 %   \bl{$A \,@\, \{\} = \;?$}

  1301 %   \end{center}  

  1302     

  1303 %   \end{frame}

  1304 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

  1305   

  1306 

  1307 

  1308 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1309 % \begin{frame}[c]

  1310 % \frametitle{The Meaning of a Regex}

  1311 

  1312 %  ...all the strings a regular expression can match.   

  1313 

  1314 % \begin{center}

  1315 %  \begin{tabular}{rcl}

  1316 %  \bl{$L(\ZERO)$}  & \bl{$\dn$} & \bl{$\{\}$}\\

  1317 %  \bl{$L(\ONE)$}     & \bl{$\dn$} & \bl{$\{[]\}$}\\

  1318 %  \bl{$L(c)$}            & \bl{$\dn$} & \bl{$\{[c]\}$}\\

  1319 %  \bl{$L(r_1 + r_2)$}    & \bl{$\dn$} & \bl{$L(r_1) \cup L(r_2)$}\\

  1320 %  \bl{$L(r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{$L(r_1) \,@\, L(r_2)$}\\

  1321 %  \bl{$L(r^*)$}           & \bl{$\dn$} & \\

  1322 %   \end{tabular}

  1323 % \end{center}

  1324 

  1325 % \begin{textblock}{14}(1.5,13.5)\small

  1326 % \bl{$L$} is a function from regular expressions to 

  1327 % sets of strings (languages):\smallskip\\

  1328 % \bl{\quad$L$ : Rexp $\Rightarrow$ Set$[$String$]$}

  1329 % \end{textblock}

  1330 

  1331 % \end{frame}

  1332 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1333 

  1334 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1335 % \begin{frame}[c]

  1336 % \frametitle{The Power Operation}

  1337 

  1338 % \begin{itemize}

  1339 % \item The \alert{\textbf{\boldmath$n$th Power}} of a language:

  1340 

  1341 % \begin{center}

  1342 % \begin{tabular}{lcl}

  1343 % \bl{$A^0$}    & \bl{$\dn$} & \bl{$\{[]\}$}\\

  1344 % \bl{$A^{n+1}$} & \bl{$\dn$} & \bl{$A \,@\, A^n$}

  1345 % \end{tabular}

  1346 % \end{center}\bigskip

  1347 

  1348 % \item[] For example

  1349 

  1350 % \begin{center}

  1351 % \begin{tabular}{lcl@{\hspace{10mm}}l}

  1352 % \bl{$A^4$} & \bl{$=$} & \bl{$A \,@\, A \,@\, A \,@\, A$} & \bl{$(@\,\{[]\})$}\\

  1353 % \bl{$A^1$} & \bl{$=$} & \bl{$A$} & \bl{$(@\,\{[]\})$}\\

  1354 % \bl{$A^0$} & \bl{$=$} & \bl{$\{[]\}$}\\

  1355 % \end{tabular}

  1356 % \end{center}

  1357 

  1358 % \end{itemize}  

  1359 

  1360 % \end{frame}

  1282 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

  1363 

  1283 \begin{frame}[c]

  1284   \frametitle{The Star Operation}

  1285   

  1286   \begin{itemize}

  1287   \item The \alert{\bf Kleene Star} of a \underline{language}:

  1288   \bigskip

  1289   

  1290   \begin{center}

  1291   \begin{tabular}{c}

  1292   \bl{$A\star \dn \bigcup_{0\le n} A^n$}

  1293   \end{tabular}

  1294   \end{center}\bigskip

  1295   

  1296   \item[] This expands to 

  1297   

  1298   \[

  1299   \bl{A^0 \cup A^1 \cup A^2 \cup A^3 \cup A^4 \cup \ldots}

  1300   \]

  1301   

  1302   or

  1303   

  1304   \small

  1305   \[

  1306   \bl{\{[]\} \;\cup\; A \;\cup\; A\,@\,A \;\cup\; 

  1307     A\,@\,A\,@\,A \;\cup\; A\,@\,A\,@\,A\,@\,A \cup \ldots}

  1308   \]

  1309   

  1310   \end{itemize}  

  1311   

  1312   \end{frame}

  1313   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   

  1314   

  1318 \begin{frame}[c]

  1367 % \begin{frame}[c]

  1319 \frametitle{Written Exam}

  1368 % \frametitle{Written Exam}

  1320 

  1369 

  1321 \begin{itemize}

  1370 % \begin{itemize}

  1322 \item Accounts for 80\%.\bigskip

  1371 % \item Accounts for 80\%.\bigskip

  1323 

  1372 

  1324 \item The question ``\textit{Is this relevant for

  1373 % \item The question ``\textit{Is this relevant for

  1325       the exam?}'' is very demotivating for the lecturer!\bigskip\\

  1374 %       the exam?}'' is very demotivating for the lecturer!\bigskip\\

  1326 

  1375 

  1327 \item Deal: Whatever is in the homework (and is not marked

  1376 % \item Deal: Whatever is in the homework (and is not marked

  1328       ``\textit{optional}'') is relevant for the exam.\bigskip

  1377 %       ``\textit{optional}'') is relevant for the exam.\bigskip

  1330 \item Each lecture has also a handout. There are also handouts about

  1379 % \item Each lecture has also a handout. There are also handouts about

  1331 notation and Scala.      

  1380 % notation and Scala.      

  1332 \end{itemize}

  1381 % \end{itemize}

  1333 

  1382 

  1334 \end{frame}

  1383 % \end{frame}

  1338 \begin{frame}[t]

  1387 % \begin{frame}[t]

  1339 \frametitle{Coursework}

  1388 % \frametitle{Coursework}

  1340 

  1389 

  1341 \begin{itemize}

  1390 % \begin{itemize}

  1342 \item Accounts for 20\%. Two strands. Choose \alert{\bf one}!\bigskip

  1391 % \item Accounts for 20\%. Two strands. Choose \alert{\bf one}!\bigskip

  1343 \end{itemize}

  1392 % \end{itemize}

  1344 

  1393 

  1345 \begin{columns}[t]

  1394 % \begin{columns}[t]

  1346 \begin{column}{.5\textwidth}

  1395 % \begin{column}{.5\textwidth}

  1347 \underline{\bf Strand 1}\medskip

  1396 % \underline{\bf Strand 1}\medskip

  1348 \begin{itemize}

  1397 % \begin{itemize}

  1349 \item 4 programming tasks:

  1398 % \item 4 programming tasks:

  1350 \begin{itemize}

  1399 % \begin{itemize}

  1351 \item matcher (4\%, 11.10.) 

  1400 % \item matcher (4\%, 11.10.) 

  1352 \item lexer (5\%, 04.11.)

  1401 % \item lexer (5\%, 04.11.)

  1353 \item parser (5\%, 22.11.)

  1402 % \item parser (5\%, 22.11.)

  1354 \item compiler (6\%, 13.12.)

  1403 % \item compiler (6\%, 13.12.)

  1355 \end{itemize}

  1404 % \end{itemize}

  1356 \item in any lang.~you like,\\ but I want to see the\\ code

  1405 % \item in any lang.~you like,\\ but I want to see the\\ code

  1357 \end{itemize}

  1406 % \end{itemize}

  1358 \end{column}

  1407 % \end{column}

  1359 

  1408 

  1360 \hspace{-45pt}\vrule{}\hspace{10pt}

  1409 % \hspace{-45pt}\vrule{}\hspace{10pt}

  1361 \begin{column}{.5\textwidth}

  1410 % \begin{column}{.5\textwidth}

  1362 \underline{\bf Strand 2}\smallskip\begin{itemize}

  1411 % \underline{\bf Strand 2}\smallskip\begin{itemize}

  1363 \item one task: prove the correctness of a regular expression matcher in 

  1412 % \item one task: prove the correctness of a regular expression matcher in 

  1364 the \underline{Isabelle} theorem prover

  1413 % the \underline{Isabelle} theorem prover

  1365 \item 20\%, submission on~13.12.\hspace{-5mm}\mbox{}

  1414 % \item 20\%, submission on~13.12.\hspace{-5mm}\mbox{}

  1366 \end{itemize}

  1415 % \end{itemize}

  1367 \end{column}

  1416 % \end{column}

  1368 \end{columns}\medskip

  1417 % \end{columns}\medskip

  1369 

  1418 

  1370 \small

  1419 % \small

  1371 \begin{itemize}

  1420 % \begin{itemize}

  1372 \item Solving more than one strand will {\bf not} give you more 

  1421 % \item Solving more than one strand will {\bf not} give you more 

  1373 marks.

  1422 % marks.

  1374 

  1423 

  1375 \end{itemize}

  1424 % \end{itemize}

  1376 

  1425 

  1377 \end{frame}

  1426 % \end{frame}