
diff -r e85600529ca5 -r 4794759139ea slides04.tex
--- a/slides04.tex	Sat Jun 15 09:11:11 2013 -0400
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,612 +0,0 @@
-\documentclass[dvipsnames,14pt,t]{beamer}
-\usepackage{beamerthemeplainculight}
-\usepackage[T1]{fontenc}
-\usepackage[latin1]{inputenc}
-\usepackage{mathpartir}
-\usepackage[absolute,overlay]{textpos}
-\usepackage{ifthen}
-\usepackage{tikz}
-\usepackage{pgf}
-\usepackage{calc} 
-\usepackage{ulem}
-\usepackage{courier}
-\usepackage{listings}
-\renewcommand{\uline}[1]{#1}
-\usetikzlibrary{arrows}
-\usetikzlibrary{automata}
-\usetikzlibrary{shapes}
-\usetikzlibrary{shadows}
-\usetikzlibrary{positioning}
-\usetikzlibrary{calc}
-\usepackage{graphicx} 
-
-\definecolor{javared}{rgb}{0.6,0,0} % for strings
-\definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments
-\definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords
-\definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc
-
-\lstset{language=Java,
-	basicstyle=\ttfamily,
-	keywordstyle=\color{javapurple}\bfseries,
-	stringstyle=\color{javagreen},
-	commentstyle=\color{javagreen},
-	morecomment=[s][\color{javadocblue}]{/**}{*/},
-	numbers=left,
-	numberstyle=\tiny\color{black},
-	stepnumber=1,
-	numbersep=10pt,
-	tabsize=2,
-	showspaces=false,
-	showstringspaces=false}
-
-\lstdefinelanguage{scala}{
-  morekeywords={abstract,case,catch,class,def,%
-    do,else,extends,false,final,finally,%
-    for,if,implicit,import,match,mixin,%
-    new,null,object,override,package,%
-    private,protected,requires,return,sealed,%
-    super,this,throw,trait,true,try,%
-    type,val,var,while,with,yield},
-  otherkeywords={=>,<-,<\%,<:,>:,\#,@},
-  sensitive=true,
-  morecomment=[l]{//},
-  morecomment=[n]{/*}{*/},
-  morestring=[b]",
-  morestring=[b]',
-  morestring=[b]"""
-}
-
-\lstset{language=Scala,
-	basicstyle=\ttfamily,
-	keywordstyle=\color{javapurple}\bfseries,
-	stringstyle=\color{javagreen},
-	commentstyle=\color{javagreen},
-	morecomment=[s][\color{javadocblue}]{/**}{*/},
-	numbers=left,
-	numberstyle=\tiny\color{black},
-	stepnumber=1,
-	numbersep=10pt,
-	tabsize=2,
-	showspaces=false,
-	showstringspaces=false}
-
-% beamer stuff 
-\renewcommand{\slidecaption}{AFL 04, King's College London, 17.~October 2012}
-\newcommand{\bl}[1]{\textcolor{blue}{#1}}       
-\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
-
-\begin{document}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}<1>[t]
-\frametitle{%
-  \begin{tabular}{@ {}c@ {}}
-  \\[-3mm]
-  \LARGE Automata and \\[-2mm] 
-  \LARGE Formal Languages (4)\\[3mm] 
-  \end{tabular}}
-
-  \normalsize
-  \begin{center}
-  \begin{tabular}{ll}
-  Email:  & christian.urban at kcl.ac.uk\\
-  Of$\!$fice: & S1.27 (1st floor Strand Building)\\
-  Slides: & KEATS (also home work is there)\\
-  \end{tabular}
-  \end{center}
-
-
-\end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}Last Week\end{tabular}}
-
-Last week I showed you\bigskip
-
-\begin{itemize}
-\item a tokenizer taking a list of regular expressions\bigskip
-
-\item tokenization identifies lexeme in an input stream of characters (or string)
-and cathegorizes  them into tokens
-
-\end{itemize}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}Two Rules\end{tabular}}
-
-\begin{itemize}
-\item Longest match rule (maximal munch rule): The 
-longest initial substring matched by any regular expression is taken
-as next token.\bigskip
-
-\item Rule priority:
-For a particular longest initial substring, the first regular
-expression that can match determines the token.
-
-\end{itemize}
-
-%\url{http://www.technologyreview.com/tr10/?year=2011}
-  
-%finite deterministic automata/ nondeterministic automaton
-
-%\item problem with infix operations, for example i-12
-
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\mode<presentation>{
-\begin{frame}[t]
-
-\begin{center}
-\texttt{"if true then then 42 else +"}
-\end{center}
-
-
-\begin{tabular}{@{}l}
-KEYWORD: \\
-\hspace{5mm}\texttt{"if"}, \texttt{"then"}, \texttt{"else"},\\ 
-WHITESPACE:\\
-\hspace{5mm}\texttt{" "}, \texttt{"$\backslash$n"},\\ 
-IDENT:\\
-\hspace{5mm}LETTER $\cdot$ (LETTER + DIGIT + \texttt{"\_"})$^*$\\ 
-NUM:\\
-\hspace{5mm}(NONZERODIGIT $\cdot$ DIGIT$^*$) + \texttt{"0"}\\
-OP:\\
-\hspace{5mm}\texttt{"+"}\\
-COMMENT:\\
-\hspace{5mm}\texttt{"$\slash$*"} $\cdot$ (ALL$^*$ $\cdot$ \texttt{"*$\slash$"} $\cdot$ ALL$^*$) $\cdot$ \texttt{"*$\slash$"}
-\end{tabular}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[t]
-
-\begin{center}
-\texttt{"if true then then 42 else +"}
-\end{center}
-
-\only<1>{
-\small\begin{tabular}{l}
-KEYWORD(if),\\ 
-WHITESPACE,\\ 
-IDENT(true),\\ 
-WHITESPACE,\\ 
-KEYWORD(then),\\ 
-WHITESPACE,\\ 
-KEYWORD(then),\\ 
-WHITESPACE,\\ 
-NUM(42),\\ 
-WHITESPACE,\\ 
-KEYWORD(else),\\ 
-WHITESPACE,\\ 
-OP(+)
-\end{tabular}}
-
-\only<2>{
-\small\begin{tabular}{l}
-KEYWORD(if),\\ 
-IDENT(true),\\ 
-KEYWORD(then),\\ 
-KEYWORD(then),\\ 
-NUM(42),\\ 
-KEYWORD(else),\\ 
-OP(+)
-\end{tabular}}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-
-There is one small problem with the tokenizer. How should we 
-tokenize:
-
-\begin{center}
-\texttt{"x - 3"}
-\end{center}
-
-\begin{tabular}{@{}l}
-OP:\\
-\hspace{5mm}\texttt{"+"}, \texttt{"-"}\\
-NUM:\\
-\hspace{5mm}(NONZERODIGIT $\cdot$ DIGIT$^*$) + \texttt{"0"}\\
-NUMBER:\\
-\hspace{5mm}NUM +  (\texttt{"-"} $\cdot$ NUM)\\
-\end{tabular}
-
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}Negation\end{tabular}}
-
-Assume you have an alphabet consisting of the letters \bl{a}, \bl{b} and \bl{c} only.
-Find a regular expression that matches all strings \emph{except} \bl{ab}, \bl{ac} and \bl{cba}.
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}Deterministic Finite Automata\end{tabular}}
-
-A deterministic finite automaton consists of:
-
-\begin{itemize}
-\item a finite set of states
-\item one of these states is the start state
-\item some states are accepting states, and
-\item there is transition function\medskip 
-
-\small
-which takes a state and a character as arguments and produces a new state\smallskip\\
-this function might not always be defined everywhere
-\end{itemize}
-
-\begin{center}
-\bl{$A(Q, q_0, F, \delta)$}
-\end{center}
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-\begin{center}
-\includegraphics[scale=0.7]{pics/ch3.jpg}
-\end{center}\pause
-
-\begin{itemize}
-\item start can be an accepting state
-\item it is possible that there is no accepting state
-\item all states might be accepting (but does not necessarily mean all strings are accepted)
-\end{itemize}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-\begin{center}
-\includegraphics[scale=0.7]{pics/ch3.jpg}
-\end{center}
-
-for this automaton \bl{$\delta$} is the function\\
-
-\begin{center}
-\begin{tabular}{lll}
-\bl{(q$_0$, a) $\rightarrow$ q$_1$} & \bl{(q$_1$, a) $\rightarrow$ q$_4$} & \bl{(q$_4$, a) $\rightarrow$ q$_4$}\\
-\bl{(q$_0$, b) $\rightarrow$ q$_2$} & \bl{(q$_1$, b) $\rightarrow$ q$_2$} & \bl{(q$_4$, b) $\rightarrow$ q$_4$}\\
-\end{tabular}\ldots
-\end{center}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[t]
-\frametitle{\begin{tabular}{c}Accepting a String\end{tabular}}
-
-Given
-
-\begin{center}
-\bl{$A(Q, q_0, F, \delta)$}
-\end{center}
-
-you can define
-
-\begin{center}
-\begin{tabular}{l}
-\bl{$\hat{\delta}(q, \texttt{""}) = q$}\\
-\bl{$\hat{\delta}(q, c::s) = \hat{\delta}(\delta(q, c), s)$}\\
-\end{tabular}
-\end{center}\pause
-
-Whether a string \bl{$s$} is accepted by \bl{$A$}?
-
-\begin{center}
-\hspace{5mm}\bl{$\hat{\delta}(q_0, s) \in F$}
-\end{center}
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}Non-Deterministic\\[-1mm] Finite Automata\end{tabular}}
-
-A non-deterministic finite automaton consists again of:
-
-\begin{itemize}
-\item a finite set of states
-\item one of these states is the start state
-\item some states are accepting states, and
-\item there is transition \alert{relation}\medskip 
-\end{itemize}
-
-
-\begin{center}
-\begin{tabular}{c}
-\bl{(q$_1$, a) $\rightarrow$ q$_2$}\\
-\bl{(q$_1$, a) $\rightarrow$ q$_3$}\\
-\end{tabular}
-\hspace{10mm}
-\begin{tabular}{c}
-\bl{(q$_1$, $\epsilon$) $\rightarrow$ q$_2$}\\
-\end{tabular}
-\end{center}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-\begin{center}
-\includegraphics[scale=0.7]{pics/ch5.jpg}
-\end{center}
-
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-\begin{center}
-\begin{tabular}[b]{ll}
-\bl{$\varnothing$} & \includegraphics[scale=0.7]{pics/NULL.jpg}\\\\
-\bl{$\epsilon$} & \includegraphics[scale=0.7]{pics/epsilon.jpg}\\\\
-\bl{c} & \includegraphics[scale=0.7]{pics/char.jpg}\\
-\end{tabular}
-\end{center}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-\begin{center}
-\begin{tabular}[t]{ll}
-\bl{r$_1$ $\cdot$ r$_2$} & \includegraphics[scale=0.6]{pics/seq.jpg}\\\\
-\end{tabular}
-\end{center}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-\begin{center}
-\begin{tabular}[t]{ll}
-\bl{r$_1$ + r$_2$} & \includegraphics[scale=0.7]{pics/alt.jpg}\\\\
-\end{tabular}
-\end{center}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-\begin{center}
-\begin{tabular}[b]{ll}
-\bl{r$^*$} & \includegraphics[scale=0.7]{pics/star.jpg}\\
-\end{tabular}
-\end{center}\pause\bigskip
-
-Why can't we just have an epsilon transition from the accepting states to the starting state?
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}Subset Construction\end{tabular}}
-
-
-\begin{textblock}{5}(1,2.5)
-\includegraphics[scale=0.5]{pics/ch5.jpg}
-\end{textblock}
-
-\begin{textblock}{11}(6.5,4.5)
-\begin{tabular}{r|cl}
-& a & b\\
-\hline
-$\varnothing$ \onslide<2>{\textcolor{white}{*}} & $\varnothing$ & $\varnothing$\\
-$\{0\}$ \onslide<2>{\textcolor{white}{*}} & $\{0,1,2\}$ & $\{2\}$\\
-$\{1\}$ \onslide<2>{\textcolor{white}{*}} &$\{1\}$ & $\varnothing$\\
-$\{2\}$ \onslide<2>{*} & $\varnothing$ &$\{2\}$\\
-$\{0,1\}$ \onslide<2>{\textcolor{white}{*}} &$\{0,1,2\}$ &$\{2\}$\\
-$\{0,2\}$ \onslide<2>{*}&$\{0,1,2\}$ &$\{2\}$\\
-$\{1,2\}$ \onslide<2>{*}& $\{1\}$ & $\{2\}$\\
-\onslide<2>{s:} $\{0,1,2\}$ \onslide<2>{*}&$\{0,1,2\}$ &$\{2\}$\\
-\end{tabular}
-\end{textblock}
-
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-\frametitle{\begin{tabular}{c}Regular Languages\end{tabular}}
-
-A language is \alert{regular} iff there exists
-a regular expression that recognises all its strings.\bigskip\medskip
-
-or equivalently\bigskip\medskip
-
-A language is \alert{regular} iff there exists
-a deterministic finite automaton that recognises all its strings.\bigskip\pause
-
-Why is every finite set of strings a regular language?
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-\begin{center}
-\includegraphics[scale=0.5]{pics/ch3.jpg}
-\end{center}
-
-\begin{center}
-\includegraphics[scale=0.5]{pics/ch4.jpg}\\
-minimal automaton
-\end{center}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-\begin{enumerate}
-\item Take all pairs \bl{(q, p)} with \bl{q $\not=$ p}
-\item Mark all pairs that accepting and non-accepting states
-\item For  all unmarked pairs \bl{(q, p)} and all characters \bl{c} tests wether
-\begin{center}
-\bl{($\delta$(q,c), $\delta$(p,c))}
-\end{center} 
-are marked. If yes, then also mark \bl{(q, p)} 
-\item Repeat last step until no chance.
-\item All unmarked pairs can be merged.
-\end{enumerate}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-Given the function 
-
-\begin{center}
-\bl{\begin{tabular}{r@{\hspace{1mm}}c@{\hspace{1mm}}l}
-$rev(\varnothing)$   & $\dn$ & $\varnothing$\\
-$rev(\epsilon)$         & $\dn$ & $\epsilon$\\
-$rev(c)$                      & $\dn$ & $c$\\
-$rev(r_1 + r_2)$        & $\dn$ & $rev(r_1) + rev(r_2)$\\
-$rev(r_1 \cdot r_2)$  & $\dn$ & $rev(r_2) \cdot rev(r_1)$\\
-$rev(r^*)$                   & $\dn$ & $rev(r)^*$\\
-\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}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-\begin{itemize}
-\item The star-case in our proof about the matcher needs the following lemma
-\begin{center}
-\bl{Der\,c\,A$^*$ $=$ (Der c A)\,@\, A$^*$}
-\end{center}
-\end{itemize}\bigskip\bigskip
-
-\begin{itemize}
-\item If \bl{\texttt{""} $\in$ A}, then\\ \bl{Der\,c\,(A @ B) $=$ (Der\,c\,A) @ B $\cup$ (Der\,c\,B)}\medskip
-\item If \bl{\texttt{""} $\not\in$ A}, then\\ \bl{Der\,c\,(A @ B) $=$ (Der\,c\,A) @ B}
-
-\end{itemize}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-\begin{itemize}
-\item Assuming you have the alphabet \bl{\{a, b, c\}}\bigskip
-\item Give a regular expression that can recognise all strings that have at least one \bl{b}.
-\end{itemize}
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}[c]
-
-``I hate coding. I do not want to look at code.''
-
-\end{frame}}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
-
-
-
-\end{document}
-
-%%% Local Variables:  
-%%% mode: latex
-%%% TeX-master: t
-%%% End: 
-




