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\usepackage{ifthen}+
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\usepackage{graphicx} +
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\definecolor{javared}{rgb}{0.6,0,0} % for strings+
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\definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments+
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\definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords+
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\makeatletter+
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\@empty\z@\@empty+
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\makeatother+
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\lstset{language=Java,+
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	basicstyle=\consolas,+
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	keywordstyle=\color{javapurple}\bfseries,+
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\lstdefinelanguage{scala}{+
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  morekeywords={abstract,case,catch,class,def,%+
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    do,else,extends,false,final,finally,%+
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    for,if,implicit,import,match,mixin,%+
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    new,null,object,override,package,%+
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    private,protected,requires,return,sealed,%+
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    super,this,throw,trait,true,try,%+
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    type,val,var,while,with,yield},+
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  otherkeywords={=>,<-,<\%,<:,>:,\#,@,->},+
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  morestring=[b]",+
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  morestring=[b]',+
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  morestring=[b]"""+
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}+
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\lstset{language=Scala,+
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	basicstyle=\consolas,+
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	keywordstyle=\color{javapurple}\bfseries,+
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	stringstyle=\color{javagreen},+
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	commentstyle=\color{javagreen},+
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	morecomment=[s][\color{javadocblue}]{/**}{*/},+
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	showstringspaces=false}+
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% beamer stuff +
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\renewcommand{\slidecaption}{AFL 03, King's College London, 9.~October 2013}+
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\newcommand{\bl}[1]{\textcolor{blue}{#1}}       +
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\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions+
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+
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\begin{document}+
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}<1>[t]+
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\frametitle{%+
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  \begin{tabular}{@ {}c@ {}}+
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  \\[-3mm]+
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  \LARGE Automata and \\[-2mm] +
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  \LARGE Formal Languages (3)\\[3mm] +
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  \end{tabular}}+
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+
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  %\begin{center}+
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  %\includegraphics[scale=0.3]{pics/ante1.jpg}\hspace{5mm}+
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  %\includegraphics[scale=0.31]{pics/ante2.jpg}\\+
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  %\footnotesize\textcolor{gray}{Antikythera automaton, 100 BC (Archimedes?)}+
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  %\end{center}+
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+
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\normalsize+
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  \begin{center}+
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  \begin{tabular}{lp{8cm}}+
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  Email:  & christian.urban at kcl.ac.uk\\+
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  Office: & S1.27 (1st floor Strand Building)\\+
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  Slides: & KEATS (also home work and coursework is there)\\+
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  \end{tabular}+
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  \end{center}+
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+
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+
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\end{frame}}+
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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}+
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+
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In programming languages they are often used to recognise:\medskip+
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+
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\begin{itemize}+
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\item symbols, digits+
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\item identifiers+
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\item numbers (non-leading zeros)+
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\item keywords+
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\item comments+
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\end{itemize}\bigskip+
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+
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+
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\mbox{}\hfill\bl{\url{http://www.regexper.com}}+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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\frametitle{\begin{tabular}{c}Last Week\end{tabular}}+
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+
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Last week I showed you a regular expression matcher +
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which works provably correctly in all cases.+
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+
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\begin{center}+
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\bl{$matcher\,r\,s$} \;\;if and only if \;\; \bl{$s \in L(r)$}+
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\end{center}\bigskip\bigskip +
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+
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\small+
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\textcolor{gray}{\mbox{}\hfill{}by Janusz Brzozowski (1964)}+
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+
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  +
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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\frametitle{\begin{tabular}{c}The Derivative of a Rexp\end{tabular}}+
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+
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\begin{center}+
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\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}+
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  \bl{$der\, c\, (\varnothing)$}      & \bl{$\dn$} & \bl{$\varnothing$} & \\+
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  \bl{$der\, c\, (\epsilon)$}           & \bl{$\dn$} & \bl{$\varnothing$} & \\+
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  \bl{$der\, c\, (d)$}                     & \bl{$\dn$} & \bl{if $c = d$ then $\epsilon$ else $\varnothing$} & \\+
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  \bl{$der\, c\, (r_1 + r_2)$}        & \bl{$\dn$} & \bl{$der\, c\, r_1 + der\, c\, r_2$} & \\+
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  \bl{$der\, c\, (r_1 \cdot r_2)$}  & \bl{$\dn$}  & \bl{if $nullable (r_1)$}\\+
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  & & \bl{then $(der\,c\,r_1) \cdot r_2 + der\, c\, r_2$}\\ +
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  & & \bl{else $(der\, c\, r_1) \cdot r_2$}\\+
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  \bl{$der\, c\, (r^*)$}          & \bl{$\dn$} & \bl{$(der\,c\,r) \cdot (r^*)$} &\smallskip\\\pause+
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+
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  \bl{$der\!s\, []\, r$}     & \bl{$\dn$} & \bl{$r$} & \\+
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  \bl{$der\!s\, (c\!::\!s)\, r$} & \bl{$\dn$} & \bl{$der\!s\,s\,(der\,c\,r)$} & \\+
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  \end{tabular}+
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\end{center}+
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+
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+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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\mode<presentation>{+
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\begin{frame}[c]+
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+
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+
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To see what is going on, define+
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+
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\begin{center}+
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\bl{$Der\,c\,A \dn \{ s \;|\;  c\!::\!s \in A\}$ } +
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\end{center}\bigskip\bigskip+
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+
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\small+
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For \bl{$A = \{"f\!oo", "bar", "f\!rak"\}$} then+
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+
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\begin{center}+
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\bl{\begin{tabular}{l@{\hspace{2mm}}c@{\hspace{2mm}}l}+
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$Der\,f\,A$ & $=$ & $\{"oo", "rak"\}$\\+
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$Der\,b\,A$ & $=$ &  $\{"ar"\}$\\  +
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$Der\,a\,A$ & $=$ & $\varnothing$\\+
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\end{tabular}}+
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\end{center}+
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 +
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+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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\frametitle{\begin{tabular}{c}The Idea of the Algorithm\end{tabular}}+
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+
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If we want to recognise the string \bl{$"abc"$} with regular expression \bl{$r$}+
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then\medskip+
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+
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\begin{enumerate}+
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\item \bl{$Der\,a\,(L(r))$}\pause+
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\item \bl{$Der\,b\,(Der\,a\,(L(r)))$}\pause+
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\item \bl{$Der\,c\,(Der\,b\,(Der\,a\,(L(r))))$}\pause+
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\item finally we test whether the empty string is in this set\pause\medskip+
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\end{enumerate}+
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+
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The matching algorithm works similarly, just over regular expression instead of sets.+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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+
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Input: string \bl{$"abc"$} and regular expression \bl{$r$} +
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+
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\begin{enumerate}+
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\item \bl{$der\,a\,r$}+
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\item \bl{$der\,b\,(der\,a\,r)$}+
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\item \bl{$der\,c\,(der\,b\,(der\,a\,r))$}\bigskip\pause+
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\item finally check whether the last regular expression can match the empty string+
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\end{enumerate}+
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+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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+
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We proved already+
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+
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\begin{center}+
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\bl{$nullable(r)$} \;if and only if\;  \bl{$"" \in L(r)$}+
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\end{center}+
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+
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by induction on the regular expression.\bigskip\pause+
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+
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\begin{center}+
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\huge\bf \alert{Any Questions?}+
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\end{center}+
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+
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+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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+
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We need to prove+
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+
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\begin{center}+
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\bl{$L(der\,c\,r) = Der\,c\,(L(r))$}+
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\end{center}+
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+
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by induction on the regular expression.+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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\frametitle{\begin{tabular}{c}Proofs about Rexps\end{tabular}}+
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+
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\begin{itemize}+
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\item \bl{$P$} holds for \bl{$\varnothing$}, \bl{$\epsilon$} and \bl{c}\bigskip+
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\item \bl{$P$} holds for \bl{$r_1 + r_2$} under the assumption that \bl{$P$} already+
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holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip+
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\item \bl{$P$} holds for \bl{$r_1 \cdot r_2$} under the assumption that \bl{$P$} already+
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holds for \bl{$r_1$} and \bl{$r_2$}.\bigskip+
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\item \bl{$P$} holds for \bl{$r^*$} under the assumption that \bl{$P$} already+
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holds for \bl{$r$}.+
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\end{itemize}+
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+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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\frametitle{\begin{tabular}{c}Proofs about Natural\\[-1mm] Numbers and Strings\end{tabular}}+
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+
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\begin{itemize}+
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\item \bl{$P$} holds for \bl{$0$} and+
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\item \bl{$P$} holds for \bl{$n + 1$} under the assumption that \bl{$P$} already+
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holds for \bl{$n$}+
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\end{itemize}\bigskip+
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+
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\begin{itemize}+
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\item \bl{$P$} holds for \bl{$""$} and+
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\item \bl{$P$} holds for \bl{$c\!::\!s$} under the assumption that \bl{$P$} already+
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holds for \bl{$s$}+
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\end{itemize}+
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+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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\frametitle{\begin{tabular}{c}Languages\end{tabular}}+
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+
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A \alert{language} is a set of strings.\bigskip+
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+
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A \alert{regular expression} specifies a language.\bigskip+
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+
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A language is \alert{regular} iff there exists+
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a regular expression that recognises all its strings.\bigskip\bigskip\pause+
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+
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\textcolor{gray}{not all languages are regular, e.g.~\bl{$a^nb^n$}.}+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[t]+
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\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}+
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+
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\begin{center}+
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   \begin{tabular}{@ {}rrl@ {\hspace{13mm}}l}+
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  \bl{$r$} & \bl{$::=$}  & \bl{$\varnothing$}  & null\\+
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         & \bl{$\mid$} & \bl{$\epsilon$}        & empty string / "" / $[]$\\+
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         & \bl{$\mid$} & \bl{$c$}                         & character\\+
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         & \bl{$\mid$} & \bl{$r_1 \cdot r_2$} & sequence\\+
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         & \bl{$\mid$} & \bl{$r_1 + r_2$}  & alternative / choice\\+
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         & \bl{$\mid$} & \bl{$r^*$}                   & star (zero or more)\\+
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  \end{tabular}+
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  \end{center}+
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  +
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How about ranges \bl{$[a\mbox{-}z]$}, \bl{$r^+$} and \bl{$\sim{}r$}? Do they increase the+
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set of languages we can recognise?+
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  +
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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\frametitle{\begin{tabular}{c}Negation of Regular Expr's\end{tabular}}+
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+
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\begin{itemize}+
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\item \bl{$\sim{}r$}  \hspace{6mm} (everything that \bl{r} cannot recognise)\medskip+
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\item \bl{$L(\sim{}r) \dn UNIV - L(r)$}\medskip+
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\item \bl{$nullable (r) \dn not (nullable(r))$}\medskip+
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\item \bl{$der\,c\,(\sim{}r) \dn \;\sim{}(der\,c\,r)$}+
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\end{itemize}\bigskip\pause+
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+
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Used often for comments:+
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+
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\[+
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\bl{/ \cdot * \cdot (\sim{}([a\mbox{-}z]^* \cdot * \cdot / \cdot [a\mbox{-}z]^*)) \cdot * \cdot /}+
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\]+
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+
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+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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\frametitle{\begin{tabular}{c}Regular Exp's for Lexing\end{tabular}}+
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+
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Lexing separates strings into ``words'' / components.+
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+
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\begin{itemize}+
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\item Identifiers (non-empty strings of letters or digits, starting with a letter)+
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\item Numbers (non-empty sequences of digits omitting leading zeros)+
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\item Keywords (else, if, while, \ldots)+
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\item White space (a non-empty sequence of blanks, newlines and tabs)+
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\item Comments+
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\end{itemize}+
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+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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\frametitle{\begin{tabular}{c}Automata\end{tabular}}+
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+
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A deterministic finite automaton consists of:+
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+
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\begin{itemize}+
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\item a set of states+
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\item one of these states is the start state+
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\item some states are accepting states, and+
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\item there is transition function\medskip +
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+
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\small+
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which takes a state as argument and a character and produces a new state\smallskip\\+
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this function might not always be defined+
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\end{itemize}+
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+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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\mode<presentation>{+
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\begin{frame}[c]+
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\frametitle{\begin{tabular}{c}Automata\end{tabular}}+
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+
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A deterministic finite automaton consists of:+
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+
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\begin{itemize}+
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\item a set of states+
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\item one of these states is the start state+
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\item some states are accepting states, and+
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\item there is transition function\medskip +
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+
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\small+
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which takes a state as argument and a character and produces a new state\smallskip\\+
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this function might not always be defined+
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\end{itemize}+
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+
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\end{frame}}+
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   +
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+
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\end{document}+
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+
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%%% Local Variables:  +
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%%% TeX-master: t+
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%%% End: +
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