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\documentclass[dvipsnames,14pt,t]{beamer}
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\usepackage{beamerthemeplainculight}
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\usepackage[T1]{fontenc}
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\usepackage[latin1]{inputenc}
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\usepackage{mathpartir}
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\usepackage[absolute,overlay]{textpos}
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\usepackage{ifthen}
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\usepackage{tikz}
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\usepackage{pgf}
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\usepackage{calc}
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\usepackage{ulem}
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\usepackage{courier}
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\usepackage{listings}
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\renewcommand{\uline}[1]{#1}
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\usetikzlibrary{arrows}
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\usetikzlibrary{automata}
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\usetikzlibrary{shapes}
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\usetikzlibrary{shadows}
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\usetikzlibrary{positioning}
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\usetikzlibrary{calc}
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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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\definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc
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\lstset{language=Java,
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basicstyle=\ttfamily,
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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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numbers=left,
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numberstyle=\tiny\color{black},
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stepnumber=1,
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numbersep=10pt,
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tabsize=2,
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showspaces=false,
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showstringspaces=false}
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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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sensitive=true,
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morecomment=[l]{//},
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morecomment=[n]{/*}{*/},
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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=\ttfamily,
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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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numbers=left,
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numberstyle=\tiny\color{black},
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stepnumber=1,
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numbersep=10pt,
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tabsize=2,
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showspaces=false,
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showstringspaces=false}
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% beamer stuff
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\renewcommand{\slidecaption}{AFL 03, King's College London, 10.~October 2012}
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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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\begin{document}
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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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%\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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\normalsize
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\begin{center}
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\begin{tabular}{ll}
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Email: & christian.urban at kcl.ac.uk\\
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Of$\!$fice: & S1.27 (1st floor Strand Building)\\
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Slides: & KEATS (also home work is there)\\
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& \alert{\bf (I have put a temporary link in there.)}\\
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\end{tabular}
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\end{center}
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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}Last Week\end{tabular}}
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Last week I showed you
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\begin{itemize}
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\item one simple-minded regular expression matcher (which however does not work in all cases), and\bigskip
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\item one which works provably in all cases
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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}
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\end{itemize}
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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}The Derivative of a Rexp\end{tabular}}
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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$^*$)}\\
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\end{tabular}
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\end{center}
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``the regular expression after \bl{c} has been recognised''
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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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For this we defined the set \bl{Der c A} as
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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}
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which is called the semantic derivative of a set
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and proved
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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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\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}The Idea of the Algorithm\end{tabular}}
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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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\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)))}
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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 set\pause\medskip
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\end{enumerate}
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The matching algorithm works similarly, just over regular expression than sets.
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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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Input: string \bl{abc} and regular expression \bl{r}
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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))}\pause
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\item finally check whether the latter regular expression can match the empty string
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\end{enumerate}
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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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We need to prove
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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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by induction on the regular expression.
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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}Proofs about Rexp\end{tabular}}
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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$_2$}.
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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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\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}Proofs about Natural Numbers\\ and Strings\end{tabular}}
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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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\begin{itemize}
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\item \bl{$P$} holds for \bl{\texttt{""}} 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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\end{frame}}
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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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\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}\bigskip\pause
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\end{center}
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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}Languages\end{tabular}}
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A \alert{language} is a set of strings.\bigskip
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A \alert{regular expression} specifies a set of strings or language.\bigskip
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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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\textcolor{gray}{not all languages are regular, e.g.~\bl{a$^n$b$^n$}.}
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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}[t]
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\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}
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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}\bigskip
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\end{center}
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How about ranges \bl{[a-z]}, \bl{r$^\text{+}$} and \bl{!r}?
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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}Negation of Regular Expr's\end{tabular}}
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\begin{itemize}
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\item \bl{!r} \hspace{6mm} (everything that \bl{r} cannot recognise)\medskip
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\item \bl{$L$(!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\,(!r) $\dn$ !(der\,c\,r)}
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\end{itemize}
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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}Regular Exp's for Lexing\end{tabular}}
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Lexing separates strings into ``words'' / components.
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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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\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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A deterministic finite automaton consists of:
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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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\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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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: t
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%%% End:
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