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(*<*)
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theory Slides1
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imports "~~/src/HOL/Library/LaTeXsugar"
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begin
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notation (latex output)
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set ("_") and
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Cons ("_::/_" [66,65] 65)
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(*>*)
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text_raw {*
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%\renewcommand{\slidecaption}{Cambridge, 9 November 2010}
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\renewcommand{\slidecaption}{Nijmegen, 25 August 2011}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}
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\frametitle{%
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\begin{tabular}{@ {}c@ {}}
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\Large A Formalisation of the\\[-4mm]
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\Large Myhill-Nerode Theorem based on\\[-4mm]
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\Large \alert<2>{Regular Expressions}\\[-4mm]
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\Large (Proof Pearl)\\[0mm]
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\end{tabular}}
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\begin{center}
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\begin{tabular}{c@ {\hspace{15mm}}c}
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\includegraphics[scale=0.034]{chunhan.jpg} &
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\includegraphics[scale=0.034]{xingyuan.jpg}\\[-5mm]
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\end{tabular}
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\end{center}
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\begin{center}
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\small joint work with Chunhan Wu and Xingyuan Zhang from the PLA
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University of Science and Technology in Nanjing
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\end{center}
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\begin{center}
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\small Christian Urban\\
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TU Munich
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\end{center}
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\end{frame}}
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*}
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\mode<presentation>{
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\frametitle{}
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\begin{textblock}{12.9}(1.5,2.0)
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\begin{block}{}
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\begin{minipage}{12.4cm}\raggedright
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\large {\bf Motivation:}\\[2mm]I want to teach \alert{students} with
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theorem\\ provers (especially for inductions).
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\end{minipage}
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\end{block}
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\end{textblock}\pause
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\mbox{}\\[35mm]\mbox{}
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\begin{itemize}
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\item \only<2>{\smath{\text{fib}}, \smath{\text{even}} and \smath{\text{odd}}}%
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\only<3->{\sout{\smath{\text{fib}}, \smath{\text{even}} and \smath{\text{odd}}}}\medskip
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\item<3-> formal language theory \\
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\mbox{}\;\;@{text "\<Rightarrow>"} nice textbooks: Kozen, Hopcroft \& Ullman\ldots
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\end{itemize}
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\end{frame}}
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
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\begin{center}
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\huge\bf\textcolor{gray}{in Nuprl}
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\end{center}
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\begin{itemize}
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\item Constable, Jackson, Naumov, Uribe\medskip
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\item \alert{18 months} for automata theory from Hopcroft \& Ullman chapters 1--11 (including Myhill-Nerode)
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
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\begin{center}
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\huge\bf\textcolor{gray}{in Coq}
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\end{center}
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\begin{itemize}
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\item Filli\^atre, Briais, Braibant and others
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\item multi-year effort; a number of results in automata theory, e.g.\medskip
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\begin{itemize}
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\item Kleene's thm.~by Filli\^atre (\alert{``rather big''})
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\item automata theory by Briais (5400 loc)
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\item Braibant ATBR library, including Myhill-Nerode ($>\!\!\!>$2000 loc)
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\item Mirkin's partial derivative automaton construction (10600 loc)
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\end{itemize}
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}[t]
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\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
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\mbox{}\\[-10mm]\mbox{}
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\begin{center}
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\huge\bf\textcolor{gray}{in HOL}
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\end{center}
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\begin{itemize}
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\item automata @{text "\<Rightarrow>"} graphs, matrices, functions
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\item<2-> combining automata/graphs
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\onslide<2->{
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&
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\onslide<3->{\raisebox{1.1mm}{\bf\Large$\;\Rightarrow\,$}}
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&
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\onslide<3->{\begin{tikzpicture}[scale=1]
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\end{tikzpicture}}
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\end{tabular}
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\end{center}\medskip
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\only<4-5>{
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\begin{tabular}{@ {}l@ {}}
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disjoint union:\\[2mm]
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\smath{A_1\uplus A_2 \dn \{(1, x)\,|\, x \in A_1\} \,\cup\, \{(2, y)\,|\, y \in A_2\}}
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\end{tabular}}
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\end{itemize}
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\only<5>{
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\begin{textblock}{13.9}(0.7,7.7)
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\medskip
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\begin{minipage}{14cm}\raggedright
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Problems with definition for regularity (Slind):\bigskip\\
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\smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}\bigskip
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\end{minipage}
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\end{textblock}}
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\medskip
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\only<6->{\underline{A solution}:\;\;use \smath{\text{nat}} \;@{text "\<Rightarrow>"}\; state nodes\medskip}
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\only<7->{You have to \alert{rename} states!}
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\end{frame}}
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*}
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\mode<presentation>{
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\begin{frame}[t]
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\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
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\mbox{}\\[-10mm]\mbox{}
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\begin{center}
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\huge\bf\textcolor{gray}{in HOL}
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\end{center}
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\begin{itemize}
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\item Kozen's ``paper'' proof of Myhill-Nerode:\\
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\hspace{2cm}requires absence of \alert{inaccessible states}
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\end{itemize}\bigskip\bigskip
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\begin{center}
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\smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}
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\end{center}
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\end{frame}}
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*}
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\mode<presentation>{
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\begin{frame}[t]
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\frametitle{}
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\mbox{}\\[25mm]\mbox{}
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\begin{textblock}{13.9}(0.7,1.2)
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\begin{block}{}
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{\bf Definition:}\smallskip\\
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A language \smath{A} is \alert{regular}, provided there exists a\\
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\alert{regular expression} that matches all strings of \smath{A}.
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\end{textblock}\pause
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{\noindent\large\bf\alert{\ldots{}and forget about automata}}\bigskip\bigskip\pause
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Infrastructure for free. But do we lose anything?\pause
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\begin{itemize}
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\item pumping lemma\pause
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\item closure under complementation\pause
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\item \only<6>{regular expression matching}%
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\only<7->{\sout{regular expression matching} \;\;(@{text "\<Rightarrow>"}Owens et al)}
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\item<8-> most textbooks are about automata
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\end{itemize}
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\end{frame}}
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*}
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE The Myhill-Nerode Theorem}
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\begin{itemize}
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\item provides necessary and suf\!ficient conditions\\ for a language
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being regular\\ \textcolor{gray}{(pumping lemma only necessary)}\bigskip
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\item key is the equivalence relation:\smallskip
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\begin{center}
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\smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
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\end{itemize}
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*}
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE The Myhill-Nerode Theorem}
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\begin{center}
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\only<1>{%
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\only<2->{%
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\end{center}
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\begin{itemize}
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\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
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\end{itemize}
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\begin{textblock}{5}(2.1,5.3)
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\node at (0,0) [single arrow, fill=red,text=white, minimum height=2cm]
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{$U\!N\!IV$};
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\draw (-0.3,-1.1) node {\begin{tabular}{l}set of all\\[-1mm] strings\end{tabular}};
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\end{tikzpicture}
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\begin{tikzpicture}
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\node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]
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{@{text "\<lbrakk>x\<rbrakk>"}$_{\approx_{A}}$};
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\draw (0.9,-1.1) node {\begin{tabular}{l}an equivalence class\end{tabular}};
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\end{tikzpicture}
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\begin{block}{}
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\begin{minipage}{11.4cm}\raggedright
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Two directions:\medskip\\
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\begin{tabular}{@ {}ll}
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1.)\;finite $\Rightarrow$ regular\\
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\;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_A) \Rightarrow \exists r.\;A = {\cal L}(r)}\\[3mm]
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2.)\;regular $\Rightarrow$ finite\\
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\;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}
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*}
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE Initial and Final {\sout{\textcolor{gray}{States}}}}
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\begin{textblock}{8}(10, 2)
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\textcolor{black}{Equivalence Classes}
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\begin{center}
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\begin{tikzpicture}[scale=3]
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\draw[very thick] (0.5,0.5) circle (.6cm);
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\clip[draw] (0.5,0.5) circle (.6cm);
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\draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
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\only<2->{\draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);}
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\only<3->{\draw[red, fill] (0.2, 0.2) rectangle (0.4, 0.4);
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\draw[red, fill] (0.4, 0.8) rectangle (0.6, 1.0);
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\draw[red, fill] (0.6, 0.0) rectangle (0.8, 0.2);
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\draw[red, fill] (0.8, 0.4) rectangle (1.0, 0.6);}
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\end{tikzpicture}
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\end{center}
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\begin{itemize}
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\item \smath{\text{finals}\,A\,\dn \{[\!|x|\!]_{\approx_{A}}\;|\;x \in A\}}
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\smallskip
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\item we can prove: \smath{A = \bigcup \text{finals}\,A}
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\begin{textblock}{5}(2.1,4.6)
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\begin{tikzpicture}
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\node at (0,0) [single arrow, fill=blue,text=white, minimum height=2cm]
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{$[] \in X$};
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\end{tikzpicture}
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\end{textblock}}
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\only<3->{%
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\begin{textblock}{5}(10,7.4)
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\begin{tikzpicture}
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\node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]
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{a final};
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\end{tikzpicture}
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\end{textblock}}
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\end{frame}}
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*}
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\mode<presentation>{
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\begin{frame}<-1>[c]
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\frametitle{\begin{tabular}{@ {}l}\LARGE%
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Transitions between Eq-Classes\end{tabular}}
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\begin{center}
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\begin{tikzpicture}[scale=3]
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\draw[very thick] (0.5,0.5) circle (.6cm);
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\clip[draw] (0.5,0.5) circle (.6cm);
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\draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
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\draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);
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\draw[blue, fill] (0.8, 0.4) rectangle (1.0, 0.6);
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\draw[white] (0.1,0.7) node (X) {$X$};
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\draw[white] (0.9,0.5) node (Y) {$Y$};
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\draw[blue, ->, line width = 2mm, bend left=45] (X) -- (Y);
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\node [inner sep=1pt,label=above:\textcolor{blue}{$c$}] at ($ (X)!.5!(Y) $) {};
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\end{tikzpicture}
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\end{center}
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\begin{center}
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\smath{X \stackrel{c}{\longrightarrow} Y \;\dn\; X ; c \subseteq Y}
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\end{center}
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\onslide<8>{
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\begin{tabular}{c}
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\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
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\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
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\node[state,initial] (q_0) {$R_1$};
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\end{tikzpicture}
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\end{tabular}}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{\LARGE Systems of Equations}
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Inspired by a method of Brzozowski\;'64:\bigskip\bigskip
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\begin{center}
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\begin{tabular}{@ {\hspace{-20mm}}c}
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\\[-13mm]
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\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
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\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
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%\draw[help lines] (0,0) grid (3,2);
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\node[state,initial] (p_0) {$X_1$};
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\node[state,accepting] (p_1) [right of=q_0] {$X_2$};
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\path[->] (p_0) edge [bend left] node {a} (p_1)
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edge [loop above] node {b} ()
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(p_1) edge [loop above] node {a} ()
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edge [bend left] node {b} (p_0);
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\end{tikzpicture}\\
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\\[-13mm]
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\end{tabular}
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\end{center}
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487 |
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\begin{center}
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\begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
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& \smath{X_1} & \smath{=} & \smath{X_1;b + X_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
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491 |
& \smath{X_2} & \smath{=} & \smath{X_1;a + X_2;a}\medskip\\
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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}<1-2,4->[t]
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\small
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\begin{center}
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\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
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\onslide<1->{\smath{X_1}} & \onslide<1->{\smath{=}}
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|
509 |
& \onslide<1->{\smath{X_1; b + X_2; b + \lambda;[]}}\\
|
|
|
510 |
\onslide<1->{\smath{X_2}} & \onslide<1->{\smath{=}}
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& \onslide<1->{\smath{X_1; a + X_2; a}}\\
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512 |
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513 |
& & & \onslide<2->{by Arden}\\
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\onslide<2->{\smath{X_1}} & \onslide<2->{\smath{=}}
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& \onslide<2->{\smath{X_1; b + X_2; b + \lambda;[]}}\\
|
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|
517 |
\onslide<2->{\smath{X_2}} & \onslide<2->{\smath{=}}
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& \only<2->{\smath{X_1; a\cdot a^\star}}\\
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204
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519 |
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& & & \onslide<4->{by Arden}\\
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521 |
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213
|
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\onslide<4->{\smath{X_1}} & \onslide<4->{\smath{=}}
|
|
|
523 |
& \onslide<4->{\smath{X_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
524 |
\onslide<4->{\smath{X_2}} & \onslide<4->{\smath{=}}
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|
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& \onslide<4->{\smath{X_1; a\cdot a^\star}}\\
|
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204
|
526 |
|
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|
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& & & \onslide<5->{by substitution}\\
|
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528 |
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213
|
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\onslide<5->{\smath{X_1}} & \onslide<5->{\smath{=}}
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|
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& \onslide<5->{\smath{X_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
531 |
\onslide<5->{\smath{X_2}} & \onslide<5->{\smath{=}}
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532 |
& \onslide<5->{\smath{X_1; a\cdot a^\star}}\\
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204
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533 |
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|
534 |
& & & \onslide<6->{by Arden}\\
|
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535 |
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213
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\onslide<6->{\smath{X_1}} & \onslide<6->{\smath{=}}
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& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
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213
|
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\onslide<6->{\smath{X_2}} & \onslide<6->{\smath{=}}
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& \onslide<6->{\smath{X_1; a\cdot a^\star}}\\
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|
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& & & \onslide<7->{by substitution}\\
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\onslide<7->{\smath{X_1}} & \onslide<7->{\smath{=}}
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204
|
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& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
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213
|
545 |
\onslide<7->{\smath{X_2}} & \onslide<7->{\smath{=}}
|
|
204
|
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& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
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|
547 |
\cdot a\cdot a^\star}}\\
|
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|
548 |
\end{tabular}
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\end{center}
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|
|
550 |
|
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551 |
\only<8->{
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\begin{textblock}{6}(2.5,4)
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\begin{block}{}
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\begin{minipage}{8cm}\raggedright
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555 |
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\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
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|
557 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
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%\draw[help lines] (0,0) grid (3,2);
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\node[state,initial] (p_0) {$X_1$};
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\node[state,accepting] (p_1) [right of=q_0] {$X_2$};
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\path[->] (p_0) edge [bend left] node {a} (p_1)
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edge [loop above] node {b} ()
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(p_1) edge [loop above] node {a} ()
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edge [bend left] node {b} (p_0);
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\end{tikzpicture}
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\end{minipage}
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\end{block}
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\end{textblock}}
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\only<1,2>{%
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\begin{textblock}{3}(0.6,1.2)
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\begin{tikzpicture}
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\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
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{\textcolor{red}{a}};
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\end{tikzpicture}
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\end{textblock}}
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\only<2>{%
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\begin{textblock}{3}(0.6,3.6)
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\begin{tikzpicture}
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\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
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{\textcolor{red}{a}};
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\end{tikzpicture}
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\end{textblock}}
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\only<4>{%
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\begin{textblock}{3}(0.6,2.9)
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\begin{tikzpicture}
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\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
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{\textcolor{red}{a}};
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\end{tikzpicture}
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\end{textblock}}
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\only<4>{%
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\begin{textblock}{3}(0.6,5.3)
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\begin{tikzpicture}
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\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
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{\textcolor{red}{a}};
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\end{tikzpicture}
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601 |
\end{textblock}}
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\only<5>{%
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603 |
\begin{textblock}{3}(1.0,5.6)
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\begin{tikzpicture}
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605 |
\node at (0,0) (A) {};
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606 |
\node at (0,1) (B) {};
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607 |
\draw[<-, line width=2mm, red] (B) to (A);
|
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|
608 |
\end{tikzpicture}
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609 |
\end{textblock}}
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\only<5,6>{%
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611 |
\begin{textblock}{3}(0.6,7.7)
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\begin{tikzpicture}
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613 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
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614 |
{\textcolor{red}{a}};
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\end{tikzpicture}
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|
616 |
\end{textblock}}
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\only<6>{%
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618 |
\begin{textblock}{3}(0.6,10.1)
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\begin{tikzpicture}
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620 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
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|
621 |
{\textcolor{red}{a}};
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|
622 |
\end{tikzpicture}
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|
623 |
\end{textblock}}
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624 |
\only<7>{%
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625 |
\begin{textblock}{3}(1.0,10.3)
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\begin{tikzpicture}
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627 |
\node at (0,0) (A) {};
|
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|
628 |
\node at (0,1) (B) {};
|
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|
629 |
\draw[->, line width=2mm, red] (B) to (A);
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630 |
\end{tikzpicture}
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631 |
\end{textblock}}
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632 |
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\end{frame}}
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634 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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635 |
*}
|
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636 |
|
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637 |
|
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|
638 |
text_raw {*
|
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639 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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640 |
\mode<presentation>{
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641 |
\begin{frame}[c]
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213
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\frametitle{\LARGE The Other Direction}
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204
|
643 |
|
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|
644 |
One has to prove
|
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645 |
|
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|
646 |
\begin{center}
|
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|
647 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}
|
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|
648 |
\end{center}
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|
649 |
|
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|
650 |
by induction on \smath{r}. Not trivial, but after a bit
|
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208
|
651 |
of thinking, one can find a \alert{refined} relation:\bigskip
|
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204
|
652 |
|
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208
|
653 |
|
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204
|
654 |
\begin{center}
|
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208
|
655 |
\mbox{\begin{tabular}{c@ {\hspace{7mm}}c@ {\hspace{7mm}}c}
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|
656 |
\begin{tikzpicture}[scale=1.1]
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|
657 |
%Circle
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|
658 |
\draw[thick] (0,0) circle (1.1);
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659 |
\end{tikzpicture}
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|
660 |
&
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|
661 |
\begin{tikzpicture}[scale=1.1]
|
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|
662 |
%Circle
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|
663 |
\draw[thick] (0,0) circle (1.1);
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|
664 |
%Main rays
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|
665 |
\foreach \a in {0, 90,...,359}
|
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|
666 |
\draw[very thick] (0, 0) -- (\a:1.1);
|
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|
667 |
\foreach \a / \l in {45/1, 135/2, 225/3, 315/4}
|
|
|
668 |
\draw (\a: 0.65) node {\small$a_\l$};
|
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|
669 |
\end{tikzpicture}
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|
670 |
&
|
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|
671 |
\begin{tikzpicture}[scale=1.1]
|
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|
672 |
%Circle
|
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|
673 |
\draw[red, thick] (0,0) circle (1.1);
|
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|
674 |
%Main rays
|
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|
675 |
\foreach \a in {0, 45,...,359}
|
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|
676 |
\draw[red, very thick] (0, 0) -- (\a:1.1);
|
|
|
677 |
\foreach \a / \l in {22.5/1.1, 67.5/1.2, 112.5/2.1, 157.5/2.2, 202.4/3.1, 247.5/3.2, 292.5/4.1, 337.5/4.2}
|
|
|
678 |
\draw (\a: 0.77) node {\textcolor{red}{\footnotesize$a_{\l}$}};
|
|
|
679 |
\end{tikzpicture}\\
|
|
|
680 |
\small\smath{U\!N\!IV} &
|
|
|
681 |
\small\smath{U\!N\!IV /\!/ \approx_{{\cal L}(r)}} &
|
|
|
682 |
\small\smath{U\!N\!IV /\!/ \alert{R}}
|
|
|
683 |
\end{tabular}}
|
|
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|
684 |
\end{center}
|
|
|
685 |
|
|
213
|
686 |
\begin{textblock}{5}(9.8,2.6)
|
|
|
687 |
\begin{tikzpicture}
|
|
|
688 |
\node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}};
|
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|
689 |
\end{tikzpicture}
|
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|
690 |
\end{textblock}
|
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204
|
691 |
|
|
|
692 |
|
|
|
693 |
\end{frame}}
|
|
|
694 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
695 |
*}
|
|
|
696 |
|
|
|
697 |
text_raw {*
|
|
|
698 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
699 |
\mode<presentation>{
|
|
213
|
700 |
\begin{frame}[t]
|
|
|
701 |
\frametitle{\LARGE Partial Derivatives}
|
|
|
702 |
|
|
|
703 |
\begin{itemize}
|
|
|
704 |
\item \ldots (set of) regular expressions after a string
|
|
|
705 |
has been parsed\\[10mm]
|
|
|
706 |
|
|
|
707 |
|
|
|
708 |
\item \alt<1>{\smath{\text{pders x r \mbox{$=$} pders y r}}}
|
|
215
|
709 |
{\smath{\underbrace{\text{pders x r \mbox{$=$} pders y r}}_{R}}}
|
|
213
|
710 |
refines \textcolor{blue}{x $\approx_{{\cal L}(\text{r})}$ y}\\[16mm]\pause
|
|
215
|
711 |
\item \smath{\text{finite} (U\!N\!IV /\!/ R)} \bigskip\pause
|
|
213
|
712 |
\item Therefore \smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}. Qed.
|
|
|
713 |
\end{itemize}
|
|
|
714 |
|
|
|
715 |
\only<2->{%
|
|
|
716 |
\begin{textblock}{5}(3.8,8.3)
|
|
|
717 |
\begin{tikzpicture}
|
|
|
718 |
\node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}};
|
|
|
719 |
\draw (2.2,0) node {Antimirov '95};
|
|
|
720 |
\end{tikzpicture}
|
|
|
721 |
\end{textblock}}
|
|
|
722 |
|
|
|
723 |
\end{frame}}
|
|
|
724 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
725 |
*}
|
|
|
726 |
|
|
|
727 |
|
|
|
728 |
|
|
|
729 |
text_raw {*
|
|
|
730 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
731 |
\mode<presentation>{
|
|
204
|
732 |
\begin{frame}[c]
|
|
|
733 |
\frametitle{\LARGE What Have We Achieved?}
|
|
|
734 |
|
|
|
735 |
\begin{itemize}
|
|
208
|
736 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
|
|
204
|
737 |
\bigskip\pause
|
|
213
|
738 |
\item regular languages are closed under complementation; this is now easy
|
|
204
|
739 |
\begin{center}
|
|
208
|
740 |
\smath{U\!N\!IV /\!/ \approx_A \;\;=\;\; U\!N\!IV /\!/ \approx_{\overline{A}}}
|
|
204
|
741 |
\end{center}\pause\bigskip
|
|
|
742 |
|
|
208
|
743 |
\item non-regularity (\smath{a^nb^n})
|
|
|
744 |
|
|
|
745 |
\begin{quote}
|
|
|
746 |
\begin{minipage}{8.8cm}
|
|
|
747 |
\begin{block}{}
|
|
|
748 |
\begin{minipage}{8.6cm}
|
|
213
|
749 |
If there exists a sufficiently large set \smath{B} (for example infinitely large),
|
|
208
|
750 |
such that
|
|
204
|
751 |
|
|
208
|
752 |
\begin{center}
|
|
|
753 |
\smath{\forall x,y \in B.\; x \not= y \;\Rightarrow\; x \not\approx_{A} y}.
|
|
|
754 |
\end{center}
|
|
|
755 |
|
|
|
756 |
then \smath{A} is not regular.
|
|
|
757 |
\end{minipage}
|
|
|
758 |
\end{block}
|
|
|
759 |
\end{minipage}\medskip\pause
|
|
|
760 |
|
|
215
|
761 |
\small(\smath{B \dn \bigcup_n a^n})
|
|
208
|
762 |
\end{quote}
|
|
|
763 |
|
|
204
|
764 |
\end{itemize}
|
|
|
765 |
|
|
|
766 |
\only<2>{
|
|
|
767 |
\begin{textblock}{10}(4,14)
|
|
|
768 |
\small
|
|
208
|
769 |
\smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
|
|
204
|
770 |
\end{textblock}
|
|
|
771 |
}
|
|
|
772 |
|
|
|
773 |
|
|
|
774 |
|
|
|
775 |
\end{frame}}
|
|
|
776 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
777 |
*}
|
|
|
778 |
|
|
|
779 |
|
|
|
780 |
text_raw {*
|
|
|
781 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
782 |
\mode<presentation>{
|
|
|
783 |
\begin{frame}[c]
|
|
|
784 |
\frametitle{\LARGE Conclusion}
|
|
|
785 |
|
|
|
786 |
\begin{itemize}
|
|
211
|
787 |
\item We have never seen a proof of Myhill-Nerode based on
|
|
208
|
788 |
regular expressions.\smallskip\pause
|
|
204
|
789 |
|
|
208
|
790 |
\item great source of examples (inductions)\smallskip\pause
|
|
204
|
791 |
|
|
208
|
792 |
\item no need to fight the theorem prover:\\
|
|
|
793 |
\begin{itemize}
|
|
|
794 |
\item first direction (790 loc)\\
|
|
|
795 |
\item second direction (400 / 390 loc)\pause
|
|
|
796 |
\end{itemize}\smallskip
|
|
204
|
797 |
|
|
213
|
798 |
\item I have \alert{\bf not} yet used it in teaching for undergraduates.\pause
|
|
204
|
799 |
|
|
|
800 |
\end{itemize}
|
|
|
801 |
|
|
208
|
802 |
\only<5->{
|
|
|
803 |
\begin{textblock}{13.8}(1,4)
|
|
|
804 |
\begin{block}{}\mbox{}\hspace{3mm}
|
|
|
805 |
\begin{minipage}{11cm}\raggedright
|
|
|
806 |
\large
|
|
|
807 |
|
|
213
|
808 |
{\bf Bold Claim: }\alert{(not proved!)}\medskip
|
|
208
|
809 |
|
|
|
810 |
{\bf 95\%} of regular language theory can be done without
|
|
213
|
811 |
automata!\medskip\\\ldots and this is much more tasteful ;o)
|
|
208
|
812 |
|
|
|
813 |
\end{minipage}
|
|
|
814 |
\end{block}
|
|
|
815 |
\end{textblock}}
|
|
|
816 |
|
|
|
817 |
\end{frame}}
|
|
|
818 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
819 |
*}
|
|
|
820 |
|
|
|
821 |
text_raw {*
|
|
|
822 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
823 |
\mode<presentation>{
|
|
213
|
824 |
\begin{frame}[b]
|
|
|
825 |
\frametitle{\mbox{}\\[2cm]\textcolor{red}{Thank you!\\[5mm]Questions?}}
|
|
208
|
826 |
|
|
204
|
827 |
\end{frame}}
|
|
|
828 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
829 |
*}
|
|
|
830 |
|
|
|
831 |
(*<*)
|
|
|
832 |
end
|
|
|
833 |
(*>*) |