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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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\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 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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\mode<presentation>{
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\begin{frame}[c]
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\frametitle{}
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\begin{textblock}{12.9}(1.5,3.2)
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\begin{block}{}
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\begin{minipage}{12.4cm}\raggedright
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\large I want to teach \alert{students} with\\
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theorem provers (especially 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
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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, 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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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 ($>\!\!\!>$2000 loc)
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\item Mirkin's partial derivative automaton construction (10600 loc)
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\end{itemize}
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\end{itemize}
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\end{frame}}
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*}
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text_raw {*
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\mode<presentation>{
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\begin{frame}[t]
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\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
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\mbox{}\\[-10mm]\mbox{}
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\begin{center}
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\huge\bf\textcolor{gray}{in HOL}
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\end{center}
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\begin{itemize}
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\item automata @{text "\<Rightarrow>"} graphs, matrices, functions
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\item<2-> combining automata/graphs
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&
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\onslide<3->{\raisebox{1.1mm}{\bf\Large$\;\Rightarrow\,$}}
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&
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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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\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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\medskip
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\only<6->{A solution:\;\;\smath{\text{nat}} \;@{text "\<Rightarrow>"}\; state nodes\medskip}
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\only<7->{You have to \alert{\underline{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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\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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{\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. 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}}
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\item<8-> most textbooks are about automata
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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)}\medskip
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\item will help with closure properties of regular languages\bigskip\pause
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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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*}
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\frametitle{\LARGE The Myhill-Nerode Theorem}
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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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{$U\!N\!IV$};
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\draw (-0.3,-1.1) node {\begin{tabular}{l}set of all\\[-1mm] strings\end{tabular}};
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\end{tikzpicture}
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\end{textblock}
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\only<2->{%
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\begin{textblock}{5}(9.4,7.2)
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\begin{tikzpicture}
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\node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]
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{@{text "\<lbrakk>x\<rbrakk>"}$_{\approx_{A}}$};
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\draw (0.9,-1.1) node {\begin{tabular}{l}equivalence class\end{tabular}};
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\end{tikzpicture}
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\end{textblock}}
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\only<3->{
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\begin{textblock}{11.9}(1.7,3)
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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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\end{tabular}
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\end{minipage}
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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{\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{final}_A\,X \dn \{[\!|x|\!]_{\approx_{A}}\;|\;x \in A\}}
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\smallskip
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\item we can prove: \smath{A = \bigcup \{X.\;\text{final}_A\,X\}}
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\end{itemize}
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\only<2->{%
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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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text_raw {*
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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 {$X$};
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\draw[white] (0.9,0.5) node {$Y$};
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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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459 |
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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463 |
\begin{frame}[c]
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\frametitle{\LARGE Systems of Equations}
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465 |
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211
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466 |
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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470 |
\\[-13mm]
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471 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
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472 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
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473 |
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%\draw[help lines] (0,0) grid (3,2);
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475 |
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\node[state,initial] (p_0) {$R_1$};
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477 |
\node[state,accepting] (p_1) [right of=q_0] {$R_2$};
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478 |
|
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479 |
\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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484 |
\\[-13mm]
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485 |
\end{tabular}
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486 |
\end{center}
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487 |
|
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488 |
\begin{center}
|
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|
489 |
\begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
|
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|
490 |
& \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
|
|
|
491 |
& \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\
|
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|
492 |
\end{tabular}
|
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493 |
\end{center}
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494 |
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495 |
\end{frame}}
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496 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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497 |
*}
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498 |
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499 |
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text_raw {*
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502 |
\mode<presentation>{
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211
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\begin{frame}<1-2,4->[t]
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204
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\small
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505 |
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506 |
\begin{center}
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|
507 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
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|
508 |
\onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
|
|
|
509 |
& \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
510 |
\onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
|
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|
511 |
& \onslide<1->{\smath{R_1; a + R_2; a}}\\
|
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|
512 |
|
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|
513 |
& & & \onslide<2->{by Arden}\\
|
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|
514 |
|
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|
515 |
\onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
|
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|
516 |
& \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
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|
517 |
\onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
|
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& \only<2->{\smath{R_1; a\cdot a^\star}}\\
|
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204
|
519 |
|
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|
520 |
& & & \onslide<4->{by Arden}\\
|
|
|
521 |
|
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|
522 |
\onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
|
|
|
523 |
& \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
524 |
\onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
|
|
|
525 |
& \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
526 |
|
|
|
527 |
& & & \onslide<5->{by substitution}\\
|
|
|
528 |
|
|
|
529 |
\onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
|
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|
530 |
& \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
531 |
\onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
|
|
|
532 |
& \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
533 |
|
|
|
534 |
& & & \onslide<6->{by Arden}\\
|
|
|
535 |
|
|
|
536 |
\onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
|
|
|
537 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
538 |
\onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
|
|
|
539 |
& \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
540 |
|
|
|
541 |
& & & \onslide<7->{by substitution}\\
|
|
|
542 |
|
|
|
543 |
\onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
|
|
|
544 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
545 |
\onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
|
|
|
546 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
|
|
547 |
\cdot a\cdot a^\star}}\\
|
|
|
548 |
\end{tabular}
|
|
|
549 |
\end{center}
|
|
|
550 |
|
|
|
551 |
\only<8->{
|
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|
552 |
\begin{textblock}{6}(2.5,4)
|
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|
553 |
\begin{block}{}
|
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|
554 |
\begin{minipage}{8cm}\raggedright
|
|
|
555 |
|
|
|
556 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
|
|
|
557 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
|
558 |
|
|
|
559 |
%\draw[help lines] (0,0) grid (3,2);
|
|
|
560 |
|
|
|
561 |
\node[state,initial] (p_0) {$R_1$};
|
|
|
562 |
\node[state,accepting] (p_1) [right of=q_0] {$R_2$};
|
|
|
563 |
|
|
|
564 |
\path[->] (p_0) edge [bend left] node {a} (p_1)
|
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|
565 |
edge [loop above] node {b} ()
|
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|
566 |
(p_1) edge [loop above] node {a} ()
|
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|
567 |
edge [bend left] node {b} (p_0);
|
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|
568 |
\end{tikzpicture}
|
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|
569 |
|
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|
570 |
\end{minipage}
|
|
|
571 |
\end{block}
|
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572 |
\end{textblock}}
|
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|
573 |
|
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|
574 |
\end{frame}}
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575 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
576 |
*}
|
|
|
577 |
|
|
|
578 |
|
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|
579 |
text_raw {*
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|
580 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
581 |
\mode<presentation>{
|
|
|
582 |
\begin{frame}[c]
|
|
|
583 |
\frametitle{\LARGE Other Direction}
|
|
|
584 |
|
|
|
585 |
One has to prove
|
|
|
586 |
|
|
|
587 |
\begin{center}
|
|
208
|
588 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}
|
|
204
|
589 |
\end{center}
|
|
|
590 |
|
|
|
591 |
by induction on \smath{r}. Not trivial, but after a bit
|
|
208
|
592 |
of thinking, one can find a \alert{refined} relation:\bigskip
|
|
204
|
593 |
|
|
208
|
594 |
|
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204
|
595 |
\begin{center}
|
|
208
|
596 |
\mbox{\begin{tabular}{c@ {\hspace{7mm}}c@ {\hspace{7mm}}c}
|
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|
597 |
\begin{tikzpicture}[scale=1.1]
|
|
|
598 |
%Circle
|
|
|
599 |
\draw[thick] (0,0) circle (1.1);
|
|
|
600 |
\end{tikzpicture}
|
|
|
601 |
&
|
|
|
602 |
\begin{tikzpicture}[scale=1.1]
|
|
|
603 |
%Circle
|
|
|
604 |
\draw[thick] (0,0) circle (1.1);
|
|
|
605 |
%Main rays
|
|
|
606 |
\foreach \a in {0, 90,...,359}
|
|
|
607 |
\draw[very thick] (0, 0) -- (\a:1.1);
|
|
|
608 |
\foreach \a / \l in {45/1, 135/2, 225/3, 315/4}
|
|
|
609 |
\draw (\a: 0.65) node {\small$a_\l$};
|
|
|
610 |
\end{tikzpicture}
|
|
|
611 |
&
|
|
|
612 |
\begin{tikzpicture}[scale=1.1]
|
|
|
613 |
%Circle
|
|
|
614 |
\draw[red, thick] (0,0) circle (1.1);
|
|
|
615 |
%Main rays
|
|
|
616 |
\foreach \a in {0, 45,...,359}
|
|
|
617 |
\draw[red, very thick] (0, 0) -- (\a:1.1);
|
|
|
618 |
\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}
|
|
|
619 |
\draw (\a: 0.77) node {\textcolor{red}{\footnotesize$a_{\l}$}};
|
|
|
620 |
\end{tikzpicture}\\
|
|
|
621 |
\small\smath{U\!N\!IV} &
|
|
|
622 |
\small\smath{U\!N\!IV /\!/ \approx_{{\cal L}(r)}} &
|
|
|
623 |
\small\smath{U\!N\!IV /\!/ \alert{R}}
|
|
|
624 |
\end{tabular}}
|
|
204
|
625 |
\end{center}
|
|
|
626 |
|
|
|
627 |
|
|
|
628 |
|
|
|
629 |
|
|
|
630 |
\end{frame}}
|
|
|
631 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
632 |
*}
|
|
|
633 |
|
|
|
634 |
text_raw {*
|
|
|
635 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
636 |
\mode<presentation>{
|
|
|
637 |
\begin{frame}[c]
|
|
|
638 |
\frametitle{\LARGE What Have We Achieved?}
|
|
|
639 |
|
|
|
640 |
\begin{itemize}
|
|
208
|
641 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
|
|
204
|
642 |
\bigskip\pause
|
|
|
643 |
\item regular languages are closed under complementation; this is easy
|
|
|
644 |
\begin{center}
|
|
208
|
645 |
\smath{U\!N\!IV /\!/ \approx_A \;\;=\;\; U\!N\!IV /\!/ \approx_{\overline{A}}}
|
|
204
|
646 |
\end{center}\pause\bigskip
|
|
|
647 |
|
|
208
|
648 |
\item non-regularity (\smath{a^nb^n})
|
|
|
649 |
|
|
|
650 |
\begin{quote}
|
|
|
651 |
\begin{minipage}{8.8cm}
|
|
|
652 |
\begin{block}{}
|
|
|
653 |
\begin{minipage}{8.6cm}
|
|
|
654 |
If there exists a sufficiently large set \smath{B} (for example infinite),
|
|
|
655 |
such that
|
|
204
|
656 |
|
|
208
|
657 |
\begin{center}
|
|
|
658 |
\smath{\forall x,y \in B.\; x \not= y \;\Rightarrow\; x \not\approx_{A} y}.
|
|
|
659 |
\end{center}
|
|
|
660 |
|
|
|
661 |
then \smath{A} is not regular.
|
|
|
662 |
\end{minipage}
|
|
|
663 |
\end{block}
|
|
|
664 |
\end{minipage}\medskip\pause
|
|
|
665 |
|
|
|
666 |
\small(\smath{A \dn \bigcup_i a^i})
|
|
|
667 |
\end{quote}
|
|
|
668 |
|
|
204
|
669 |
\end{itemize}
|
|
|
670 |
|
|
|
671 |
\only<2>{
|
|
|
672 |
\begin{textblock}{10}(4,14)
|
|
|
673 |
\small
|
|
208
|
674 |
\smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
|
|
204
|
675 |
\end{textblock}
|
|
|
676 |
}
|
|
|
677 |
|
|
|
678 |
|
|
|
679 |
|
|
|
680 |
\end{frame}}
|
|
|
681 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
682 |
*}
|
|
|
683 |
|
|
|
684 |
|
|
|
685 |
text_raw {*
|
|
|
686 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
687 |
\mode<presentation>{
|
|
|
688 |
\begin{frame}[c]
|
|
|
689 |
\frametitle{\LARGE Conclusion}
|
|
|
690 |
|
|
|
691 |
\begin{itemize}
|
|
211
|
692 |
\item We have never seen a proof of Myhill-Nerode based on
|
|
208
|
693 |
regular expressions.\smallskip\pause
|
|
204
|
694 |
|
|
208
|
695 |
\item great source of examples (inductions)\smallskip\pause
|
|
204
|
696 |
|
|
208
|
697 |
\item no need to fight the theorem prover:\\
|
|
|
698 |
\begin{itemize}
|
|
|
699 |
\item first direction (790 loc)\\
|
|
|
700 |
\item second direction (400 / 390 loc)\pause
|
|
|
701 |
\end{itemize}\smallskip
|
|
204
|
702 |
|
|
208
|
703 |
\item I have \alert{\bf not} yet used it for teaching of undergraduates.\pause
|
|
204
|
704 |
|
|
|
705 |
\end{itemize}
|
|
|
706 |
|
|
208
|
707 |
\only<5->{
|
|
|
708 |
\begin{textblock}{13.8}(1,4)
|
|
|
709 |
\begin{block}{}\mbox{}\hspace{3mm}
|
|
|
710 |
\begin{minipage}{11cm}\raggedright
|
|
|
711 |
\large
|
|
|
712 |
|
|
|
713 |
{\bf Bold Claim }\alert{(not proved!)}\medskip
|
|
|
714 |
|
|
|
715 |
{\bf 95\%} of regular language theory can be done without
|
|
|
716 |
automata\medskip\\\ldots this is much more tasteful. ;o)
|
|
|
717 |
|
|
|
718 |
\end{minipage}
|
|
|
719 |
\end{block}
|
|
|
720 |
\end{textblock}}
|
|
|
721 |
|
|
|
722 |
\end{frame}}
|
|
|
723 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
724 |
*}
|
|
|
725 |
|
|
|
726 |
text_raw {*
|
|
|
727 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
728 |
\mode<presentation>{
|
|
|
729 |
\begin{frame}[c]
|
|
|
730 |
\frametitle{\mbox{}\\[2cm]\textcolor{red}{Questions?}}
|
|
|
731 |
|
|
|
732 |
|
|
|
733 |
|
|
204
|
734 |
\end{frame}}
|
|
|
735 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
736 |
*}
|
|
|
737 |
|
|
|
738 |
(*<*)
|
|
|
739 |
end
|
|
|
740 |
(*>*) |