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(*<*)
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theory Slides2
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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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\renewcommand{\slidecaption}{St Andrews, 19 November 2011}
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\newcommand{\sout}[1]{\tikz[baseline=(X.base), inner sep=-0.1pt, outer sep=0pt]
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\node [cross out,red, ultra thick, draw] (X) {\textcolor{black}{#1}};}
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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 Formalising\\[-3mm]
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\LARGE Regular Language Theory\\[-3mm]
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\LARGE with Regular Expressions,\\[-3mm]
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\LARGE \alert<2>{Only}\\[0mm]
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\end{tabular}}
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\begin{center}
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Christian Urban\\
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\small King's College London
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\end{center}\bigskip
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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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\end{frame}}
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*}
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text_raw {*
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\frametitle{}
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\includegraphics[scale=0.5]{roy.jpg}\medskip
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Roy intertwined with my scientific life on many occasions, most
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notably:\bigskip
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\begin{itemize}
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\item he admitted me for M.Phil.~in St Andrews and\\
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made me like theory\smallskip
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\item sent me to Cambridge for Ph.D.\bigskip
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\item made me appreciate precision in proofs
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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{}
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\begin{tabular}{c@ {\hspace{2mm}}c}
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\\[6mm]
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\begin{tabular}{c}
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\includegraphics[scale=0.11]{harper.jpg}\\[-2mm]
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{\footnotesize Bob Harper}\\[-2.5mm]
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{\footnotesize (CMU)}
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\end{tabular}
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\begin{tabular}{c}
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\includegraphics[scale=0.37]{pfenning.jpg}\\[-2mm]
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{\footnotesize Frank Pfenning}\\[-2.5mm]
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{\footnotesize (CMU)}
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\end{tabular} &
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\begin{tabular}{p{6cm}}
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\raggedright
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\color{gray}{published a proof in\\ {\bf ACM Transactions on Computational Logic}, 2005,
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$\sim$31pp}
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\end{tabular}\\
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\pause
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\\[0mm]
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\begin{tabular}{c}
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\includegraphics[scale=0.36]{appel.jpg}\\[-2mm]
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{\footnotesize Andrew Appel}\\[-2.5mm]
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{\footnotesize (Princeton)}
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\end{tabular} &
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\begin{tabular}{p{6cm}}
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\raggedright
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\color{gray}{relied on their proof in a\\ {\bf security} critical application}
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\end{tabular}
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\end{tabular}\medskip\pause
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\small
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\begin{minipage}{1.0\textwidth}
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(I also found an {\bf error} in my Ph.D.-thesis about cut-elimination
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examined by Henk Barendregt and Andy Pitts.)
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\end{minipage}
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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{}\\[-15mm]\mbox{}
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\begin{center}
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\huge\bf\textcolor{gray}{in Theorem Provers}\\
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\footnotesize\textcolor{gray}{e.g.~Isabelle, Coq, HOL4, \ldots}
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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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\begin{center}
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%\draw[step=2mm] (-1,-1) grid (1,1);
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\draw[rounded corners=1mm, very thick] (-1.0,-0.3) rectangle (-0.2,0.3);
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\draw[rounded corners=1mm, very thick] ( 0.2,-0.3) rectangle ( 1.0,0.3);
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\node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\draw (-0.6,0.0) node {\small$A_1$};
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\draw ( 0.6,0.0) node {\small$A_2$};
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\end{tikzpicture}}
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&
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\onslide<3->{\raisebox{1.1mm}{\bf\Large$\;\Rightarrow\,$}}
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&
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\onslide<3->{\begin{tikzpicture}[scale=1]
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%\draw[step=2mm] (-1,-1) grid (1,1);
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\node (A) at (-1.0,0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\node (C) at (-0.2, 0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\node (D) at (-0.2,-0.13) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\node (E) at (1.0, 0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\node (F) at (1.0,-0.0) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\node (G) at (1.0,-0.2) [circle, very thick, draw, fill=white, inner sep=0.4mm] {};
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\draw (C) to [red, very thick, bend left=45] (B);
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\draw (D) to [red, very thick, bend right=45] (B);
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\draw (-0.6,0.0) node {\small$A_1$};
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\draw ( 0.6,0.0) node {\small$A_2$};
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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}{@ {\hspace{-5mm}}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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\begin{block}{}
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\medskip
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\begin{minipage}{14cm}\raggedright
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Problems with definition for regularity:\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{block}
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\end{textblock}}
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\medskip
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\only<6->{\underline{A solution}:\;\;use \smath{\text{nat}}s \;@{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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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{}\\[-15mm]\mbox{}
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\begin{center}
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\huge\bf\textcolor{gray}{in Theorem Provers}\\
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\footnotesize\textcolor{gray}{e.g.~Isabelle, Coq, HOL4, \ldots}
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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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text_raw {*
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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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\begin{minipage}{13.4cm}\raggedright
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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{minipage}
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\end{block}
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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?\medskip\pause
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\begin{minipage}{1.1\textwidth}
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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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{\footnotesize(@{text "\<Rightarrow>"}Brozowski'64, Owens et al '09)}}
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\item<8-> most textbooks are about automata
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\end{itemize}
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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:\medskip
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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{center}
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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{\LARGE The Myhill-Nerode Theorem}
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\begin{center}
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\only<1>{%
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\draw[very thick] (0.5,0.5) circle (.6cm);
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\end{tikzpicture}}%
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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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\begin{tikzpicture}
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\node at (0,0) [single arrow, fill=red,text=white, minimum height=2cm]
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{$U\!N\!IV$};
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\draw (-0.3,-1.1) node {\begin{tabular}{l}set of all\\[-1mm] strings\end{tabular}};
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\end{tikzpicture}
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\end{textblock}
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\only<2->{%
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\begin{textblock}{5}(9.1,7.2)
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\begin{tikzpicture}
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\node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]
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{@{text "\<lbrakk>x\<rbrakk>"}$_{\approx_{A}}$};
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\draw (0.9,-1.1) node {\begin{tabular}{l}an 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{block}
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\end{textblock}}
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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}[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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\end{textblock}
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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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\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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*}
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text_raw {*
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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) $) {};
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\end{tikzpicture}
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\end{center}
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436 |
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\begin{center}
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\smath{X \stackrel{c}{\longrightarrow} Y \;\dn\; X ; c \subseteq Y}
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439 |
\end{center}
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440 |
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\onslide<8>{
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442 |
\begin{tabular}{c}
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\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
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\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
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\node[state,initial] (q_0) {$R_1$};
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446 |
\end{tikzpicture}
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\end{tabular}}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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457 |
\begin{frame}[c]
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458 |
\frametitle{\LARGE Systems of Equations}
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459 |
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460 |
Inspired by a method of Brzozowski\;'64:\bigskip\bigskip
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461 |
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462 |
\begin{center}
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463 |
\begin{tabular}{@ {\hspace{-20mm}}c}
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464 |
\\[-13mm]
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\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
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\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
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467 |
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%\draw[help lines] (0,0) grid (3,2);
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469 |
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470 |
\node[state,initial] (p_0) {$X_1$};
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471 |
\node[state,accepting] (p_1) [right of=q_0] {$X_2$};
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472 |
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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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478 |
\\[-13mm]
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479 |
\end{tabular}
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480 |
\end{center}
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481 |
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482 |
\begin{center}
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|
483 |
\begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
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484 |
& \smath{X_1} & \smath{=} & \smath{X_1;b + X_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
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|
485 |
& \smath{X_2} & \smath{=} & \smath{X_1;a + X_2;a}\medskip\\
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486 |
\end{tabular}
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487 |
\end{center}
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488 |
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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491 |
*}
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492 |
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493 |
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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496 |
\mode<presentation>{
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497 |
\begin{frame}<1-2,4->[t]
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498 |
\small
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499 |
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500 |
\begin{center}
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501 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
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|
502 |
\onslide<1->{\smath{X_1}} & \onslide<1->{\smath{=}}
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|
503 |
& \onslide<1->{\smath{X_1; b + X_2; b + \lambda;[]}}\\
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504 |
\onslide<1->{\smath{X_2}} & \onslide<1->{\smath{=}}
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|
505 |
& \onslide<1->{\smath{X_1; a + X_2; a}}\\
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|
506 |
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507 |
& & & \onslide<2->{by Arden}\\
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508 |
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|
509 |
\onslide<2->{\smath{X_1}} & \onslide<2->{\smath{=}}
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|
510 |
& \onslide<2->{\smath{X_1; b + X_2; b + \lambda;[]}}\\
|
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|
511 |
\onslide<2->{\smath{X_2}} & \onslide<2->{\smath{=}}
|
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|
512 |
& \only<2->{\smath{X_1; a\cdot a^\star}}\\
|
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|
513 |
|
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|
514 |
& & & \onslide<4->{by Arden}\\
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515 |
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|
516 |
\onslide<4->{\smath{X_1}} & \onslide<4->{\smath{=}}
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|
517 |
& \onslide<4->{\smath{X_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
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|
518 |
\onslide<4->{\smath{X_2}} & \onslide<4->{\smath{=}}
|
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|
519 |
& \onslide<4->{\smath{X_1; a\cdot a^\star}}\\
|
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|
520 |
|
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|
521 |
& & & \onslide<5->{by substitution}\\
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|
522 |
|
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|
523 |
\onslide<5->{\smath{X_1}} & \onslide<5->{\smath{=}}
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|
524 |
& \onslide<5->{\smath{X_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
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|
525 |
\onslide<5->{\smath{X_2}} & \onslide<5->{\smath{=}}
|
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|
526 |
& \onslide<5->{\smath{X_1; a\cdot a^\star}}\\
|
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527 |
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|
528 |
& & & \onslide<6->{by Arden}\\
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529 |
|
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|
530 |
\onslide<6->{\smath{X_1}} & \onslide<6->{\smath{=}}
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|
531 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
532 |
\onslide<6->{\smath{X_2}} & \onslide<6->{\smath{=}}
|
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|
533 |
& \onslide<6->{\smath{X_1; a\cdot a^\star}}\\
|
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|
534 |
|
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|
535 |
& & & \onslide<7->{by substitution}\\
|
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|
536 |
|
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|
537 |
\onslide<7->{\smath{X_1}} & \onslide<7->{\smath{=}}
|
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|
538 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
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|
539 |
\onslide<7->{\smath{X_2}} & \onslide<7->{\smath{=}}
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|
540 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
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|
541 |
\cdot a\cdot a^\star}}\\
|
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542 |
\end{tabular}
|
|
|
543 |
\end{center}
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544 |
|
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|
545 |
\only<8->{
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546 |
\begin{textblock}{6}(2.5,4)
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547 |
\begin{block}{}
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548 |
\begin{minipage}{8cm}\raggedright
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|
549 |
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550 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
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|
551 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
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552 |
|
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553 |
%\draw[help lines] (0,0) grid (3,2);
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554 |
|
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555 |
\node[state,initial] (p_0) {$X_1$};
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556 |
\node[state,accepting] (p_1) [right of=q_0] {$X_2$};
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557 |
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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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560 |
(p_1) edge [loop above] node {a} ()
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561 |
edge [bend left] node {b} (p_0);
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562 |
\end{tikzpicture}
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563 |
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\end{minipage}
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565 |
\end{block}
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566 |
\end{textblock}}
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567 |
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568 |
\only<1,2>{%
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569 |
\begin{textblock}{3}(0.6,1.2)
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570 |
\begin{tikzpicture}
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571 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
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572 |
{\textcolor{red}{a}};
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\end{tikzpicture}
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574 |
\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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581 |
\end{textblock}}
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582 |
\only<4>{%
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583 |
\begin{textblock}{3}(0.6,2.9)
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584 |
\begin{tikzpicture}
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585 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
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586 |
{\textcolor{red}{a}};
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\end{tikzpicture}
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588 |
\end{textblock}}
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\only<4>{%
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590 |
\begin{textblock}{3}(0.6,5.3)
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\begin{tikzpicture}
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592 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
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593 |
{\textcolor{red}{a}};
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594 |
\end{tikzpicture}
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595 |
\end{textblock}}
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596 |
\only<5>{%
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597 |
\begin{textblock}{3}(1.0,5.6)
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\begin{tikzpicture}
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599 |
\node at (0,0) (A) {};
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600 |
\node at (0,1) (B) {};
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601 |
\draw[<-, line width=2mm, red] (B) to (A);
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602 |
\end{tikzpicture}
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603 |
\end{textblock}}
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|
604 |
\only<5,6>{%
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605 |
\begin{textblock}{3}(0.6,7.7)
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606 |
\begin{tikzpicture}
|
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|
607 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
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|
608 |
{\textcolor{red}{a}};
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|
609 |
\end{tikzpicture}
|
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|
610 |
\end{textblock}}
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|
611 |
\only<6>{%
|
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|
612 |
\begin{textblock}{3}(0.6,10.1)
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613 |
\begin{tikzpicture}
|
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|
614 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
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|
615 |
{\textcolor{red}{a}};
|
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|
616 |
\end{tikzpicture}
|
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|
617 |
\end{textblock}}
|
|
|
618 |
\only<7>{%
|
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|
619 |
\begin{textblock}{3}(1.0,10.3)
|
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|
620 |
\begin{tikzpicture}
|
|
|
621 |
\node at (0,0) (A) {};
|
|
|
622 |
\node at (0,1) (B) {};
|
|
|
623 |
\draw[->, line width=2mm, red] (B) to (A);
|
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|
624 |
\end{tikzpicture}
|
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|
625 |
\end{textblock}}
|
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|
626 |
|
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|
627 |
\end{frame}}
|
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628 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
629 |
*}
|
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630 |
|
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631 |
|
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632 |
text_raw {*
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633 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
634 |
\mode<presentation>{
|
|
|
635 |
\begin{frame}[c]
|
|
|
636 |
\frametitle{\LARGE The Other Direction}
|
|
|
637 |
|
|
|
638 |
One has to prove
|
|
|
639 |
|
|
|
640 |
\begin{center}
|
|
|
641 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}
|
|
|
642 |
\end{center}
|
|
|
643 |
|
|
|
644 |
by induction on \smath{r}. Not trivial, but after a bit
|
|
|
645 |
of thinking, one can find a \alert{refined} relation:\bigskip
|
|
|
646 |
|
|
|
647 |
|
|
|
648 |
\begin{center}
|
|
|
649 |
\mbox{\begin{tabular}{c@ {\hspace{7mm}}c@ {\hspace{7mm}}c}
|
|
|
650 |
\begin{tikzpicture}[scale=1.1]
|
|
|
651 |
%Circle
|
|
|
652 |
\draw[thick] (0,0) circle (1.1);
|
|
|
653 |
\end{tikzpicture}
|
|
|
654 |
&
|
|
|
655 |
\begin{tikzpicture}[scale=1.1]
|
|
|
656 |
%Circle
|
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|
657 |
\draw[thick] (0,0) circle (1.1);
|
|
|
658 |
%Main rays
|
|
|
659 |
\foreach \a in {0, 90,...,359}
|
|
|
660 |
\draw[very thick] (0, 0) -- (\a:1.1);
|
|
|
661 |
\foreach \a / \l in {45/1, 135/2, 225/3, 315/4}
|
|
|
662 |
\draw (\a: 0.65) node {\small$a_\l$};
|
|
|
663 |
\end{tikzpicture}
|
|
|
664 |
&
|
|
|
665 |
\begin{tikzpicture}[scale=1.1]
|
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|
666 |
%Circle
|
|
|
667 |
\draw[red, thick] (0,0) circle (1.1);
|
|
|
668 |
%Main rays
|
|
|
669 |
\foreach \a in {0, 45,...,359}
|
|
|
670 |
\draw[red, very thick] (0, 0) -- (\a:1.1);
|
|
|
671 |
\foreach \a / \l in {22.5/1.1, 67.5/1.2, 112.5/2.1, 157.5/2.2, 202.4/3.1, 247.5/3.2, 292.5/4.1, 337.5/4.2}
|
|
|
672 |
\draw (\a: 0.77) node {\textcolor{red}{\footnotesize$a_{\l}$}};
|
|
|
673 |
\end{tikzpicture}\\
|
|
|
674 |
\small\smath{U\!N\!IV} &
|
|
|
675 |
\small\smath{U\!N\!IV /\!/ \approx_{{\cal L}(r)}} &
|
|
|
676 |
\small\smath{U\!N\!IV /\!/ \alert{R}}
|
|
|
677 |
\end{tabular}}
|
|
|
678 |
\end{center}
|
|
|
679 |
|
|
|
680 |
\begin{textblock}{5}(9.8,2.6)
|
|
|
681 |
\begin{tikzpicture}
|
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|
682 |
\node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}};
|
|
|
683 |
\end{tikzpicture}
|
|
|
684 |
\end{textblock}
|
|
|
685 |
|
|
|
686 |
|
|
|
687 |
\end{frame}}
|
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|
688 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
689 |
*}
|
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|
690 |
|
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|
691 |
text_raw {*
|
|
|
692 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
693 |
\mode<presentation>{
|
|
|
694 |
\begin{frame}[t]
|
|
|
695 |
\frametitle{\LARGE\begin{tabular}{c}Derivatives of RExps\end{tabular}}
|
|
|
696 |
|
|
|
697 |
\begin{itemize}
|
|
|
698 |
\item introduced by Brozowski~'64
|
|
|
699 |
\item a regular expressions after a character has been parsed\\[-18mm]\mbox{}
|
|
|
700 |
\end{itemize}
|
|
|
701 |
|
|
|
702 |
\only<1>{%
|
|
|
703 |
\textcolor{blue}{%
|
|
|
704 |
\begin{center}
|
|
|
705 |
\begin{tabular}{@ {}lc@ {\hspace{3mm}}l@ {}}
|
|
|
706 |
der c $\varnothing$ & $\dn$ & $\varnothing$\\
|
|
|
707 |
der c [] & $\dn$ & $\varnothing$\\
|
|
|
708 |
der c d & $\dn$ & if c $=$ d then [] else $\varnothing$\\
|
|
|
709 |
der c ($r_1 + r_2$) & $\dn$ & (der c $r_1$) $+$ (der c $r_2$)\\
|
|
|
710 |
der c ($r^\star$) & $\dn$ & (der c $r$) $\cdot$ $r^\star$\\
|
|
|
711 |
der c ($r_1 \cdot r_2$) & $\dn$ & if nullable $r_1$\\
|
|
|
712 |
& & then (der c $r_1$) $\cdot$ $r_2$ $+$ (der c $r_2$)\\
|
|
|
713 |
& & else (der c $r_1$) $\cdot$ $r_2$\\
|
|
|
714 |
\end{tabular}
|
|
|
715 |
\end{center}}}
|
|
|
716 |
\only<2>{%
|
|
|
717 |
\textcolor{blue}{%
|
|
|
718 |
\begin{center}
|
|
|
719 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
|
|
|
720 |
pder c $\varnothing$ & $\dn$ & \alert{$\{\}$}\\
|
|
|
721 |
pder c [] & $\dn$ & \alert{$\{\}$}\\
|
|
|
722 |
pder c d & $\dn$ & if c $=$ d then $\{$[]$\}$ else $\{\}$\\
|
|
|
723 |
pder c ($r_1 + r_2$) & $\dn$ & (pder c $r_1$) \alert{$\cup$} (der c $r_2$)\\
|
|
|
724 |
pder c ($r^\star$) & $\dn$ & (pder c $r$) $\cdot$ $r^\star$\\
|
|
|
725 |
pder c ($r_1 \cdot r_2$) & $\dn$ & if nullable $r_1$\\
|
|
|
726 |
& & then (pder c $r_1$) $\cdot$ $r_2$ \alert{$\cup$} (pder c $r_2$)\\
|
|
|
727 |
& & else (pder c $r_1$) $\cdot$ $r_2$\\
|
|
|
728 |
\end{tabular}
|
|
|
729 |
\end{center}}}
|
|
|
730 |
|
|
|
731 |
\only<2>{
|
|
|
732 |
\begin{textblock}{6}(8.5,4.7)
|
|
|
733 |
\begin{block}{}
|
|
|
734 |
\begin{quote}
|
|
|
735 |
\begin{minipage}{6cm}\raggedright
|
|
|
736 |
\begin{itemize}
|
|
|
737 |
\item partial derivatives
|
|
|
738 |
\item by Antimirov~'95
|
|
|
739 |
\end{itemize}
|
|
|
740 |
\end{minipage}
|
|
|
741 |
\end{quote}
|
|
|
742 |
\end{block}
|
|
|
743 |
\end{textblock}}
|
|
|
744 |
|
|
|
745 |
\end{frame}}
|
|
|
746 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
747 |
*}
|
|
|
748 |
|
|
|
749 |
|
|
|
750 |
text_raw {*
|
|
|
751 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
752 |
\mode<presentation>{
|
|
|
753 |
\begin{frame}[t]
|
|
|
754 |
\frametitle{\LARGE Partial Derivatives}
|
|
|
755 |
|
|
|
756 |
\mbox{}\\[0mm]\mbox{}
|
|
|
757 |
|
|
|
758 |
\begin{itemize}
|
|
|
759 |
|
|
|
760 |
\item \alt<1>{\smath{\text{pders $x$ $r$ \mbox{$=$} pders $y$ $r$}}}
|
|
|
761 |
{\smath{\underbrace{\text{pders $x$ $r$ \mbox{$=$} pders $y$ $r$}}_{R}}}
|
|
|
762 |
refines \textcolor{blue}{$x$ $\approx_{{\cal L}(r)}$ $y$}\\[16mm]\pause
|
|
|
763 |
\item \smath{\text{finite} (U\!N\!IV /\!/ R)} \bigskip\pause
|
|
|
764 |
\item Therefore \smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}. Qed.
|
|
|
765 |
\end{itemize}
|
|
|
766 |
|
|
|
767 |
\only<2->{%
|
|
|
768 |
\begin{textblock}{5}(3.9,7.2)
|
|
|
769 |
\begin{tikzpicture}
|
|
|
770 |
\node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}};
|
|
|
771 |
\draw (2.2,0) node {Antimirov '95};
|
|
|
772 |
\end{tikzpicture}
|
|
|
773 |
\end{textblock}}
|
|
|
774 |
|
|
|
775 |
\end{frame}}
|
|
|
776 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
777 |
*}
|
|
|
778 |
|
|
|
779 |
|
|
|
780 |
|
|
|
781 |
text_raw {*
|
|
|
782 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
783 |
\mode<presentation>{
|
|
|
784 |
\begin{frame}[t]
|
|
|
785 |
\frametitle{\LARGE What Have We Achieved?}
|
|
|
786 |
|
|
|
787 |
\begin{itemize}
|
|
|
788 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
|
|
|
789 |
\medskip\pause
|
|
|
790 |
\item regular languages are closed under complementation; this is now easy
|
|
|
791 |
\begin{center}
|
|
|
792 |
\smath{U\!N\!IV /\!/ \approx_A \;\;=\;\; U\!N\!IV /\!/ \approx_{\overline{A}}}
|
|
|
793 |
\end{center}\pause\medskip
|
|
|
794 |
|
|
|
795 |
\item non-regularity (\smath{a^nb^n})\medskip\pause\pause
|
|
|
796 |
|
|
|
797 |
\item take \alert{\bf any} language; build the language of substrings\\
|
|
|
798 |
\pause
|
|
|
799 |
|
|
|
800 |
then this language \alert{\bf is} regular\;\; (\smath{a^nb^n} $\Rightarrow$ \smath{a^\star{}b^\star})
|
|
|
801 |
|
|
|
802 |
\end{itemize}
|
|
|
803 |
|
|
|
804 |
\only<2>{
|
|
|
805 |
\begin{textblock}{10}(4,14)
|
|
|
806 |
\small
|
|
|
807 |
\smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
|
|
|
808 |
\end{textblock}}
|
|
|
809 |
|
|
|
810 |
\only<4>{
|
|
|
811 |
\begin{textblock}{5}(2,8.6)
|
|
|
812 |
\begin{minipage}{8.8cm}
|
|
|
813 |
\begin{block}{}
|
|
|
814 |
\begin{minipage}{8.6cm}
|
|
|
815 |
If there exists a sufficiently large set \smath{B} (for example infinitely large),
|
|
|
816 |
such that
|
|
|
817 |
|
|
|
818 |
\begin{center}
|
|
|
819 |
\smath{\forall x,y \in B.\; x \not= y \;\Rightarrow\; x \not\approx_{A} y}.
|
|
|
820 |
\end{center}
|
|
|
821 |
|
|
|
822 |
then \smath{A} is not regular.\hspace{1.3cm}\small(\smath{B \dn \bigcup_n a^n})
|
|
|
823 |
\end{minipage}
|
|
|
824 |
\end{block}
|
|
|
825 |
\end{minipage}
|
|
|
826 |
\end{textblock}
|
|
|
827 |
}
|
|
|
828 |
|
|
|
829 |
\end{frame}}
|
|
|
830 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
831 |
*}
|
|
|
832 |
|
|
|
833 |
|
|
|
834 |
text_raw {*
|
|
|
835 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
836 |
\mode<presentation>{
|
|
|
837 |
\begin{frame}[c]
|
|
|
838 |
\frametitle{\LARGE Conclusion}
|
|
|
839 |
|
|
|
840 |
\begin{itemize}
|
|
|
841 |
\item We have never seen a proof of Myhill-Nerode based on
|
|
|
842 |
regular expressions.\smallskip\pause
|
|
|
843 |
|
|
|
844 |
\item great source of examples (inductions)\smallskip\pause
|
|
|
845 |
|
|
|
846 |
\item no need to fight the theorem prover:\\
|
|
|
847 |
\begin{itemize}
|
|
|
848 |
\item first direction (790 loc)\\
|
|
|
849 |
\item second direction (400 / 390 loc)
|
|
|
850 |
\end{itemize}
|
|
|
851 |
\end{itemize}
|
|
|
852 |
|
|
|
853 |
\end{frame}}
|
|
|
854 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
855 |
*}
|
|
|
856 |
|
|
|
857 |
text_raw {*
|
|
|
858 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
859 |
\mode<presentation>{
|
|
|
860 |
\begin{frame}[b]
|
|
|
861 |
\frametitle{\mbox{}\\[2cm]\textcolor{red}{Thank you!\\[5mm]Questions?}}
|
|
|
862 |
|
|
|
863 |
\end{frame}}
|
|
|
864 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
865 |
*}
|
|
|
866 |
|
|
|
867 |
(*<*)
|
|
|
868 |
end
|
|
|
869 |
(*>*) |