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(*<*)
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theory Slides5
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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}{London, 29 August 2012}
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\newcommand{\bl}[1]{\textcolor{blue}{#1}}
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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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\\[-3mm]
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\LARGE The Myhill-Nerode Theorem\\[-3mm]
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\LARGE in a Theorem Prover\\[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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\only<2->{
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\begin{textblock}{6}(9,5.3)
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\alert{\bf Isabelle/HOL}
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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{}
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\mbox{}\\[2mm]
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\begin{itemize}
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\item my background is in
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\begin{itemize}
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\item \normalsize programming languages and theorem provers
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\item \normalsize develop Nominal Isabelle
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\end{itemize}\bigskip\bigskip\bigskip\bigskip\bigskip
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\item<1->to formalise and mechanically check proofs from
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programming language research, TCS \textcolor{gray}{and OS}\bigskip
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\item<2->we found out that the variable convention can lead to
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faulty proofs\ldots
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\end{itemize}
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\onslide<2->{
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\begin{center}
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\begin{block}{}
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\color{gray}
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\footnotesize
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{\bf\mbox{}\hspace{-1.5mm}Variable Convention:}\\[0mm]
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If $M_1,\ldots,M_n$ occur in a certain mathematical context
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(e.g. definition, proof), then in these terms all bound variables
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are chosen to be different from the free variables.\hfill Henk Barendregt
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\end{block}
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\end{center}}
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\only<1->{
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\begin{textblock}{6}(10.9,3.5)
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\includegraphics[scale=0.23]{isabelle1.png}
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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{}
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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 on LF 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\\ (proof-carrying code)}
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\end{tabular}
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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 {*
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\tikzstyle{every node}=[node distance=25mm,text height=1.5ex, text depth=.25ex]
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\tikzstyle{node1}=[rectangle, minimum size=10mm, rounded corners=3mm, very thick,
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draw=black!50, top color=white, bottom color=black!20]
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\tikzstyle{node2}=[rectangle, minimum size=12mm, rounded corners=3mm, very thick,
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draw=red!70, top color=white, bottom color=red!50!black!20]
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<2->[squeeze]
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\frametitle{}
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\begin{columns}
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\begin{column}{0.8\textwidth}
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\begin{textblock}{0}(1,2)
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\begin{tikzpicture}
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\matrix[ampersand replacement=\&,column sep=7mm, row sep=5mm]
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{ \&[-10mm]
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\node (def1) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}}; \&
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\node (proof1) [node1] {\large Proof}; \&
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\node (alg1) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}}; \\
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\onslide<4->{\node {\begin{tabular}{c}\small 1st\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
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\onslide<4->{\node (def2) [node2] {\large Spec$^\text{+ex}$};} \&
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\onslide<4->{\node (proof2) [node1] {\large Proof};} \&
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\onslide<4->{\node (alg2) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\
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\onslide<5->{\node {\begin{tabular}{c}\small 2nd\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
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\onslide<5->{\node (def3) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \&
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\onslide<5->{\node (proof3) [node1] {\large Proof};} \&
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\onslide<5->{\node (alg3) [node2] {\large Alg$^\text{-ex}$};} \\
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\onslide<6->{\node {\begin{tabular}{c}\small 3rd\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
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\onslide<6->{\node (def4) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \&
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\onslide<6->{\node (proof4) [node2] {\large\hspace{1mm}Proof\hspace{1mm}};} \&
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\onslide<6->{\node (alg4) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\
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};
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\draw[->,black!50,line width=2mm] (proof1) -- (def1);
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\draw[->,black!50,line width=2mm] (proof1) -- (alg1);
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\onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (def2);}
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\onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (alg2);}
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\onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (def3);}
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\onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (alg3);}
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\onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (def4);}
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\onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (alg4);}
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\onslide<3->{\draw[white,line width=1mm] (1.1,3.2) -- (0.9,2.85) -- (1.1,2.35) -- (0.9,2.0);}
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\end{tikzpicture}
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\end{textblock}
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\end{column}
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\end{columns}
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\begin{textblock}{3}(12,3.6)
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\onslide<4->{
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\begin{tikzpicture}
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\node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{2h};
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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}[c]
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\frametitle{}
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\begin{itemize}
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\item I also found fixable errors in my Ph.D.-thesis about cut-elimination
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(examined by Henk Barendregt and Andy Pitts)\bigskip
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\item found flaws in a proof about a classic OS scheduling algorithm
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--- helped us to implement\\ it correctly and ef$\!$ficiently\\
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{\small\textcolor{gray}{(the existing literature ``proved''
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correct an incorrect algorithm; used in the Mars Pathfinder mission)}}
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\end{itemize}\bigskip\bigskip\pause
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{\bf Conclusion:}\smallskip
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Pencil-and-paper proofs in TCS are not foolproof,
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not even expertproof.
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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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\small Scott Aaronson (Berkeley/MIT):\\[-7mm]\mbox{}
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\begin{center}
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\begin{block}{}
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\color{gray}
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\small
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``I still remember having to grade hundreds of exams where the
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students started out by assuming what had to be proved, or filled
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page after page with gibberish in the hope that, somewhere in the
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mess, they might accidentally have said something
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correct. \ldots{}innumerable examples of ``parrot proofs'' ---
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NP-completeness reductions done in the wrong direction, arguments
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that look more like LSD trips than coherent chains of logic \ldots{}''
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\end{block}
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\end{center}\pause
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\begin{tabular}{@ {}c@ {}}
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Tobias Nipkow calls this the ``London Underground Phenomenon'':
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\end{tabular}
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\begin{center}
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\begin{tabular}{ccc}
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students & \;\;\raisebox{-8mm}{\includegraphics[scale=0.16]{gap.jpg}}\;\; & proofs
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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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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{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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\mode<presentation>{
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\begin{frame}<1->[t]
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\frametitle{Regular Expressions}
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\begin{textblock}{6}(2,4)
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\begin{tabular}{@ {}rrl}
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\bl{r} & \bl{$::=$} & \bl{$\varnothing$}\\
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& \bl{$\mid$} & \bl{[]}\\
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& \bl{$\mid$} & \bl{c}\\
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& \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$}\\
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& \bl{$\mid$} & \bl{r$_1$ + r$_2$}\\
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& \bl{$\mid$} & \bl{r$^*$}\\
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\end{tabular}
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\end{textblock}
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\begin{textblock}{6}(8,3.5)
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\includegraphics[scale=0.35]{Screen1.png}
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\end{textblock}
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\begin{textblock}{6}(10.2,2.8)
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\footnotesize Isabelle:
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\end{textblock}
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\begin{textblock}{6}(7,12)
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\footnotesize\textcolor{gray}{students have seen them and can be motivated about them}
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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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\mode<presentation>{
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\begin{frame}<1->[t]
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\mbox{}\\[-2mm]
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\small
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\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}}
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\bl{nullable ($\varnothing$)} & \bl{$=$} & \bl{false} &\\
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\bl{nullable ([])} & \bl{$=$} & \bl{true} &\\
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\bl{nullable (c)} & \bl{$=$} & \bl{false} &\\
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\bl{nullable (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) $\vee$ (nullable r$_2$)} & \\
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\bl{nullable (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) $\wedge$ (nullable r$_2$)} & \\
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\bl{nullable (r$^*$)} & \bl{$=$} & \bl{true} & \\
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\end{tabular}\medskip\pause
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\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
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\bl{der c ($\varnothing$)} & \bl{$=$} & \bl{$\varnothing$} & \\
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\bl{der c ([])} & \bl{$=$} & \bl{$\varnothing$} & \\
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\bl{der c (d)} & \bl{$=$} & \bl{if c $=$ d then [] else $\varnothing$} & \\
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\bl{der c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\
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\bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{((der c r$_1$) $\cdot$ r$_2$) + } & \\
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& & \bl{\hspace{3mm}(if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\
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\bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ (r$^*$)} &\smallskip\\\pause
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\bl{derivative [] r} & \bl{$=$} & \bl{r} & \\
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\bl{derivative (c::s) r} & \bl{$=$} & \bl{derivative s (der c r)} & \\
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\end{tabular}\medskip
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\bl{matches r s $=$ nullable (derivative s r)}
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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 Regular Expression Matching\\[-2mm] in Education}
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\begin{itemize}
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\item Harper in JFP'99: ``Functional Pearl: Proof- Directed Debugging''\medskip
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382 |
\item Yi in JFP'06: ``Educational Pearl: `Proof-Directed Debugging' revisited
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for a first-order version''\medskip\bigskip\bigskip\pause
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\item Owens et al in JFP'09: ``Regular-expression derivatives re-examined''
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\bigskip
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386 |
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\begin{quote}\small
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``Unfortunately, regular expression derivatives have been lost in the
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sands of time, and few computer scientists are aware of them.''
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\end{quote}
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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{}\\[-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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410 |
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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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\begin{tabular}{ccc}
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\begin{tikzpicture}[scale=1]
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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 (B) at ( 0.2,0.0) [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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\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 (B) at ( 0.2,0.0) [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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459 |
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\draw [very thick, red] (C) to [bend left=45] (B);
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\draw [very thick, red] (D) to [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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465 |
\end{tikzpicture}}
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466 |
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|
467 |
\end{tabular}
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468 |
\end{center}\medskip
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469 |
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|
470 |
\only<4-5>{
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|
471 |
\begin{tabular}{@ {\hspace{-5mm}}l@ {}}
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|
472 |
disjoint union:\\[2mm]
|
|
|
473 |
\smath{A_1\uplus A_2 \dn \{(1, x)\,|\, x \in A_1\} \,\cup\, \{(2, y)\,|\, y \in A_2\}}
|
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|
474 |
\end{tabular}}
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475 |
\end{itemize}
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|
476 |
|
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|
477 |
\only<5>{
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|
478 |
\begin{textblock}{13.9}(0.7,7.7)
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479 |
\begin{block}{}
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|
480 |
\medskip
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|
481 |
\begin{minipage}{14cm}\raggedright
|
|
|
482 |
Problems with definition for regularity:\bigskip\\
|
|
|
483 |
\smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}\bigskip
|
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|
484 |
\end{minipage}
|
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|
485 |
\end{block}
|
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|
486 |
\end{textblock}}
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|
487 |
\medskip
|
|
|
488 |
|
|
|
489 |
\only<6->{\underline{A solution}:\;\;use \smath{\text{nat}}s \;@{text "\<Rightarrow>"}\; state nodes\medskip}
|
|
|
490 |
|
|
|
491 |
\only<7->{You have to \alert{rename} states!}
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|
492 |
|
|
|
493 |
\end{frame}}
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494 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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495 |
*}
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496 |
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497 |
text_raw {*
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|
499 |
\mode<presentation>{
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|
500 |
\begin{frame}[t]
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|
501 |
\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
|
|
|
502 |
\mbox{}\\[-15mm]\mbox{}
|
|
|
503 |
|
|
|
504 |
\begin{center}
|
|
|
505 |
\huge\bf\textcolor{gray}{in Theorem Provers}\\
|
|
|
506 |
\footnotesize\textcolor{gray}{e.g.~Isabelle, Coq, HOL4, \ldots}
|
|
|
507 |
\end{center}
|
|
|
508 |
|
|
|
509 |
\begin{itemize}
|
|
|
510 |
\item Kozen's paper-proof of Myhill-Nerode:\\
|
|
|
511 |
requires absence of \alert{inaccessible states}
|
|
|
512 |
\item complementation of automata only works for \alert{complete} automata
|
|
|
513 |
(need sink states)\medskip
|
|
|
514 |
\end{itemize}\bigskip\bigskip
|
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|
515 |
|
|
|
516 |
\begin{center}
|
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|
517 |
\smath{\;\text{is\_regular}(A) \dn \exists M.\;\text{is\_dfa}(M) \wedge {\cal L} (M) = A}
|
|
|
518 |
\end{center}
|
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519 |
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520 |
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|
521 |
\end{frame}}
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522 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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523 |
*}
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524 |
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525 |
text_raw {*
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526 |
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527 |
\mode<presentation>{
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|
|
528 |
\begin{frame}[t]
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|
529 |
\frametitle{}
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|
530 |
\mbox{}\\[25mm]\mbox{}
|
|
|
531 |
|
|
|
532 |
\begin{textblock}{13.9}(0.7,1.2)
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|
533 |
\begin{block}{}
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|
534 |
\begin{minipage}{13.4cm}\raggedright
|
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|
535 |
{\bf Definition:}\smallskip\\
|
|
|
536 |
|
|
|
537 |
A language \smath{A} is \alert{regular}, provided there exists a\\
|
|
|
538 |
\alert{regular expression} that matches all strings of \smath{A}.
|
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|
539 |
\end{minipage}
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|
540 |
\end{block}
|
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|
541 |
\end{textblock}\pause
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|
542 |
|
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|
543 |
{\noindent\large\bf\alert{\ldots{}and forget about automata}}\bigskip\bigskip\pause
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|
544 |
|
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|
545 |
Infrastructure for free. But do we lose anything?\medskip\pause
|
|
|
546 |
|
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|
547 |
\begin{minipage}{1.1\textwidth}
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|
548 |
\begin{itemize}
|
|
|
549 |
\item pumping lemma\pause
|
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|
550 |
\item closure under complementation\pause
|
|
|
551 |
\item \only<6>{regular expression matching}%
|
|
|
552 |
\only<7->{\sout{regular expression matching}
|
|
|
553 |
{\footnotesize(@{text "\<Rightarrow>"}Brzozowski'64, Owens et al '09)}}
|
|
|
554 |
\item<8-> most textbooks are about automata
|
|
|
555 |
\end{itemize}
|
|
|
556 |
\end{minipage}
|
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|
557 |
|
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|
558 |
|
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|
559 |
\end{frame}}
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560 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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561 |
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562 |
*}
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text_raw {*
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567 |
\mode<presentation>{
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568 |
\begin{frame}[c]
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|
569 |
\frametitle{\LARGE The Myhill-Nerode Theorem}
|
|
|
570 |
|
|
|
571 |
\begin{itemize}
|
|
|
572 |
\item provides necessary and suf\!ficient conditions\\ for a language
|
|
|
573 |
being regular\\ \textcolor{gray}{(pumping lemma only necessary)}\bigskip
|
|
|
574 |
|
|
|
575 |
\item key is the equivalence relation:\medskip
|
|
|
576 |
\begin{center}
|
|
|
577 |
\smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
|
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|
578 |
\end{center}
|
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|
579 |
\end{itemize}
|
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|
580 |
|
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|
581 |
|
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|
582 |
\end{frame}}
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583 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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584 |
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585 |
*}
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587 |
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\mode<presentation>{
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|
590 |
\begin{frame}[c]
|
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|
591 |
\frametitle{\LARGE The Myhill-Nerode Theorem}
|
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|
592 |
|
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|
593 |
\begin{center}
|
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|
594 |
\only<1>{%
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\begin{tikzpicture}[scale=3]
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|
596 |
\draw[very thick] (0.5,0.5) circle (.6cm);
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597 |
\end{tikzpicture}}%
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598 |
\only<2->{%
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599 |
\begin{tikzpicture}[scale=3]
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600 |
\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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602 |
\draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
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\end{tikzpicture}}
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|
604 |
\end{center}
|
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|
605 |
|
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|
606 |
\begin{itemize}
|
|
|
607 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
|
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|
608 |
\end{itemize}
|
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|
609 |
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610 |
\begin{textblock}{5}(2.1,5.3)
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611 |
\begin{tikzpicture}
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612 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=2cm]
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613 |
{$U\!N\!IV$};
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614 |
\draw (-0.3,-1.1) node {\begin{tabular}{l}set of all\\[-1mm] strings\end{tabular}};
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615 |
\end{tikzpicture}
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616 |
\end{textblock}
|
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|
617 |
|
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|
618 |
\only<2->{%
|
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|
619 |
\begin{textblock}{5}(9.1,7.2)
|
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|
620 |
\begin{tikzpicture}
|
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|
621 |
\node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]
|
|
|
622 |
{@{text "\<lbrakk>s\<rbrakk>"}$_{\approx_{A}}$};
|
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|
623 |
\draw (0.9,-1.1) node {\begin{tabular}{l}an equivalence class\end{tabular}};
|
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|
624 |
\end{tikzpicture}
|
|
|
625 |
\end{textblock}}
|
|
|
626 |
|
|
|
627 |
\only<3->{
|
|
|
628 |
\begin{textblock}{11.9}(1.7,3)
|
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|
629 |
\begin{block}{}
|
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|
630 |
\begin{minipage}{11.4cm}\raggedright
|
|
|
631 |
Two directions:\medskip\\
|
|
|
632 |
\begin{tabular}{@ {}ll}
|
|
|
633 |
1.)\;finite $\Rightarrow$ regular\\
|
|
|
634 |
\;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_A) \Rightarrow \exists r.\;A = {\cal L}(r)}\\[3mm]
|
|
|
635 |
2.)\;regular $\Rightarrow$ finite\\
|
|
|
636 |
\;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}
|
|
|
637 |
\end{tabular}
|
|
|
638 |
|
|
|
639 |
\end{minipage}
|
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|
640 |
\end{block}
|
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|
641 |
\end{textblock}}
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642 |
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643 |
\end{frame}}
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644 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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645 |
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646 |
*}
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647 |
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648 |
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649 |
text_raw {*
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|
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|
651 |
\mode<presentation>{
|
|
|
652 |
\begin{frame}[c]
|
|
|
653 |
\frametitle{\LARGE Initial and Final {\sout{\textcolor{gray}{States}}}}
|
|
|
654 |
|
|
|
655 |
\begin{textblock}{8}(10, 2)
|
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|
656 |
\textcolor{black}{Equivalence Classes}
|
|
|
657 |
\end{textblock}
|
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|
658 |
|
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|
659 |
|
|
|
660 |
\begin{center}
|
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|
661 |
\begin{tikzpicture}[scale=3]
|
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|
662 |
\draw[very thick] (0.5,0.5) circle (.6cm);
|
|
|
663 |
\clip[draw] (0.5,0.5) circle (.6cm);
|
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|
664 |
\draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
|
|
|
665 |
\only<2->{\draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);}
|
|
|
666 |
\only<3->{\draw[red, fill] (0.2, 0.2) rectangle (0.4, 0.4);
|
|
|
667 |
\draw[red, fill] (0.4, 0.8) rectangle (0.6, 1.0);
|
|
|
668 |
\draw[red, fill] (0.6, 0.0) rectangle (0.8, 0.2);
|
|
|
669 |
\draw[red, fill] (0.8, 0.4) rectangle (1.0, 0.6);}
|
|
|
670 |
\end{tikzpicture}
|
|
|
671 |
\end{center}
|
|
|
672 |
|
|
|
673 |
\begin{itemize}
|
|
|
674 |
\item \smath{\text{finals}\,A\,\dn \{[\!|s|\!]_{\approx_{A}}\;|\;s \in A\}}
|
|
|
675 |
\smallskip
|
|
|
676 |
\item we can prove: \smath{A = \bigcup \text{finals}\,A}
|
|
|
677 |
\end{itemize}
|
|
|
678 |
|
|
|
679 |
\only<2->{%
|
|
|
680 |
\begin{textblock}{5}(2.1,4.6)
|
|
|
681 |
\begin{tikzpicture}
|
|
|
682 |
\node at (0,0) [single arrow, fill=blue,text=white, minimum height=2cm]
|
|
|
683 |
{$[] \in X$};
|
|
|
684 |
\end{tikzpicture}
|
|
|
685 |
\end{textblock}}
|
|
|
686 |
|
|
|
687 |
\only<3->{%
|
|
|
688 |
\begin{textblock}{5}(10,7.4)
|
|
|
689 |
\begin{tikzpicture}
|
|
|
690 |
\node at (0,0) [shape border rotate=180,single arrow, fill=red,text=white, minimum height=2cm]
|
|
|
691 |
{a final};
|
|
|
692 |
\end{tikzpicture}
|
|
|
693 |
\end{textblock}}
|
|
|
694 |
|
|
|
695 |
\end{frame}}
|
|
|
696 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
697 |
*}
|
|
|
698 |
|
|
|
699 |
|
|
|
700 |
text_raw {*
|
|
|
701 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
702 |
\mode<presentation>{
|
|
|
703 |
\begin{frame}<-1>[c]
|
|
|
704 |
\frametitle{\begin{tabular}{@ {}l}\LARGE%
|
|
|
705 |
Transitions between Eq-Classes\end{tabular}}
|
|
|
706 |
|
|
|
707 |
\begin{center}
|
|
|
708 |
\begin{tikzpicture}[scale=3]
|
|
|
709 |
\draw[very thick] (0.5,0.5) circle (.6cm);
|
|
|
710 |
\clip[draw] (0.5,0.5) circle (.6cm);
|
|
|
711 |
\draw[step=2mm, very thick] (-1.4,-1.4) grid (1.4,1.4);
|
|
|
712 |
\draw[blue, fill] (0.0, 0.6) rectangle (0.2, 0.8);
|
|
|
713 |
\draw[blue, fill] (0.8, 0.4) rectangle (1.0, 0.6);
|
|
|
714 |
\draw[white] (0.1,0.7) node (X) {$X$};
|
|
|
715 |
\draw[white] (0.9,0.5) node (Y) {$Y$};
|
|
|
716 |
\draw[blue, ->, line width = 2mm, bend left=45] (X) -- (Y);
|
|
|
717 |
\node [inner sep=1pt,label=above:\textcolor{blue}{$c$}] at ($ (X)!.5!(Y) $) {};
|
|
|
718 |
\end{tikzpicture}
|
|
|
719 |
\end{center}
|
|
|
720 |
|
|
|
721 |
\begin{center}
|
|
|
722 |
\smath{X \stackrel{c}{\longrightarrow} Y \;\dn\; X ; c \subseteq Y}
|
|
|
723 |
\end{center}
|
|
|
724 |
|
|
|
725 |
\onslide<8>{
|
|
|
726 |
\begin{tabular}{c}
|
|
|
727 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
|
|
|
728 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
|
729 |
\node[state,initial] (q_0) {$R_1$};
|
|
|
730 |
\end{tikzpicture}
|
|
|
731 |
\end{tabular}}
|
|
|
732 |
|
|
|
733 |
\end{frame}}
|
|
|
734 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
735 |
*}
|
|
|
736 |
|
|
|
737 |
|
|
|
738 |
text_raw {*
|
|
|
739 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
740 |
\mode<presentation>{
|
|
|
741 |
\begin{frame}[c]
|
|
|
742 |
\frametitle{\LARGE Systems of Equations}
|
|
|
743 |
|
|
|
744 |
Inspired by a method of Brzozowski\;'64:\bigskip\bigskip
|
|
|
745 |
|
|
|
746 |
\begin{center}
|
|
|
747 |
\begin{tabular}{@ {\hspace{-20mm}}c}
|
|
|
748 |
\\[-13mm]
|
|
|
749 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
|
|
|
750 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
|
751 |
|
|
|
752 |
%\draw[help lines] (0,0) grid (3,2);
|
|
|
753 |
|
|
|
754 |
\node[state,initial] (p_0) {$X_1$};
|
|
|
755 |
\node[state,accepting] (p_1) [right of=q_0] {$X_2$};
|
|
|
756 |
|
|
|
757 |
\path[->] (p_0) edge [bend left] node {a} (p_1)
|
|
|
758 |
edge [loop above] node {b} ()
|
|
|
759 |
(p_1) edge [loop above] node {a} ()
|
|
|
760 |
edge [bend left] node {b} (p_0);
|
|
|
761 |
\end{tikzpicture}\\
|
|
|
762 |
\\[-13mm]
|
|
|
763 |
\end{tabular}
|
|
|
764 |
\end{center}
|
|
|
765 |
|
|
|
766 |
\begin{center}
|
|
|
767 |
\begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
|
|
|
768 |
& \smath{X_1} & \smath{=} & \smath{X_1;b + X_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
|
|
|
769 |
& \smath{X_2} & \smath{=} & \smath{X_1;a + X_2;a}\medskip\\
|
|
|
770 |
\end{tabular}
|
|
|
771 |
\end{center}
|
|
|
772 |
|
|
|
773 |
\end{frame}}
|
|
|
774 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
775 |
*}
|
|
|
776 |
|
|
|
777 |
text_raw {*
|
|
|
778 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
779 |
\mode<presentation>{
|
|
|
780 |
\begin{frame}<1>[t]
|
|
|
781 |
\small
|
|
|
782 |
|
|
|
783 |
\begin{center}
|
|
|
784 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
|
785 |
\onslide<1->{\smath{X_1}} & \onslide<1->{\smath{=}}
|
|
|
786 |
& \onslide<1->{\smath{X_1; b + X_2; b + \lambda;[]}}\\
|
|
|
787 |
\onslide<1->{\smath{X_2}} & \onslide<1->{\smath{=}}
|
|
|
788 |
& \onslide<1->{\smath{X_1; a + X_2; a}}\\
|
|
|
789 |
|
|
|
790 |
& & & \onslide<2->{by Arden}\\
|
|
|
791 |
|
|
|
792 |
\onslide<2->{\smath{X_1}} & \onslide<2->{\smath{=}}
|
|
|
793 |
& \onslide<2->{\smath{X_1; b + X_2; b + \lambda;[]}}\\
|
|
|
794 |
\onslide<2->{\smath{X_2}} & \onslide<2->{\smath{=}}
|
|
|
795 |
& \only<2->{\smath{X_1; a\cdot a^\star}}\\
|
|
|
796 |
|
|
|
797 |
& & & \onslide<4->{by Arden}\\
|
|
|
798 |
|
|
|
799 |
\onslide<4->{\smath{X_1}} & \onslide<4->{\smath{=}}
|
|
|
800 |
& \onslide<4->{\smath{X_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
801 |
\onslide<4->{\smath{X_2}} & \onslide<4->{\smath{=}}
|
|
|
802 |
& \onslide<4->{\smath{X_1; a\cdot a^\star}}\\
|
|
|
803 |
|
|
|
804 |
& & & \onslide<5->{by substitution}\\
|
|
|
805 |
|
|
|
806 |
\onslide<5->{\smath{X_1}} & \onslide<5->{\smath{=}}
|
|
|
807 |
& \onslide<5->{\smath{X_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
808 |
\onslide<5->{\smath{X_2}} & \onslide<5->{\smath{=}}
|
|
|
809 |
& \onslide<5->{\smath{X_1; a\cdot a^\star}}\\
|
|
|
810 |
|
|
|
811 |
& & & \onslide<6->{by Arden}\\
|
|
|
812 |
|
|
|
813 |
\onslide<6->{\smath{X_1}} & \onslide<6->{\smath{=}}
|
|
|
814 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
815 |
\onslide<6->{\smath{X_2}} & \onslide<6->{\smath{=}}
|
|
|
816 |
& \onslide<6->{\smath{X_1; a\cdot a^\star}}\\
|
|
|
817 |
|
|
|
818 |
& & & \onslide<7->{by substitution}\\
|
|
|
819 |
|
|
|
820 |
\onslide<7->{\smath{X_1}} & \onslide<7->{\smath{=}}
|
|
|
821 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
822 |
\onslide<7->{\smath{X_2}} & \onslide<7->{\smath{=}}
|
|
|
823 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
|
|
824 |
\cdot a\cdot a^\star}}\\
|
|
|
825 |
\end{tabular}
|
|
|
826 |
\end{center}
|
|
|
827 |
|
|
|
828 |
\only<8->{
|
|
|
829 |
\begin{textblock}{6}(2.5,4)
|
|
|
830 |
\begin{block}{}
|
|
|
831 |
\begin{minipage}{8cm}\raggedright
|
|
|
832 |
|
|
|
833 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
|
|
|
834 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
|
835 |
|
|
|
836 |
%\draw[help lines] (0,0) grid (3,2);
|
|
|
837 |
|
|
|
838 |
\node[state,initial] (p_0) {$X_1$};
|
|
|
839 |
\node[state,accepting] (p_1) [right of=q_0] {$X_2$};
|
|
|
840 |
|
|
|
841 |
\path[->] (p_0) edge [bend left] node {a} (p_1)
|
|
|
842 |
edge [loop above] node {b} ()
|
|
|
843 |
(p_1) edge [loop above] node {a} ()
|
|
|
844 |
edge [bend left] node {b} (p_0);
|
|
|
845 |
\end{tikzpicture}
|
|
|
846 |
|
|
|
847 |
\end{minipage}
|
|
|
848 |
\end{block}
|
|
|
849 |
\end{textblock}}
|
|
|
850 |
|
|
|
851 |
\only<1,2>{%
|
|
|
852 |
\begin{textblock}{3}(0.6,1.2)
|
|
|
853 |
\begin{tikzpicture}
|
|
|
854 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
|
|
|
855 |
{\textcolor{red}{a}};
|
|
|
856 |
\end{tikzpicture}
|
|
|
857 |
\end{textblock}}
|
|
|
858 |
\only<2>{%
|
|
|
859 |
\begin{textblock}{3}(0.6,3.6)
|
|
|
860 |
\begin{tikzpicture}
|
|
|
861 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
|
|
|
862 |
{\textcolor{red}{a}};
|
|
|
863 |
\end{tikzpicture}
|
|
|
864 |
\end{textblock}}
|
|
|
865 |
\only<4>{%
|
|
|
866 |
\begin{textblock}{3}(0.6,2.9)
|
|
|
867 |
\begin{tikzpicture}
|
|
|
868 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
|
|
|
869 |
{\textcolor{red}{a}};
|
|
|
870 |
\end{tikzpicture}
|
|
|
871 |
\end{textblock}}
|
|
|
872 |
\only<4>{%
|
|
|
873 |
\begin{textblock}{3}(0.6,5.3)
|
|
|
874 |
\begin{tikzpicture}
|
|
|
875 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
|
|
|
876 |
{\textcolor{red}{a}};
|
|
|
877 |
\end{tikzpicture}
|
|
|
878 |
\end{textblock}}
|
|
|
879 |
\only<5>{%
|
|
|
880 |
\begin{textblock}{3}(1.0,5.6)
|
|
|
881 |
\begin{tikzpicture}
|
|
|
882 |
\node at (0,0) (A) {};
|
|
|
883 |
\node at (0,1) (B) {};
|
|
|
884 |
\draw[<-, line width=2mm, red] (B) to (A);
|
|
|
885 |
\end{tikzpicture}
|
|
|
886 |
\end{textblock}}
|
|
|
887 |
\only<5,6>{%
|
|
|
888 |
\begin{textblock}{3}(0.6,7.7)
|
|
|
889 |
\begin{tikzpicture}
|
|
|
890 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
|
|
|
891 |
{\textcolor{red}{a}};
|
|
|
892 |
\end{tikzpicture}
|
|
|
893 |
\end{textblock}}
|
|
|
894 |
\only<6>{%
|
|
|
895 |
\begin{textblock}{3}(0.6,10.1)
|
|
|
896 |
\begin{tikzpicture}
|
|
|
897 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
|
|
|
898 |
{\textcolor{red}{a}};
|
|
|
899 |
\end{tikzpicture}
|
|
|
900 |
\end{textblock}}
|
|
|
901 |
\only<7>{%
|
|
|
902 |
\begin{textblock}{3}(1.0,10.3)
|
|
|
903 |
\begin{tikzpicture}
|
|
|
904 |
\node at (0,0) (A) {};
|
|
|
905 |
\node at (0,1) (B) {};
|
|
|
906 |
\draw[->, line width=2mm, red] (B) to (A);
|
|
|
907 |
\end{tikzpicture}
|
|
|
908 |
\end{textblock}}
|
|
|
909 |
|
|
|
910 |
\end{frame}}
|
|
|
911 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
912 |
*}
|
|
|
913 |
|
|
|
914 |
|
|
|
915 |
text_raw {*
|
|
|
916 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
917 |
\mode<presentation>{
|
|
|
918 |
\begin{frame}[c]
|
|
|
919 |
\frametitle{\LARGE A Variant of Arden's Lemma}
|
|
|
920 |
|
|
|
921 |
{\bf Arden's Lemma:}\smallskip
|
|
|
922 |
|
|
|
923 |
If \smath{[] \not\in A} then
|
|
|
924 |
\begin{center}
|
|
|
925 |
\smath{X = X; A + \text{something}}
|
|
|
926 |
\end{center}
|
|
|
927 |
has the (unique) solution
|
|
|
928 |
\begin{center}
|
|
|
929 |
\smath{X = \text{something} ; A^\star}
|
|
|
930 |
\end{center}
|
|
|
931 |
|
|
|
932 |
|
|
|
933 |
\end{frame}}
|
|
|
934 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
935 |
*}
|
|
|
936 |
|
|
|
937 |
|
|
|
938 |
text_raw {*
|
|
|
939 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
940 |
\mode<presentation>{
|
|
|
941 |
\begin{frame}<1-2,4->[t]
|
|
|
942 |
\small
|
|
|
943 |
|
|
|
944 |
\begin{center}
|
|
|
945 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
|
946 |
\onslide<1->{\smath{X_1}} & \onslide<1->{\smath{=}}
|
|
|
947 |
& \onslide<1->{\smath{X_1; b + X_2; b + \lambda;[]}}\\
|
|
|
948 |
\onslide<1->{\smath{X_2}} & \onslide<1->{\smath{=}}
|
|
|
949 |
& \onslide<1->{\smath{X_1; a + X_2; a}}\\
|
|
|
950 |
|
|
|
951 |
& & & \onslide<2->{by Arden}\\
|
|
|
952 |
|
|
|
953 |
\onslide<2->{\smath{X_1}} & \onslide<2->{\smath{=}}
|
|
|
954 |
& \onslide<2->{\smath{X_1; b + X_2; b + \lambda;[]}}\\
|
|
|
955 |
\onslide<2->{\smath{X_2}} & \onslide<2->{\smath{=}}
|
|
|
956 |
& \only<2->{\smath{X_1; a\cdot a^\star}}\\
|
|
|
957 |
|
|
|
958 |
& & & \onslide<4->{by Arden}\\
|
|
|
959 |
|
|
|
960 |
\onslide<4->{\smath{X_1}} & \onslide<4->{\smath{=}}
|
|
|
961 |
& \onslide<4->{\smath{X_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
962 |
\onslide<4->{\smath{X_2}} & \onslide<4->{\smath{=}}
|
|
|
963 |
& \onslide<4->{\smath{X_1; a\cdot a^\star}}\\
|
|
|
964 |
|
|
|
965 |
& & & \onslide<5->{by substitution}\\
|
|
|
966 |
|
|
|
967 |
\onslide<5->{\smath{X_1}} & \onslide<5->{\smath{=}}
|
|
|
968 |
& \onslide<5->{\smath{X_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
969 |
\onslide<5->{\smath{X_2}} & \onslide<5->{\smath{=}}
|
|
|
970 |
& \onslide<5->{\smath{X_1; a\cdot a^\star}}\\
|
|
|
971 |
|
|
|
972 |
& & & \onslide<6->{by Arden}\\
|
|
|
973 |
|
|
|
974 |
\onslide<6->{\smath{X_1}} & \onslide<6->{\smath{=}}
|
|
|
975 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
976 |
\onslide<6->{\smath{X_2}} & \onslide<6->{\smath{=}}
|
|
|
977 |
& \onslide<6->{\smath{X_1; a\cdot a^\star}}\\
|
|
|
978 |
|
|
|
979 |
& & & \onslide<7->{by substitution}\\
|
|
|
980 |
|
|
|
981 |
\onslide<7->{\smath{X_1}} & \onslide<7->{\smath{=}}
|
|
|
982 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
983 |
\onslide<7->{\smath{X_2}} & \onslide<7->{\smath{=}}
|
|
|
984 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
|
|
985 |
\cdot a\cdot a^\star}}\\
|
|
|
986 |
\end{tabular}
|
|
|
987 |
\end{center}
|
|
|
988 |
|
|
|
989 |
\only<8->{
|
|
|
990 |
\begin{textblock}{6}(2.5,4)
|
|
|
991 |
\begin{block}{}
|
|
|
992 |
\begin{minipage}{8cm}\raggedright
|
|
|
993 |
|
|
|
994 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
|
|
|
995 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
|
996 |
|
|
|
997 |
%\draw[help lines] (0,0) grid (3,2);
|
|
|
998 |
|
|
|
999 |
\node[state,initial] (p_0) {$X_1$};
|
|
|
1000 |
\node[state,accepting] (p_1) [right of=q_0] {$X_2$};
|
|
|
1001 |
|
|
|
1002 |
\path[->] (p_0) edge [bend left] node {a} (p_1)
|
|
|
1003 |
edge [loop above] node {b} ()
|
|
|
1004 |
(p_1) edge [loop above] node {a} ()
|
|
|
1005 |
edge [bend left] node {b} (p_0);
|
|
|
1006 |
\end{tikzpicture}
|
|
|
1007 |
|
|
|
1008 |
\end{minipage}
|
|
|
1009 |
\end{block}
|
|
|
1010 |
\end{textblock}}
|
|
|
1011 |
|
|
|
1012 |
\only<1,2>{%
|
|
|
1013 |
\begin{textblock}{3}(0.6,1.2)
|
|
|
1014 |
\begin{tikzpicture}
|
|
|
1015 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
|
|
|
1016 |
{\textcolor{red}{a}};
|
|
|
1017 |
\end{tikzpicture}
|
|
|
1018 |
\end{textblock}}
|
|
|
1019 |
\only<2>{%
|
|
|
1020 |
\begin{textblock}{3}(0.6,3.6)
|
|
|
1021 |
\begin{tikzpicture}
|
|
|
1022 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
|
|
|
1023 |
{\textcolor{red}{a}};
|
|
|
1024 |
\end{tikzpicture}
|
|
|
1025 |
\end{textblock}}
|
|
|
1026 |
\only<4>{%
|
|
|
1027 |
\begin{textblock}{3}(0.6,2.9)
|
|
|
1028 |
\begin{tikzpicture}
|
|
|
1029 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
|
|
|
1030 |
{\textcolor{red}{a}};
|
|
|
1031 |
\end{tikzpicture}
|
|
|
1032 |
\end{textblock}}
|
|
|
1033 |
\only<4>{%
|
|
|
1034 |
\begin{textblock}{3}(0.6,5.3)
|
|
|
1035 |
\begin{tikzpicture}
|
|
|
1036 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
|
|
|
1037 |
{\textcolor{red}{a}};
|
|
|
1038 |
\end{tikzpicture}
|
|
|
1039 |
\end{textblock}}
|
|
|
1040 |
\only<5>{%
|
|
|
1041 |
\begin{textblock}{3}(1.0,5.6)
|
|
|
1042 |
\begin{tikzpicture}
|
|
|
1043 |
\node at (0,0) (A) {};
|
|
|
1044 |
\node at (0,1) (B) {};
|
|
|
1045 |
\draw[<-, line width=2mm, red] (B) to (A);
|
|
|
1046 |
\end{tikzpicture}
|
|
|
1047 |
\end{textblock}}
|
|
|
1048 |
\only<5,6>{%
|
|
|
1049 |
\begin{textblock}{3}(0.6,7.7)
|
|
|
1050 |
\begin{tikzpicture}
|
|
|
1051 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
|
|
|
1052 |
{\textcolor{red}{a}};
|
|
|
1053 |
\end{tikzpicture}
|
|
|
1054 |
\end{textblock}}
|
|
|
1055 |
\only<6>{%
|
|
|
1056 |
\begin{textblock}{3}(0.6,10.1)
|
|
|
1057 |
\begin{tikzpicture}
|
|
|
1058 |
\node at (0,0) [single arrow, fill=red,text=white, minimum height=0cm]
|
|
|
1059 |
{\textcolor{red}{a}};
|
|
|
1060 |
\end{tikzpicture}
|
|
|
1061 |
\end{textblock}}
|
|
|
1062 |
\only<7>{%
|
|
|
1063 |
\begin{textblock}{3}(1.0,10.3)
|
|
|
1064 |
\begin{tikzpicture}
|
|
|
1065 |
\node at (0,0) (A) {};
|
|
|
1066 |
\node at (0,1) (B) {};
|
|
|
1067 |
\draw[->, line width=2mm, red] (B) to (A);
|
|
|
1068 |
\end{tikzpicture}
|
|
|
1069 |
\end{textblock}}
|
|
|
1070 |
|
|
|
1071 |
\end{frame}}
|
|
|
1072 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1073 |
*}
|
|
|
1074 |
|
|
|
1075 |
|
|
|
1076 |
text_raw {*
|
|
|
1077 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1078 |
\mode<presentation>{
|
|
|
1079 |
\begin{frame}[c]
|
|
|
1080 |
\frametitle{\LARGE The Other Direction}
|
|
|
1081 |
|
|
|
1082 |
One has to prove
|
|
|
1083 |
|
|
|
1084 |
\begin{center}
|
|
|
1085 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}
|
|
|
1086 |
\end{center}
|
|
|
1087 |
|
|
|
1088 |
by induction on \smath{r}. Not trivial, but after a bit
|
|
|
1089 |
of thinking, one can find a \alert{refined} relation:\bigskip
|
|
|
1090 |
|
|
|
1091 |
|
|
|
1092 |
\begin{center}
|
|
|
1093 |
\mbox{\begin{tabular}{c@ {\hspace{7mm}}c@ {\hspace{7mm}}c}
|
|
|
1094 |
\begin{tikzpicture}[scale=1.1]
|
|
|
1095 |
%Circle
|
|
|
1096 |
\draw[thick] (0,0) circle (1.1);
|
|
|
1097 |
\end{tikzpicture}
|
|
|
1098 |
&
|
|
|
1099 |
\begin{tikzpicture}[scale=1.1]
|
|
|
1100 |
%Circle
|
|
|
1101 |
\draw[thick] (0,0) circle (1.1);
|
|
|
1102 |
%Main rays
|
|
|
1103 |
\foreach \a in {0, 90,...,359}
|
|
|
1104 |
\draw[very thick] (0, 0) -- (\a:1.1);
|
|
|
1105 |
\foreach \a / \l in {45/1, 135/2, 225/3, 315/4}
|
|
|
1106 |
\draw (\a: 0.65) node {\small$a_\l$};
|
|
|
1107 |
\end{tikzpicture}
|
|
|
1108 |
&
|
|
|
1109 |
\begin{tikzpicture}[scale=1.1]
|
|
|
1110 |
%Circle
|
|
|
1111 |
\draw[red, thick] (0,0) circle (1.1);
|
|
|
1112 |
%Main rays
|
|
|
1113 |
\foreach \a in {0, 45,...,359}
|
|
|
1114 |
\draw[red, very thick] (0, 0) -- (\a:1.1);
|
|
|
1115 |
\foreach \a / \l in {22.5/1.1, 67.5/1.2, 112.5/2.1, 157.5/2.2, 202.4/3.1, 247.5/3.2, 292.5/4.1, 337.5/4.2}
|
|
|
1116 |
\draw (\a: 0.77) node {\textcolor{red}{\footnotesize$a_{\l}$}};
|
|
|
1117 |
\end{tikzpicture}\\
|
|
|
1118 |
\small\smath{U\!N\!IV} &
|
|
|
1119 |
\small\smath{U\!N\!IV /\!/ \approx_{{\cal L}(r)}} &
|
|
|
1120 |
\small\smath{U\!N\!IV /\!/ \alert{R}}
|
|
|
1121 |
\end{tabular}}
|
|
|
1122 |
\end{center}
|
|
|
1123 |
|
|
|
1124 |
\begin{textblock}{5}(9.8,2.6)
|
|
|
1125 |
\begin{tikzpicture}
|
|
|
1126 |
\node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}};
|
|
|
1127 |
\end{tikzpicture}
|
|
|
1128 |
\end{textblock}
|
|
|
1129 |
|
|
|
1130 |
|
|
|
1131 |
\end{frame}}
|
|
|
1132 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1133 |
*}
|
|
|
1134 |
|
|
|
1135 |
text_raw {*
|
|
|
1136 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1137 |
\mode<presentation>{
|
|
|
1138 |
\begin{frame}[t]
|
|
|
1139 |
\frametitle{\LARGE\begin{tabular}{c}Derivatives of RExps\end{tabular}}
|
|
|
1140 |
|
|
|
1141 |
\begin{itemize}
|
|
|
1142 |
\item introduced by Brzozowski~'64
|
|
|
1143 |
\item produces a regular expression after a character has been ``parsed''\\[-18mm]\mbox{}
|
|
|
1144 |
\end{itemize}
|
|
|
1145 |
|
|
|
1146 |
\only<1->{%
|
|
|
1147 |
\textcolor{blue}{%
|
|
|
1148 |
\begin{center}
|
|
|
1149 |
\begin{tabular}{@ {}lc@ {\hspace{3mm}}l@ {}}
|
|
|
1150 |
der c $\varnothing$ & $\dn$ & $\varnothing$\\
|
|
|
1151 |
der c [] & $\dn$ & $\varnothing$\\
|
|
|
1152 |
der c d & $\dn$ & if c $=$ d then [] else $\varnothing$\\
|
|
|
1153 |
der c ($r_1 + r_2$) & $\dn$ & (der c $r_1$) $+$ (der c $r_2$)\\
|
|
|
1154 |
der c ($r^*$) & $\dn$ & (der c $r$) $\cdot$ ($r^*$)\\
|
|
|
1155 |
der c ($r_1 \cdot r_2$) & $\dn$ & ((der c $r_1$) $\cdot$ $r_2$) +\\
|
|
|
1156 |
& & \hspace{-3mm}(if nullable $r_1$ then der c $r_2$ else $\varnothing$)\\
|
|
|
1157 |
\end{tabular}
|
|
|
1158 |
\end{center}}}
|
|
|
1159 |
|
|
|
1160 |
\only<2->{
|
|
|
1161 |
\begin{textblock}{13}(1.5,5.7)
|
|
|
1162 |
\begin{block}{}
|
|
|
1163 |
\begin{quote}
|
|
|
1164 |
\begin{minipage}{13cm}\raggedright
|
|
|
1165 |
derivatives refine \smath{x \approx_{{\cal{L}}(r)} y}\bigskip
|
|
|
1166 |
\begin{center}
|
|
|
1167 |
\only<2>{\mbox{\hspace{-22mm}}\smath{{\cal{L}}(\text{ders}~x~r) = {\cal{L}}(\text{ders}~y~r)
|
|
|
1168 |
\Longleftrightarrow x \approx_{L(r)} y}}
|
|
|
1169 |
\only<3>{\mbox{\hspace{-22mm}}\smath{\text{ders}~x~r = \text{ders}~y~r
|
|
|
1170 |
\Longrightarrow x \approx_{L(r)} y}}
|
|
|
1171 |
\end{center}\bigskip
|
|
|
1172 |
\
|
|
|
1173 |
\smath{\text{finite}(\text{ders}~A~r)}, but only modulo ACI
|
|
|
1174 |
|
|
|
1175 |
\begin{center}
|
|
|
1176 |
\begin{tabular}{@ {\hspace{-10mm}}rcl}
|
|
|
1177 |
\smath{(r_1 + r_2) + r_3} & \smath{\equiv} & \smath{r_1 + (r_2 + r_3)}\\
|
|
|
1178 |
\smath{r_1 + r_2} & \smath{\equiv} & \smath{r_2 + r_1}\\
|
|
|
1179 |
\smath{r + r} & \smath{\equiv} & \smath{r}\\
|
|
|
1180 |
\end{tabular}
|
|
|
1181 |
\end{center}
|
|
|
1182 |
\end{minipage}
|
|
|
1183 |
\end{quote}
|
|
|
1184 |
\end{block}
|
|
|
1185 |
\end{textblock}}
|
|
|
1186 |
|
|
|
1187 |
\end{frame}}
|
|
|
1188 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1189 |
*}
|
|
|
1190 |
|
|
|
1191 |
text_raw {*
|
|
|
1192 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1193 |
\mode<presentation>{
|
|
|
1194 |
\begin{frame}<2>[t]
|
|
|
1195 |
\frametitle{\LARGE\begin{tabular}{c}Partial Derivatives of RExps\end{tabular}}
|
|
|
1196 |
|
|
|
1197 |
|
|
|
1198 |
\only<2>{%
|
|
|
1199 |
\textcolor{blue}{%
|
|
|
1200 |
\begin{center}
|
|
|
1201 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
|
|
|
1202 |
pder c $\varnothing$ & $\dn$ & \alert{$\{\}$}\\
|
|
|
1203 |
pder c [] & $\dn$ & \alert{$\{\}$}\\
|
|
|
1204 |
pder c d & $\dn$ & if c $=$ d then $\{$[]$\}$ else $\{\}$\\
|
|
|
1205 |
pder c ($r_1 + r_2$) & $\dn$ & (pder c $r_1$) \alert{$\cup$} (der c $r_2$)\\
|
|
|
1206 |
pder c ($r^\star$) & $\dn$ & (pder c $r$) $\cdot$ $r^\star$\\
|
|
|
1207 |
pder c ($r_1 \cdot r_2$) & $\dn$ & (pder c $r_1$) $\cdot$ $r_2$ \alert{$\cup$}\\
|
|
|
1208 |
& & \hspace{-4mm}if nullable $r_1$ then (pder c $r_2$) else $\varnothing$\\
|
|
|
1209 |
\end{tabular}
|
|
|
1210 |
\end{center}}}
|
|
|
1211 |
|
|
|
1212 |
\only<2>{
|
|
|
1213 |
\begin{textblock}{6}(8.5,2.7)
|
|
|
1214 |
\begin{block}{}
|
|
|
1215 |
\begin{quote}
|
|
|
1216 |
\begin{minipage}{6cm}\raggedright
|
|
|
1217 |
\begin{itemize}
|
|
|
1218 |
\item partial derivatives
|
|
|
1219 |
\item by Antimirov~'95
|
|
|
1220 |
\end{itemize}
|
|
|
1221 |
\end{minipage}
|
|
|
1222 |
\end{quote}
|
|
|
1223 |
\end{block}
|
|
|
1224 |
\end{textblock}}
|
|
|
1225 |
|
|
|
1226 |
\end{frame}}
|
|
|
1227 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1228 |
*}
|
|
|
1229 |
|
|
|
1230 |
text_raw {*
|
|
|
1231 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1232 |
\mode<presentation>{
|
|
|
1233 |
\begin{frame}[t]
|
|
|
1234 |
\frametitle{\LARGE Partial Derivatives}
|
|
|
1235 |
|
|
|
1236 |
\mbox{}\\[0mm]\mbox{}
|
|
|
1237 |
|
|
|
1238 |
\begin{itemize}
|
|
|
1239 |
|
|
|
1240 |
\item \alt<1>{\smath{\text{pders $x$ $r$ \mbox{$=$} pders $y$ $r$}}}
|
|
|
1241 |
{\smath{\underbrace{\text{pders $x$ $r$ \mbox{$=$} pders $y$ $r$}}_{R}}}
|
|
|
1242 |
refines \textcolor{blue}{$x$ $\approx_{{\cal L}(r)}$ $y$}\\[16mm]\pause
|
|
|
1243 |
\item \smath{\text{finite} (U\!N\!IV /\!/ R)} \bigskip\pause
|
|
|
1244 |
\item Therefore \smath{\text{finite} (U\!N\!IV /\!/ \approx_{{\cal L}(r)})}. Qed.
|
|
|
1245 |
\end{itemize}
|
|
|
1246 |
|
|
|
1247 |
\only<2->{%
|
|
|
1248 |
\begin{textblock}{5}(3.9,7.2)
|
|
|
1249 |
\begin{tikzpicture}
|
|
|
1250 |
\node at (0,0) [shape border rotate=270,single arrow, fill=red,text=white, minimum height=0cm]{\textcolor{red}{a}};
|
|
|
1251 |
\draw (2.2,0) node {Antimirov '95};
|
|
|
1252 |
\end{tikzpicture}
|
|
|
1253 |
\end{textblock}}
|
|
|
1254 |
|
|
|
1255 |
\end{frame}}
|
|
|
1256 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1257 |
*}
|
|
|
1258 |
|
|
|
1259 |
|
|
|
1260 |
|
|
|
1261 |
text_raw {*
|
|
|
1262 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1263 |
\mode<presentation>{
|
|
|
1264 |
\begin{frame}[t]
|
|
|
1265 |
\frametitle{\LARGE What Have We Achieved?}
|
|
|
1266 |
|
|
|
1267 |
\begin{itemize}
|
|
|
1268 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_A) \;\Leftrightarrow\; A\; \text{is regular}}
|
|
|
1269 |
\medskip\pause
|
|
|
1270 |
\item regular languages are closed under complementation; this is now easy
|
|
|
1271 |
\begin{center}
|
|
|
1272 |
\smath{U\!N\!IV /\!/ \approx_A \;\;=\;\; U\!N\!IV /\!/ \approx_{\overline{A}}}
|
|
|
1273 |
\end{center}\pause\medskip
|
|
|
1274 |
|
|
|
1275 |
\item non-regularity (\smath{a^nb^n})\medskip\pause\pause
|
|
|
1276 |
|
|
|
1277 |
\item take \alert{\bf any} language\\ build the language of substrings\\
|
|
|
1278 |
\pause
|
|
|
1279 |
|
|
|
1280 |
then this language \alert{\bf is} regular\;\; (\smath{a^nb^n} $\Rightarrow$ \smath{a^\star{}b^\star})
|
|
|
1281 |
|
|
|
1282 |
\end{itemize}
|
|
|
1283 |
|
|
|
1284 |
\only<2>{
|
|
|
1285 |
\begin{textblock}{10}(4,14)
|
|
|
1286 |
\small
|
|
|
1287 |
\smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}
|
|
|
1288 |
\end{textblock}}
|
|
|
1289 |
|
|
|
1290 |
\only<4>{
|
|
|
1291 |
\begin{textblock}{5}(2,8.6)
|
|
|
1292 |
\begin{minipage}{8.8cm}
|
|
|
1293 |
\begin{block}{}
|
|
|
1294 |
\begin{minipage}{8.6cm}
|
|
|
1295 |
If there exists a sufficiently large set \smath{B} (for example infinitely large),
|
|
|
1296 |
such that
|
|
|
1297 |
|
|
|
1298 |
\begin{center}
|
|
|
1299 |
\smath{\forall x,y \in B.\; x \not= y \;\Rightarrow\; x \not\approx_{A} y}.
|
|
|
1300 |
\end{center}
|
|
|
1301 |
|
|
|
1302 |
then \smath{A} is not regular.\hspace{1.3cm}\small(\smath{B \dn \bigcup_n a^n})
|
|
|
1303 |
\end{minipage}
|
|
|
1304 |
\end{block}
|
|
|
1305 |
\end{minipage}
|
|
|
1306 |
\end{textblock}
|
|
|
1307 |
}
|
|
|
1308 |
|
|
|
1309 |
\end{frame}}
|
|
|
1310 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1311 |
*}
|
|
|
1312 |
|
|
|
1313 |
text_raw {*
|
|
|
1314 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1315 |
\mode<presentation>{
|
|
|
1316 |
\begin{frame}[c]
|
|
|
1317 |
\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
|
|
|
1318 |
|
|
|
1319 |
\begin{center}
|
|
|
1320 |
\huge\bf\textcolor{gray}{in Nuprl}
|
|
|
1321 |
\end{center}
|
|
|
1322 |
|
|
|
1323 |
\begin{itemize}
|
|
|
1324 |
\item Constable, Jackson, Naumov, Uribe\medskip
|
|
|
1325 |
\item \alert{18 months} for automata theory from Hopcroft \& Ullman chapters 1--11 (including Myhill-Nerode)
|
|
|
1326 |
\end{itemize}
|
|
|
1327 |
|
|
|
1328 |
\end{frame}}
|
|
|
1329 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1330 |
|
|
|
1331 |
*}
|
|
|
1332 |
|
|
|
1333 |
text_raw {*
|
|
|
1334 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1335 |
\mode<presentation>{
|
|
|
1336 |
\begin{frame}[c]
|
|
|
1337 |
\frametitle{\normalsize Formal language theory\ldots\hfill\mbox{}}
|
|
|
1338 |
|
|
|
1339 |
\begin{center}
|
|
|
1340 |
\huge\bf\textcolor{gray}{in Coq}
|
|
|
1341 |
\end{center}
|
|
|
1342 |
|
|
|
1343 |
\begin{itemize}
|
|
|
1344 |
\item Filli\^atre, Briais, Braibant and others
|
|
|
1345 |
\item multi-year effort; a number of results in automata theory, e.g.\medskip
|
|
|
1346 |
\begin{itemize}
|
|
|
1347 |
\item Kleene's thm.~by Filli\^atre (\alert{``rather big''})
|
|
|
1348 |
\item automata theory by Briais (5400 loc)
|
|
|
1349 |
\item Braibant ATBR library, including Myhill-Nerode\\ ($>$7000 loc)
|
|
|
1350 |
\item Mirkin's partial derivative automaton construction (10600 loc)
|
|
|
1351 |
\end{itemize}
|
|
|
1352 |
\end{itemize}
|
|
|
1353 |
|
|
|
1354 |
\end{frame}}
|
|
|
1355 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1356 |
*}
|
|
|
1357 |
|
|
|
1358 |
|
|
|
1359 |
text_raw {*
|
|
|
1360 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1361 |
\mode<presentation>{
|
|
|
1362 |
\begin{frame}[c]
|
|
|
1363 |
\frametitle{\LARGE Conclusion}
|
|
|
1364 |
|
|
|
1365 |
\begin{itemize}
|
|
|
1366 |
\item we have never seen a proof of Myhill-Nerode based on
|
|
|
1367 |
regular expressions only\smallskip\pause
|
|
|
1368 |
|
|
|
1369 |
\item great source of examples (inductions)\smallskip\pause
|
|
|
1370 |
|
|
|
1371 |
\item no need to fight the theorem prover:\\
|
|
|
1372 |
\begin{itemize}
|
|
|
1373 |
\item first direction (790 loc)\\
|
|
|
1374 |
\item second direction (400 / 390 loc)
|
|
|
1375 |
\end{itemize}
|
|
|
1376 |
|
|
|
1377 |
\item I am not saying automata are bad; just formal proofs about
|
|
|
1378 |
them are quite dif$\!$ficult\pause\bigskip\medskip
|
|
|
1379 |
|
|
|
1380 |
\item parsing with derivatives of grammars\\ (Matt Might ICFP'11)
|
|
|
1381 |
\end{itemize}
|
|
|
1382 |
|
|
|
1383 |
\end{frame}}
|
|
|
1384 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1385 |
*}
|
|
|
1386 |
|
|
|
1387 |
text_raw {*
|
|
|
1388 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1389 |
\mode<presentation>{
|
|
|
1390 |
\begin{frame}[c]
|
|
|
1391 |
\frametitle{\LARGE An Apology}
|
|
|
1392 |
|
|
|
1393 |
\begin{itemize}
|
|
|
1394 |
\item This should all of course be done co-inductively
|
|
|
1395 |
\end{itemize}
|
|
|
1396 |
|
|
|
1397 |
\footnotesize
|
|
|
1398 |
\begin{tabular}{@ {\hspace{4mm}}l}
|
|
|
1399 |
From: Jasmin Christian Blanchette\\
|
|
|
1400 |
To: isabelle-dev@mailbroy.informatik.tu-muenchen.de\\
|
|
|
1401 |
Subject: [isabelle-dev] NEWS\\
|
|
|
1402 |
Date: \alert{\bf Tue, 28 Aug 2012} 17:40:55 +0200\\
|
|
|
1403 |
\\
|
|
|
1404 |
* {\bf HOL/Codatatype}: New (co)datatype package with support for mixed,\\
|
|
|
1405 |
nested recursion and interesting non-free datatypes.\\
|
|
|
1406 |
\\
|
|
|
1407 |
* HOL/Ordinals\_and\_Cardinals: Theories of ordinals and cardinals\\
|
|
|
1408 |
(supersedes the AFP entry of the same name).\\[2mm]
|
|
|
1409 |
Kudos to Andrei and Dmitriy!\\
|
|
|
1410 |
\\
|
|
|
1411 |
Jasmin\\[-1mm]
|
|
|
1412 |
------------------------------------\\
|
|
|
1413 |
isabelle-dev mailing list\\
|
|
|
1414 |
isabelle-dev@in.tum.de\\
|
|
|
1415 |
\end{tabular}
|
|
|
1416 |
|
|
|
1417 |
\end{frame}}
|
|
|
1418 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1419 |
*}
|
|
|
1420 |
|
|
|
1421 |
|
|
|
1422 |
text_raw {*
|
|
|
1423 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1424 |
\mode<presentation>{
|
|
|
1425 |
\begin{frame}[b]
|
|
|
1426 |
\frametitle{\mbox{}\\[2cm]\textcolor{red}{Thank you very much!\\[5mm]Questions?}}
|
|
|
1427 |
|
|
|
1428 |
\end{frame}}
|
|
|
1429 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1430 |
*}
|
|
|
1431 |
|
|
|
1432 |
(*<*)
|
|
|
1433 |
end
|
|
|
1434 |
(*>*) |