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2772
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(*<*)
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theory Slides7
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imports "~~/src/HOL/Library/LaTeXsugar" "Main"
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begin
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declare [[show_question_marks = false]]
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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}{Hefei, 15.~April 2011}
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\newcommand{\abst}[2]{#1.#2}% atom-abstraction
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\newcommand{\pair}[2]{\langle #1,#2\rangle} % pairing
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\newcommand{\susp}{{\boldsymbol{\cdot}}}% for suspensions
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\newcommand{\unit}{\langle\rangle}% unit
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\newcommand{\app}[2]{#1\,#2}% application
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\newcommand{\eqprob}{\mathrel{{\approx}?}}
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\newcommand{\freshprob}{\mathrel{\#?}}
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\newcommand{\redu}[1]{\stackrel{#1}{\Longrightarrow}}% reduction
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\newcommand{\id}{\varepsilon}% identity substitution
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\newcommand{\bl}[1]{\textcolor{blue}{#1}}
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\newcommand{\gr}[1]{\textcolor{gray}{#1}}
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\newcommand{\rd}[1]{\textcolor{red}{#1}}
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\newcommand{\ok}{\includegraphics[scale=0.07]{ok.png}}
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\newcommand{\notok}{\includegraphics[scale=0.07]{notok.png}}
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\newcommand{\largenotok}{\includegraphics[scale=1]{notok.png}}
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\renewcommand{\Huge}{\fontsize{61.92}{77}\selectfont}
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\newcommand{\veryHuge}{\fontsize{74.3}{93}\selectfont}
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\newcommand{\VeryHuge}{\fontsize{89.16}{112}\selectfont}
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\newcommand{\VERYHuge}{\fontsize{107}{134}\selectfont}
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\newcommand{\LL}{$\mathbb{L}\,$}
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\pgfdeclareradialshading{smallbluesphere}{\pgfpoint{0.5mm}{0.5mm}}%
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{rgb(0mm)=(0,0,0.9);
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rgb(0.9mm)=(0,0,0.7);
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rgb(1.3mm)=(0,0,0.5);
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rgb(1.4mm)=(1,1,1)}
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\def\myitemi{\begin{pgfpicture}{-1ex}{-0.55ex}{1ex}{1ex}
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\usebeamercolor[fg]{subitem projected}
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{\pgftransformscale{0.8}\pgftext{\normalsize\pgfuseshading{bigsphere}}}
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\pgftext{%
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\usebeamerfont*{subitem projected}}
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\end{pgfpicture}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1>[t]
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\frametitle{%
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\begin{tabular}{@ {\hspace{-3mm}}c@ {}}
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\\
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\LARGE Verifying a Regular Expression\\[-1mm]
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\LARGE Matcher and Formal Language\\[-1mm]
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\LARGE Theory\\[5mm]
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\end{tabular}}
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\begin{center}
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Christian Urban\\
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\small Technical University of Munich, Germany
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\end{center}
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\begin{center}
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\small joint work with Chunhan Wu and Xingyuan Zhang from the PLA
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University of Science and Technology in Nanjing
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\end{center}
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\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{This Talk: 3 Points}
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\large
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\begin{itemize}
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\item It is easy to make mistakes.\bigskip
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\item Theorem provers can prevent mistakes, {\bf if} the problem
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is formulated so that it is suitable for theorem provers.\bigskip
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\item This re-formulation can be done, even in domains where
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we do not expect it.
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<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$_1$ + r$_2$}\\
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& \bl{$\mid$} & \bl{r$_1$ $\cdot$ 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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\only<2>{
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\begin{textblock}{9}(3.6,11.8)
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\bl{matches r s $\;\Longrightarrow\;$ true $\vee$ false}\\[3.5mm]
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\hspace{10mm}\begin{tikzpicture}
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\coordinate (m1) at (0.4,1);
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\draw (0,0.3) node (m2) {\small\color{gray}rexp};
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\path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1);
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\coordinate (s1) at (0.81,1);
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\draw (1.3,0.3) node (s2) {\small\color{gray} string};
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\path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1);
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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->[t]
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\frametitle{Specification}
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\small
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\begin{textblock}{6}(0,3.5)
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\begin{tabular}{r@ {\hspace{0.5mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l}
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\multicolumn{4}{c}{rexp $\Rightarrow$ set of strings}\bigskip\\
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&\bl{\LL ($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$}\\
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&\bl{\LL ([])} & \bl{$\dn$} & \bl{\{[]\}}\\
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&\bl{\LL (c)} & \bl{$\dn$} & \bl{\{c\}}\\
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&\bl{\LL (r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) $\cup$ \LL (r$_2$)}\\
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\rd{$\Rightarrow$} &\bl{\LL (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) ;; \LL (r$_2$)}\\
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\rd{$\Rightarrow$} &\bl{\LL (r$^*$)} & \bl{$\dn$} & \bl{(\LL (r))$^\star$}\\
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\end{tabular}
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\end{textblock}
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\begin{textblock}{9}(7.3,3)
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{\mbox{}\hspace{2cm}\footnotesize Isabelle:\smallskip}
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\includegraphics[scale=0.325]{Screen3.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}<1->[t]
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\frametitle{Version 1}
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\small
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\mbox{}\\[-8mm]\mbox{}
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\begin{center}\def\arraystretch{1.05}
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\begin{tabular}{@ {\hspace{-5mm}}l@ {\hspace{2.5mm}}c@ {\hspace{2.5mm}}l@ {}}
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\bl{match [] []} & \bl{$=$} & \bl{true}\\
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\bl{match [] (c::s)} & \bl{$=$} & \bl{false}\\
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\bl{match ($\varnothing$::rs) s} & \bl{$=$} & \bl{false}\\
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\bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\
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\bl{match (c::rs) []} & \bl{$=$} & \bl{false}\\
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\bl{match (c::rs) (d::s)} & \bl{$=$} & \bl{if c = d then match rs s else false}\\
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\bl{match (r$_1$ + r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s $\vee$ match (r$_2$::rs) s}\\
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\bl{match (r$_1$ $\cdot$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\
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\bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\
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\end{tabular}
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\end{center}
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\begin{textblock}{9}(0.2,1.6)
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\hspace{10mm}\begin{tikzpicture}
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\coordinate (m1) at (0.44,-0.5);
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\draw (0,0.3) node (m2) {\small\color{gray}\mbox{}\hspace{-9mm}list of rexps};
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\path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1);
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\coordinate (s1) at (0.86,-0.5);
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\draw (1.5,0.3) node (s2) {\small\color{gray} string};
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\path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1);
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\end{tikzpicture}
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\end{textblock}
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\begin{textblock}{9}(2.8,11.8)
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\bl{matches$_1$ r s $\;=\;$ match [r] s}
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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{Testing}
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\small
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Every good programmer should do thourough tests:
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\begin{center}
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\begin{tabular}{@ {\hspace{-20mm}}lcl}
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\bl{matches$_1$ (a$\cdot$b)$^*\;$ []} & \bl{$\mapsto$} & \bl{true}\\
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\bl{matches$_1$ (a$\cdot$b)$^*\;$ ab} & \bl{$\mapsto$} & \bl{true}\\
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\bl{matches$_1$ (a$\cdot$b)$^*\;$ aba} & \bl{$\mapsto$} & \bl{false}\\
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\bl{matches$_1$ (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\
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\bl{matches$_1$ (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\
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\onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x} & \bl{$\mapsto$} & \bl{true}}\\
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\onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x0} & \bl{$\mapsto$} & \bl{true}}\\
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\onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x3} & \bl{$\mapsto$} & \bl{false}}
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\end{tabular}
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\end{center}
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\onslide<3->
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{looks OK \ldots let's ship it to customers\hspace{5mm}
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\raisebox{-5mm}{\includegraphics[scale=0.05]{sun.png}}}
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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{Version 1}
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\only<1->{Several hours later\ldots}\pause
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\begin{center}
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\begin{tabular}{@ {\hspace{0mm}}lcl}
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\bl{matches$_1$ []$^*$ s} & \bl{$\mapsto$} & loops\\
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\onslide<4->{\bl{matches$_1$ ([] + \ldots)$^*$ s} & \bl{$\mapsto$} & loops\\}
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\end{tabular}
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\end{center}
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\small
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\onslide<3->{
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\begin{center}
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\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
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\ldots\\
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\bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\
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\ldots\\
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\bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1->[t]
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\frametitle{Testing}
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\begin{itemize}
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\item While testing is an important part in the process of programming development\pause\ldots
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\item we can only test a {\bf finite} amount of examples.\bigskip\pause
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\begin{center}
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\colorbox{cream}
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{\gr{\begin{minipage}{10cm}
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``Testing can only show the presence of errors, never their
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absence.'' (Edsger W.~Dijkstra)
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\end{minipage}}}
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\end{center}\bigskip\pause
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\item In a theorem prover we can establish properties that apply to
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{\bf all} input and {\bf all} output.
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\end{itemize}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mode<presentation>{
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\begin{frame}<1->[t]
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\frametitle{Version 2}
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\mbox{}\\[-14mm]\mbox{}
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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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324 |
\end{tabular}\medskip
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325 |
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326 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
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327 |
\bl{der c ($\varnothing$)} & \bl{$=$} & \bl{$\varnothing$} & \\
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\bl{der c ([])} & \bl{$=$} & \bl{$\varnothing$} & \\
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329 |
\bl{der c (d)} & \bl{$=$} & \bl{if c = d then [] else $\varnothing$} & \\
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330 |
\bl{der c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\
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331 |
\bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{((der c r$_1$) $\cdot$ r$_2$)} & \\
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332 |
& & \bl{\;\;+ (if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\
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333 |
\bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ r$^*$} &\smallskip\\
|
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334 |
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335 |
\bl{derivative r []} & \bl{$=$} & \bl{r} & \\
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336 |
\bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\
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337 |
\end{tabular}\medskip
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338 |
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339 |
\bl{matches$_2$ r s $=$ nullable (derivative r s)}
|
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340 |
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\begin{textblock}{6}(9.5,0.9)
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342 |
\begin{flushright}
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343 |
\color{gray}``if r matches []''
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344 |
\end{flushright}
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345 |
\end{textblock}
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347 |
\begin{textblock}{6}(9.5,6.18)
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348 |
\begin{flushright}
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349 |
\color{gray}``derivative w.r.t.~a char''
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350 |
\end{flushright}
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\end{textblock}
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352 |
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353 |
\begin{textblock}{6}(9.5,12.1)
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354 |
\begin{flushright}
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355 |
\color{gray}``deriv.~w.r.t.~a string''
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356 |
\end{flushright}
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357 |
\end{textblock}
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358 |
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\begin{textblock}{6}(9.5,13.98)
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360 |
\begin{flushright}
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361 |
\color{gray}``main''
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\end{flushright}
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\end{textblock}
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\end{frame}}
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*}
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text_raw {*
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\mode<presentation>{
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372 |
\begin{frame}<1->[t]
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\frametitle{Is the Matcher Error-Free?}
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374 |
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375 |
We expect that
|
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376 |
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377 |
\begin{center}
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378 |
\begin{tabular}{lcl}
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379 |
\bl{matches$_2$ r s = true} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}%
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380 |
\only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\in$ \LL(r)}\\
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\bl{matches$_2$ r s = false} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}%
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382 |
\only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\notin$ \LL(r)}\\
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\end{tabular}
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\end{center}
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\pause\pause\bigskip
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??? By \alert<4->{induction}, we can {\bf prove} these properties.\bigskip
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387 |
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\begin{tabular}{lrcl}
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389 |
Lemmas: & \bl{nullable (r)} & \bl{$\Longleftrightarrow$} & \bl{[] $\in$ \LL (r)}\\
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& \bl{s $\in$ \LL (der c r)} & \bl{$\Longleftrightarrow$} & \bl{(c::s) $\in$ \LL (r)}\\
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\end{tabular}
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392 |
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393 |
\only<4->{
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\begin{textblock}{3}(0.9,4.5)
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395 |
\rd{\huge$\forall$\large{}r s.}
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\end{textblock}}
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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*}
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401 |
text_raw {*
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\mode<presentation>{
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\begin{frame}<1>[c]
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405 |
\frametitle{
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406 |
\begin{tabular}{c}
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407 |
\mbox{}\\[23mm]
|
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408 |
\LARGE Demo
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409 |
\end{tabular}}
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410 |
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411 |
\end{frame}}
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412 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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413 |
*}
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414 |
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416 |
text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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418 |
\mode<presentation>{
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419 |
\begin{frame}<1->[t]
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420 |
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421 |
\mbox{}\\[-2mm]
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422 |
|
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|
423 |
\small
|
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|
424 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}}
|
|
|
425 |
\bl{nullable (NULL)} & \bl{$=$} & \bl{false} &\\
|
|
|
426 |
\bl{nullable (EMPTY)} & \bl{$=$} & \bl{true} &\\
|
|
|
427 |
\bl{nullable (CHR c)} & \bl{$=$} & \bl{false} &\\
|
|
|
428 |
\bl{nullable (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) orelse (nullable r$_2$)} & \\
|
|
|
429 |
\bl{nullable (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) andalso (nullable r$_2$)} & \\
|
|
|
430 |
\bl{nullable (STAR r)} & \bl{$=$} & \bl{true} & \\
|
|
|
431 |
\end{tabular}\medskip
|
|
|
432 |
|
|
|
433 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
|
|
|
434 |
\bl{der c (NULL)} & \bl{$=$} & \bl{NULL} & \\
|
|
|
435 |
\bl{der c (EMPTY)} & \bl{$=$} & \bl{NULL} & \\
|
|
|
436 |
\bl{der c (CHR d)} & \bl{$=$} & \bl{if c=d then EMPTY else NULL} & \\
|
|
|
437 |
\bl{der c (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (der c r$_1$) (der c r$_2$)} & \\
|
|
|
438 |
\bl{der c (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (SEQ (der c r$_1$) r$_2$)} & \\
|
|
|
439 |
& & \bl{\phantom{ALT} (if nullable r$_1$ then der c r$_2$ else NULL)}\\
|
|
|
440 |
\bl{der c (STAR r)} & \bl{$=$} & \bl{SEQ (der c r) (STAR r)} &\smallskip\\
|
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|
441 |
|
|
|
442 |
\bl{derivative r []} & \bl{$=$} & \bl{r} & \\
|
|
|
443 |
\bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\
|
|
|
444 |
\end{tabular}\medskip
|
|
|
445 |
|
|
|
446 |
\bl{matches r s $=$ nullable (derivative r s)}
|
|
|
447 |
|
|
|
448 |
\only<2>{
|
|
|
449 |
\begin{textblock}{8}(1.5,4)
|
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|
450 |
\includegraphics[scale=0.3]{approved.png}
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|
451 |
\end{textblock}}
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452 |
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453 |
\end{frame}}
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454 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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455 |
*}
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456 |
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457 |
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458 |
text_raw {*
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459 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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460 |
\mode<presentation>{
|
|
|
461 |
\begin{frame}[c]
|
|
|
462 |
\frametitle{No Automata?}
|
|
|
463 |
|
|
|
464 |
You might be wondering why I did not use any automata?
|
|
|
465 |
|
|
|
466 |
\begin{itemize}
|
|
|
467 |
\item {\bf Def.:} A \alert{regular language} is one where there is a DFA that
|
|
|
468 |
recognises it.\bigskip\pause
|
|
|
469 |
\end{itemize}
|
|
|
470 |
|
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|
471 |
|
|
|
472 |
There are many reasons why this is a good definition:\medskip
|
|
|
473 |
\begin{itemize}
|
|
|
474 |
\item pumping lemma
|
|
|
475 |
\item closure properties of regular languages\\ (e.g.~closure under complement)
|
|
|
476 |
\end{itemize}
|
|
|
477 |
|
|
|
478 |
\end{frame}}
|
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|
479 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
480 |
|
|
|
481 |
*}
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|
482 |
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483 |
text_raw {*
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484 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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485 |
\mode<presentation>{
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|
486 |
\begin{frame}[t]
|
|
|
487 |
\frametitle{Really Bad News!}
|
|
|
488 |
|
|
|
489 |
DFAs are bad news for formalisations in theorem provers. They might
|
|
|
490 |
be represented as:
|
|
|
491 |
|
|
|
492 |
\begin{itemize}
|
|
|
493 |
\item graphs
|
|
|
494 |
\item matrices
|
|
|
495 |
\item partial functions
|
|
|
496 |
\end{itemize}
|
|
|
497 |
|
|
|
498 |
All constructions are messy to reason about.\bigskip\bigskip
|
|
|
499 |
\pause
|
|
|
500 |
|
|
|
501 |
\small
|
|
|
502 |
\only<2>{
|
|
|
503 |
Constable et al needed (on and off) 18 months for a 3-person team
|
|
|
504 |
to formalise automata theory in Nuprl including Myhill-Nerode. There is
|
|
|
505 |
only very little other formalised work on regular languages I know of
|
|
|
506 |
in Coq, Isabelle and HOL.}
|
|
|
507 |
\only<3>{Typical textbook reasoning goes like: ``\ldots if \smath{M} and \smath{N} are any two
|
|
|
508 |
automata with no inaccessible states \ldots''
|
|
|
509 |
}
|
|
|
510 |
|
|
|
511 |
\end{frame}}
|
|
|
512 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
513 |
|
|
|
514 |
*}
|
|
|
515 |
|
|
|
516 |
text_raw {*
|
|
|
517 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
518 |
\mode<presentation>{
|
|
|
519 |
\begin{frame}[c]
|
|
|
520 |
\frametitle{}
|
|
|
521 |
\large
|
|
|
522 |
\begin{center}
|
|
|
523 |
\begin{tabular}{p{9cm}}
|
|
|
524 |
My point:\bigskip\\
|
|
|
525 |
|
|
|
526 |
The theory about regular languages can be reformulated
|
|
|
527 |
to be more suitable for theorem proving.
|
|
|
528 |
\end{tabular}
|
|
|
529 |
\end{center}
|
|
|
530 |
\end{frame}}
|
|
|
531 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
532 |
*}
|
|
|
533 |
|
|
|
534 |
text_raw {*
|
|
|
535 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
536 |
\mode<presentation>{
|
|
|
537 |
\begin{frame}[c]
|
|
|
538 |
\frametitle{\LARGE The Myhill-Nerode Theorem}
|
|
|
539 |
|
|
|
540 |
\begin{itemize}
|
|
|
541 |
\item provides necessary and suf\!ficient conditions for a language
|
|
|
542 |
being regular (pumping lemma only necessary)\medskip
|
|
|
543 |
|
|
|
544 |
\item will help with closure properties of regular languages\bigskip\pause
|
|
|
545 |
|
|
|
546 |
\item key is the equivalence relation:\smallskip
|
|
|
547 |
\begin{center}
|
|
|
548 |
\smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
|
|
|
549 |
\end{center}
|
|
|
550 |
\end{itemize}
|
|
|
551 |
|
|
|
552 |
\end{frame}}
|
|
|
553 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
554 |
*}
|
|
|
555 |
|
|
|
556 |
text_raw {*
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|
557 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
558 |
\mode<presentation>{
|
|
|
559 |
\begin{frame}[c]
|
|
|
560 |
\frametitle{\LARGE The Myhill-Nerode Theorem}
|
|
|
561 |
|
|
|
562 |
\mbox{}\\[5cm]
|
|
|
563 |
|
|
|
564 |
\begin{itemize}
|
|
|
565 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
|
|
|
566 |
\end{itemize}
|
|
|
567 |
|
|
|
568 |
\end{frame}}
|
|
|
569 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
570 |
|
|
|
571 |
*}
|
|
|
572 |
|
|
|
573 |
text_raw {*
|
|
|
574 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
575 |
\mode<presentation>{
|
|
|
576 |
\begin{frame}[c]
|
|
|
577 |
\frametitle{\LARGE Equivalence Classes}
|
|
|
578 |
|
|
|
579 |
\begin{itemize}
|
|
|
580 |
\item \smath{L = []}
|
|
|
581 |
\begin{center}
|
|
|
582 |
\smath{\Big\{\{[]\},\; U\!N\!IV - \{[]\}\Big\}}
|
|
|
583 |
\end{center}\bigskip\bigskip
|
|
|
584 |
|
|
|
585 |
\item \smath{L = [c]}
|
|
|
586 |
\begin{center}
|
|
|
587 |
\smath{\Big\{\{[]\},\; \{[c]\},\; U\!N\!IV - \{[], [c]\}\Big\}}
|
|
|
588 |
\end{center}\bigskip\bigskip
|
|
|
589 |
|
|
|
590 |
\item \smath{L = \varnothing}
|
|
|
591 |
\begin{center}
|
|
|
592 |
\smath{\Big\{U\!N\!IV\Big\}}
|
|
|
593 |
\end{center}
|
|
|
594 |
|
|
|
595 |
\end{itemize}
|
|
|
596 |
|
|
|
597 |
\end{frame}}
|
|
|
598 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
599 |
|
|
|
600 |
*}
|
|
|
601 |
|
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|
602 |
text_raw {*
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|
603 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
604 |
\mode<presentation>{
|
|
|
605 |
\begin{frame}[c]
|
|
|
606 |
\frametitle{\LARGE Regular Languages}
|
|
|
607 |
|
|
|
608 |
\begin{itemize}
|
|
|
609 |
\item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M}
|
|
|
610 |
such that \smath{\mathbb{L}(M) = L}\\[1.5cm]
|
|
|
611 |
|
|
|
612 |
\item Myhill-Nerode:
|
|
|
613 |
|
|
|
614 |
\begin{center}
|
|
|
615 |
\begin{tabular}{l}
|
|
|
616 |
finite $\Rightarrow$ regular\\
|
|
|
617 |
\;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r. L = \mathbb{L}(r)}\\[3mm]
|
|
|
618 |
regular $\Rightarrow$ finite\\
|
|
|
619 |
\;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
|
|
|
620 |
\end{tabular}
|
|
|
621 |
\end{center}
|
|
|
622 |
|
|
|
623 |
\end{itemize}
|
|
|
624 |
|
|
|
625 |
\end{frame}}
|
|
|
626 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
627 |
|
|
|
628 |
*}
|
|
|
629 |
|
|
|
630 |
text_raw {*
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|
|
631 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
632 |
\mode<presentation>{
|
|
|
633 |
\begin{frame}[c]
|
|
|
634 |
\frametitle{\LARGE Final States}
|
|
|
635 |
|
|
|
636 |
\mbox{}\\[3cm]
|
|
|
637 |
|
|
|
638 |
\begin{itemize}
|
|
|
639 |
\item ??? \smath{\text{final}_L\,X \dn \{[|s|]_\approx\;|\; s \in X\}}\\
|
|
|
640 |
\medskip
|
|
|
641 |
|
|
|
642 |
\item we can prove: \smath{L = \bigcup \{X\;|\;\text{final}_L\,X\}}
|
|
|
643 |
|
|
|
644 |
\end{itemize}
|
|
|
645 |
|
|
|
646 |
\end{frame}}
|
|
|
647 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
648 |
*}
|
|
|
649 |
|
|
|
650 |
text_raw {*
|
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|
651 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
652 |
\mode<presentation>{
|
|
|
653 |
\begin{frame}[c]
|
|
|
654 |
\frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes}
|
|
|
655 |
|
|
|
656 |
\smath{L = \{[c]\}}
|
|
|
657 |
|
|
|
658 |
\begin{tabular}{@ {\hspace{-7mm}}cc}
|
|
|
659 |
\begin{tabular}{c}
|
|
|
660 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
|
|
|
661 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
|
662 |
|
|
|
663 |
%\draw[help lines] (0,0) grid (3,2);
|
|
|
664 |
|
|
|
665 |
\node[state,initial] (q_0) {$R_1$};
|
|
|
666 |
\node[state,accepting] (q_1) [above right of=q_0] {$R_2$};
|
|
|
667 |
\node[state] (q_2) [below right of=q_0] {$R_3$};
|
|
|
668 |
|
|
|
669 |
\path[->] (q_0) edge node {c} (q_1)
|
|
|
670 |
edge node [swap] {$\Sigma-{c}$} (q_2)
|
|
|
671 |
(q_2) edge [loop below] node {$\Sigma$} ()
|
|
|
672 |
(q_1) edge node {$\Sigma$} (q_2);
|
|
|
673 |
\end{tikzpicture}
|
|
|
674 |
\end{tabular}
|
|
|
675 |
&
|
|
|
676 |
\begin{tabular}[t]{ll}
|
|
|
677 |
\\[-20mm]
|
|
|
678 |
\multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm]
|
|
|
679 |
|
|
|
680 |
\smath{R_1}: & \smath{\{[]\}}\\
|
|
|
681 |
\smath{R_2}: & \smath{\{[c]\}}\\
|
|
|
682 |
\smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm]
|
|
|
683 |
\multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ;; [c] \subseteq Y}}}
|
|
|
684 |
\end{tabular}
|
|
|
685 |
|
|
|
686 |
\end{tabular}
|
|
|
687 |
|
|
|
688 |
\end{frame}}
|
|
|
689 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
690 |
*}
|
|
|
691 |
|
|
|
692 |
|
|
|
693 |
text_raw {*
|
|
|
694 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
695 |
\mode<presentation>{
|
|
|
696 |
\begin{frame}[c]
|
|
|
697 |
\frametitle{\LARGE Systems of Equations}
|
|
|
698 |
|
|
|
699 |
Inspired by a method of Brzozowski\;'64, we can build an equational system
|
|
|
700 |
characterising the equivalence classes:
|
|
|
701 |
|
|
|
702 |
\begin{center}
|
|
|
703 |
\begin{tabular}{@ {\hspace{-20mm}}c}
|
|
|
704 |
\\[-13mm]
|
|
|
705 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
|
|
|
706 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
|
707 |
|
|
|
708 |
%\draw[help lines] (0,0) grid (3,2);
|
|
|
709 |
|
|
|
710 |
\node[state,initial] (p_0) {$R_1$};
|
|
|
711 |
\node[state,accepting] (p_1) [right of=q_0] {$R_2$};
|
|
|
712 |
|
|
|
713 |
\path[->] (p_0) edge [bend left] node {a} (p_1)
|
|
|
714 |
edge [loop above] node {b} ()
|
|
|
715 |
(p_1) edge [loop above] node {a} ()
|
|
|
716 |
edge [bend left] node {b} (p_0);
|
|
|
717 |
\end{tikzpicture}\\
|
|
|
718 |
\\[-13mm]
|
|
|
719 |
\end{tabular}
|
|
|
720 |
\end{center}
|
|
|
721 |
|
|
|
722 |
\begin{center}
|
|
|
723 |
\begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
|
|
|
724 |
& \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
|
|
|
725 |
& \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\
|
|
|
726 |
\onslide<3->{we can prove}
|
|
|
727 |
& \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}}
|
|
|
728 |
& \onslide<3->{\smath{R_1; \mathbb{L}(b) \,\cup\, R_2;\mathbb{L}(b) \,\cup\, \{[]\};\{[]\}}}\\
|
|
|
729 |
& \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}}
|
|
|
730 |
& \onslide<3->{\smath{R_1; \mathbb{L}(a) \,\cup\, R_2;\mathbb{L}(a)}}\\
|
|
|
731 |
\end{tabular}
|
|
|
732 |
\end{center}
|
|
|
733 |
|
|
|
734 |
\end{frame}}
|
|
|
735 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
736 |
*}
|
|
|
737 |
|
|
|
738 |
|
|
|
739 |
text_raw {*
|
|
|
740 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
741 |
\mode<presentation>{
|
|
|
742 |
\begin{frame}<1>[t]
|
|
|
743 |
\small
|
|
|
744 |
|
|
|
745 |
\begin{center}
|
|
|
746 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
|
747 |
\onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
|
|
|
748 |
& \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
749 |
\onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
|
|
|
750 |
& \onslide<1->{\smath{R_1; a + R_2; a}}\\
|
|
|
751 |
|
|
|
752 |
& & & \onslide<2->{by Arden}\\
|
|
|
753 |
|
|
|
754 |
\onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
|
|
|
755 |
& \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
756 |
\onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
|
|
|
757 |
& \only<2>{\smath{R_1; a + R_2; a}}%
|
|
|
758 |
\only<3->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
759 |
|
|
|
760 |
& & & \onslide<4->{by Arden}\\
|
|
|
761 |
|
|
|
762 |
\onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
|
|
|
763 |
& \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
764 |
\onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
|
|
|
765 |
& \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
766 |
|
|
|
767 |
& & & \onslide<5->{by substitution}\\
|
|
|
768 |
|
|
|
769 |
\onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
|
|
|
770 |
& \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
771 |
\onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
|
|
|
772 |
& \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
773 |
|
|
|
774 |
& & & \onslide<6->{by Arden}\\
|
|
|
775 |
|
|
|
776 |
\onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
|
|
|
777 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
778 |
\onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
|
|
|
779 |
& \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
780 |
|
|
|
781 |
& & & \onslide<7->{by substitution}\\
|
|
|
782 |
|
|
|
783 |
\onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
|
|
|
784 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
785 |
\onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
|
|
|
786 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
|
|
787 |
\cdot a\cdot a^\star}}\\
|
|
|
788 |
\end{tabular}
|
|
|
789 |
\end{center}
|
|
|
790 |
|
|
|
791 |
\end{frame}}
|
|
|
792 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
793 |
*}
|
|
|
794 |
|
|
|
795 |
text_raw {*
|
|
|
796 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
797 |
\mode<presentation>{
|
|
|
798 |
\begin{frame}[c]
|
|
|
799 |
\frametitle{\LARGE A Variant of Arden's Lemma}
|
|
|
800 |
|
|
|
801 |
{\bf Arden's Lemma:}\smallskip
|
|
|
802 |
|
|
|
803 |
If \smath{[] \not\in A} then
|
|
|
804 |
\begin{center}
|
|
|
805 |
\smath{X = X; A + \text{something}}
|
|
|
806 |
\end{center}
|
|
|
807 |
has the (unique) solution
|
|
|
808 |
\begin{center}
|
|
|
809 |
\smath{X = \text{something} ; A^\star}
|
|
|
810 |
\end{center}
|
|
|
811 |
|
|
|
812 |
|
|
|
813 |
\end{frame}}
|
|
|
814 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
815 |
*}
|
|
|
816 |
|
|
|
817 |
|
|
|
818 |
text_raw {*
|
|
|
819 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
820 |
\mode<presentation>{
|
|
|
821 |
\begin{frame}<1->[t]
|
|
|
822 |
\small
|
|
|
823 |
|
|
|
824 |
\begin{center}
|
|
|
825 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
|
826 |
\onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
|
|
|
827 |
& \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
828 |
\onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
|
|
|
829 |
& \onslide<1->{\smath{R_1; a + R_2; a}}\\
|
|
|
830 |
|
|
|
831 |
& & & \onslide<2->{by Arden}\\
|
|
|
832 |
|
|
|
833 |
\onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
|
|
|
834 |
& \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
|
835 |
\onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
|
|
|
836 |
& \only<2>{\smath{R_1; a + R_2; a}}%
|
|
|
837 |
\only<3->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
838 |
|
|
|
839 |
& & & \onslide<4->{by Arden}\\
|
|
|
840 |
|
|
|
841 |
\onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
|
|
|
842 |
& \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
843 |
\onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
|
|
|
844 |
& \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
845 |
|
|
|
846 |
& & & \onslide<5->{by substitution}\\
|
|
|
847 |
|
|
|
848 |
\onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
|
|
|
849 |
& \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
|
850 |
\onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
|
|
|
851 |
& \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
852 |
|
|
|
853 |
& & & \onslide<6->{by Arden}\\
|
|
|
854 |
|
|
|
855 |
\onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
|
|
|
856 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
857 |
\onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
|
|
|
858 |
& \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
|
|
|
859 |
|
|
|
860 |
& & & \onslide<7->{by substitution}\\
|
|
|
861 |
|
|
|
862 |
\onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
|
|
|
863 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
|
864 |
\onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
|
|
|
865 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
|
|
866 |
\cdot a\cdot a^\star}}\\
|
|
|
867 |
\end{tabular}
|
|
|
868 |
\end{center}
|
|
|
869 |
|
|
|
870 |
\only<8->{
|
|
|
871 |
\begin{textblock}{6}(2.5,4)
|
|
|
872 |
\begin{block}{}
|
|
|
873 |
\begin{minipage}{8cm}\raggedright
|
|
|
874 |
|
|
|
875 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
|
|
|
876 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
|
877 |
|
|
|
878 |
%\draw[help lines] (0,0) grid (3,2);
|
|
|
879 |
|
|
|
880 |
\node[state,initial] (p_0) {$R_1$};
|
|
|
881 |
\node[state,accepting] (p_1) [right of=q_0] {$R_2$};
|
|
|
882 |
|
|
|
883 |
\path[->] (p_0) edge [bend left] node {a} (p_1)
|
|
|
884 |
edge [loop above] node {b} ()
|
|
|
885 |
(p_1) edge [loop above] node {a} ()
|
|
|
886 |
edge [bend left] node {b} (p_0);
|
|
|
887 |
\end{tikzpicture}
|
|
|
888 |
|
|
|
889 |
\end{minipage}
|
|
|
890 |
\end{block}
|
|
|
891 |
\end{textblock}}
|
|
|
892 |
|
|
|
893 |
\end{frame}}
|
|
|
894 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
895 |
*}
|
|
|
896 |
|
|
|
897 |
|
|
|
898 |
text_raw {*
|
|
|
899 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
900 |
\mode<presentation>{
|
|
|
901 |
\begin{frame}[c]
|
|
|
902 |
\frametitle{\LARGE The Equ's Solving Algorithm}
|
|
|
903 |
|
|
|
904 |
\begin{itemize}
|
|
|
905 |
\item The algorithm must terminate: Arden makes one equation smaller;
|
|
|
906 |
substitution deletes one variable from the right-hand sides.\bigskip
|
|
|
907 |
|
|
|
908 |
\item We need to maintain the invariant that Arden is applicable
|
|
|
909 |
(if \smath{[] \not\in A} then \ldots):\medskip
|
|
|
910 |
|
|
|
911 |
\begin{center}\small
|
|
|
912 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
|
913 |
\smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
|
|
|
914 |
\smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\
|
|
|
915 |
|
|
|
916 |
& & & by Arden\\
|
|
|
917 |
|
|
|
918 |
\smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
|
|
|
919 |
\smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\
|
|
|
920 |
\end{tabular}
|
|
|
921 |
\end{center}
|
|
|
922 |
|
|
|
923 |
\end{itemize}
|
|
|
924 |
|
|
|
925 |
|
|
|
926 |
\end{frame}}
|
|
|
927 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
928 |
*}
|
|
|
929 |
|
|
|
930 |
|
|
|
931 |
text_raw {*
|
|
|
932 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
933 |
\mode<presentation>{
|
|
|
934 |
\begin{frame}[c]
|
|
|
935 |
\frametitle{\LARGE Other Direction}
|
|
|
936 |
|
|
|
937 |
One has to prove
|
|
|
938 |
|
|
|
939 |
\begin{center}
|
|
|
940 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
|
|
|
941 |
\end{center}
|
|
|
942 |
|
|
|
943 |
by induction on \smath{r}. Not trivial, but after a bit
|
|
|
944 |
of thinking, one can prove that if
|
|
|
945 |
|
|
|
946 |
\begin{center}
|
|
|
947 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm}
|
|
|
948 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})}
|
|
|
949 |
\end{center}
|
|
|
950 |
|
|
|
951 |
then
|
|
|
952 |
|
|
|
953 |
\begin{center}
|
|
|
954 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})}
|
|
|
955 |
\end{center}
|
|
|
956 |
|
|
|
957 |
|
|
|
958 |
|
|
|
959 |
\end{frame}}
|
|
|
960 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
961 |
*}
|
|
|
962 |
|
|
|
963 |
|
|
|
964 |
text_raw {*
|
|
|
965 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
966 |
\mode<presentation>{
|
|
|
967 |
\begin{frame}[c]
|
|
|
968 |
\frametitle{\LARGE What Have We Achieved?}
|
|
|
969 |
|
|
|
970 |
\begin{itemize}
|
|
|
971 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
|
|
|
972 |
\bigskip\pause
|
|
|
973 |
\item regular languages are closed under complementation; this is now easy\medskip
|
|
|
974 |
\begin{center}
|
|
|
975 |
\smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}}
|
|
|
976 |
\end{center}
|
|
|
977 |
\end{itemize}
|
|
|
978 |
|
|
|
979 |
|
|
|
980 |
\end{frame}}
|
|
|
981 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
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|
982 |
*}
|
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|
983 |
|
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|
984 |
text_raw {*
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|
985 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
|
986 |
\mode<presentation>{
|
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|
987 |
\begin{frame}[c]
|
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|
988 |
\frametitle{\LARGE Examples}
|
|
|
989 |
|
|
|
990 |
\begin{itemize}
|
|
|
991 |
\item \smath{L \equiv \Sigma^\star 0 \Sigma} is regular
|
|
|
992 |
\begin{quote}\small
|
|
|
993 |
\begin{tabular}{lcl}
|
|
|
994 |
\smath{A_1} & \smath{=} & \smath{\Sigma^\star 00}\\
|
|
|
995 |
\smath{A_2} & \smath{=} & \smath{\Sigma^\star 01}\\
|
|
|
996 |
\smath{A_3} & \smath{=} & \smath{\Sigma^\star 10 \cup \{0\}}\\
|
|
|
997 |
\smath{A_4} & \smath{=} & \smath{\Sigma^\star 11 \cup \{1\} \cup \{[]\}}\\
|
|
|
998 |
\end{tabular}
|
|
|
999 |
\end{quote}
|
|
|
1000 |
|
|
|
1001 |
\item \smath{L \equiv \{ 0^n 1^n \,|\, n \ge 0\}} is not regular
|
|
|
1002 |
\begin{quote}\small
|
|
|
1003 |
\begin{tabular}{lcl}
|
|
|
1004 |
\smath{B_0} & \smath{=} & \smath{\{0^n 1^n \,|\, n \ge 0\}}\\
|
|
|
1005 |
\smath{B_1} & \smath{=} & \smath{\{0^n 1^{(n-1)} \,|\, n \ge 1\}}\\
|
|
|
1006 |
\smath{B_2} & \smath{=} & \smath{\{0^n 1^{(n-2)} \,|\, n \ge 2\}}\\
|
|
|
1007 |
\smath{B_3} & \smath{=} & \smath{\{0^n 1^{(n-3)} \,|\, n \ge 3\}}\\
|
|
|
1008 |
& \smath{\vdots} &\\
|
|
|
1009 |
\end{tabular}
|
|
|
1010 |
\end{quote}
|
|
|
1011 |
\end{itemize}
|
|
|
1012 |
|
|
|
1013 |
\end{frame}}
|
|
|
1014 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1015 |
*}
|
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|
1016 |
|
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|
1017 |
|
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|
1018 |
text_raw {*
|
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|
1019 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1020 |
\mode<presentation>{
|
|
|
1021 |
\begin{frame}[c]
|
|
|
1022 |
\frametitle{\LARGE What We Have Not Achieved}
|
|
|
1023 |
|
|
|
1024 |
\begin{itemize}
|
|
|
1025 |
\item regular expressions are not good if you look for a minimal
|
|
|
1026 |
one for a language (DFAs have this notion)\pause\bigskip
|
|
|
1027 |
|
|
|
1028 |
\item Is there anything to be said about context free languages:\medskip
|
|
|
1029 |
|
|
|
1030 |
\begin{quote}
|
|
|
1031 |
A context free language is where every string can be recognised by
|
|
|
1032 |
a pushdown automaton.\bigskip
|
|
|
1033 |
\end{quote}
|
|
|
1034 |
\end{itemize}
|
|
|
1035 |
|
|
|
1036 |
\textcolor{gray}{\footnotesize Yes. Derivatives also work for c-f grammars. Ongoing work.}
|
|
|
1037 |
|
|
|
1038 |
\end{frame}}
|
|
|
1039 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1040 |
*}
|
|
|
1041 |
|
|
|
1042 |
|
|
|
1043 |
text_raw {*
|
|
|
1044 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1045 |
\mode<presentation>{
|
|
|
1046 |
\begin{frame}[c]
|
|
|
1047 |
\frametitle{\LARGE Conclusion}
|
|
|
1048 |
|
|
|
1049 |
\begin{itemize}
|
|
|
1050 |
\item We formalised the Myhill-Nerode theorem based on
|
|
|
1051 |
regular expressions only (DFAs are difficult to deal with in a theorem prover).\smallskip
|
|
|
1052 |
|
|
|
1053 |
\item Seems to be a common theme: algorithms need to be reformulated
|
|
|
1054 |
to better suit formal treatment.\smallskip
|
|
|
1055 |
|
|
|
1056 |
\item The most interesting aspect is that we are able to
|
|
|
1057 |
implement the matcher directly inside the theorem prover
|
|
|
1058 |
(ongoing work).\smallskip
|
|
|
1059 |
|
|
|
1060 |
\item Parsing is a vast field which seem to offer new results.
|
|
|
1061 |
\end{itemize}
|
|
|
1062 |
|
|
|
1063 |
\end{frame}}
|
|
|
1064 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1065 |
*}
|
|
|
1066 |
|
|
|
1067 |
text_raw {*
|
|
|
1068 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1069 |
\mode<presentation>{
|
|
|
1070 |
\begin{frame}<1>[b]
|
|
|
1071 |
\frametitle{
|
|
|
1072 |
\begin{tabular}{c}
|
|
|
1073 |
\mbox{}\\[13mm]
|
|
|
1074 |
\alert{\LARGE Thank you very much!}\\
|
|
|
1075 |
\alert{\Large Questions?}
|
|
|
1076 |
\end{tabular}}
|
|
|
1077 |
|
|
|
1078 |
\end{frame}}
|
|
|
1079 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
1080 |
*}
|
|
|
1081 |
|
|
|
1082 |
|
|
|
1083 |
|
|
|
1084 |
(*<*)
|
|
|
1085 |
end
|
|
|
1086 |
(*>*) |