diff -r fb201e383f1b -r da575186d492 Slides/Slides7.thy --- a/Slides/Slides7.thy Tue Feb 19 05:38:46 2013 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1246 +0,0 @@ -(*<*) -theory Slides7 -imports "~~/src/HOL/Library/LaTeXsugar" "Main" -begin - -declare [[show_question_marks = false]] - -notation (latex output) - set ("_") and - Cons ("_::/_" [66,65] 65) - -(*>*) - -text_raw {* - \renewcommand{\slidecaption}{Beijing, 29.~April 2011} - - \newcommand{\abst}[2]{#1.#2}% atom-abstraction - \newcommand{\pair}[2]{\langle #1,#2\rangle} % pairing - \newcommand{\susp}{{\boldsymbol{\cdot}}}% for suspensions - \newcommand{\unit}{\langle\rangle}% unit - \newcommand{\app}[2]{#1\,#2}% application - \newcommand{\eqprob}{\mathrel{{\approx}?}} - \newcommand{\freshprob}{\mathrel{\#?}} - \newcommand{\redu}[1]{\stackrel{#1}{\Longrightarrow}}% reduction - \newcommand{\id}{\varepsilon}% identity substitution - - \newcommand{\bl}[1]{\textcolor{blue}{#1}} - \newcommand{\gr}[1]{\textcolor{gray}{#1}} - \newcommand{\rd}[1]{\textcolor{red}{#1}} - - \newcommand{\ok}{\includegraphics[scale=0.07]{ok.png}} - \newcommand{\notok}{\includegraphics[scale=0.07]{notok.png}} - \newcommand{\largenotok}{\includegraphics[scale=1]{notok.png}} - - \renewcommand{\Huge}{\fontsize{61.92}{77}\selectfont} - \newcommand{\veryHuge}{\fontsize{74.3}{93}\selectfont} - \newcommand{\VeryHuge}{\fontsize{89.16}{112}\selectfont} - \newcommand{\VERYHuge}{\fontsize{107}{134}\selectfont} - - \newcommand{\LL}{$\mathbb{L}\,$} - - - \pgfdeclareradialshading{smallbluesphere}{\pgfpoint{0.5mm}{0.5mm}}% - {rgb(0mm)=(0,0,0.9); - rgb(0.9mm)=(0,0,0.7); - rgb(1.3mm)=(0,0,0.5); - rgb(1.4mm)=(1,1,1)} - - \def\myitemi{\begin{pgfpicture}{-1ex}{-0.55ex}{1ex}{1ex} - \usebeamercolor[fg]{subitem projected} - {\pgftransformscale{0.8}\pgftext{\normalsize\pgfuseshading{bigsphere}}} - \pgftext{% - \usebeamerfont*{subitem projected}} - \end{pgfpicture}} - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[t] - \frametitle{% - \begin{tabular}{@ {\hspace{-3mm}}c@ {}} - \\ - \LARGE Verifying a Regular Expression\\[-1mm] - \LARGE Matcher and Formal Language\\[-1mm] - \LARGE Theory\\[5mm] - \end{tabular}} - \begin{center} - Christian Urban\\ - \small Technical University of Munich, Germany - \end{center} - - - \begin{center} - \small joint work with Chunhan Wu and Xingyuan Zhang from the PLA - University of Science and Technology in Nanjing - \end{center} - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{This Talk: 3 Points} - \large - \begin{itemize} - \item It is easy to make mistakes.\bigskip - \item Theorem provers can prevent mistakes, {\bf if} the problem - is formulated so that it is suitable for theorem provers.\bigskip - \item This re-formulation can be done, even in domains where - we least expect it. - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1->[t] - \frametitle{Regular Expressions} - - \begin{textblock}{6}(2,4) - \begin{tabular}{@ {}rrl} - \bl{r} & \bl{$::=$} & \bl{$\varnothing$}\\ - & \bl{$\mid$} & \bl{[]}\\ - & \bl{$\mid$} & \bl{c}\\ - & \bl{$\mid$} & \bl{r$_1$ + r$_2$}\\ - & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$}\\ - & \bl{$\mid$} & \bl{r$^*$}\\ - \end{tabular} - \end{textblock} - - \begin{textblock}{6}(8,3.5) - \includegraphics[scale=0.35]{Screen1.png} - \end{textblock} - - \begin{textblock}{6}(10.2,2.8) - \footnotesize Isabelle: - \end{textblock} - - \only<2>{ - \begin{textblock}{9}(3.6,11.8) - \bl{matches r s $\;\Longrightarrow\;$ true $\vee$ false}\\[3.5mm] - - \hspace{10mm}\begin{tikzpicture} - \coordinate (m1) at (0.4,1); - \draw (0,0.3) node (m2) {\small\color{gray}rexp}; - \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1); - - \coordinate (s1) at (0.81,1); - \draw (1.3,0.3) node (s2) {\small\color{gray} string}; - \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1); - \end{tikzpicture} - \end{textblock}} - - - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1->[t] - \frametitle{Specification} - - \small - \begin{textblock}{6}(0,3.5) - \begin{tabular}{r@ {\hspace{0.5mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l} - \multicolumn{4}{c}{rexp $\Rightarrow$ set of strings}\bigskip\\ - &\bl{\LL ($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$}\\ - &\bl{\LL ([])} & \bl{$\dn$} & \bl{\{[]\}}\\ - &\bl{\LL (c)} & \bl{$\dn$} & \bl{\{c\}}\\ - &\bl{\LL (r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) $\cup$ \LL (r$_2$)}\\ - \rd{$\Rightarrow$} &\bl{\LL (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) ;; \LL (r$_2$)}\\ - \rd{$\Rightarrow$} &\bl{\LL (r$^*$)} & \bl{$\dn$} & \bl{(\LL (r))$^\star$}\\ - \end{tabular} - \end{textblock} - - \begin{textblock}{9}(7.3,3) - {\mbox{}\hspace{2cm}\footnotesize Isabelle:\smallskip} - \includegraphics[scale=0.325]{Screen3.png} - \end{textblock} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1->[t] - \frametitle{Version 1} - \small - \mbox{}\\[-8mm]\mbox{} - - \begin{center}\def\arraystretch{1.05} - \begin{tabular}{@ {\hspace{-5mm}}l@ {\hspace{2.5mm}}c@ {\hspace{2.5mm}}l@ {}} - \bl{match [] []} & \bl{$=$} & \bl{true}\\ - \bl{match [] (c::s)} & \bl{$=$} & \bl{false}\\ - \bl{match ($\varnothing$::rs) s} & \bl{$=$} & \bl{false}\\ - \bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\ - \bl{match (c::rs) []} & \bl{$=$} & \bl{false}\\ - \bl{match (c::rs) (d::s)} & \bl{$=$} & \bl{if c = d then match rs s else false}\\ - \bl{match (r$_1$ + r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s $\vee$ match (r$_2$::rs) s}\\ - \bl{match (r$_1$ $\cdot$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\ - \bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\ - \end{tabular} - \end{center} - - \begin{textblock}{9}(0.2,1.6) - \hspace{10mm}\begin{tikzpicture} - \coordinate (m1) at (0.44,-0.5); - \draw (0,0.3) node (m2) {\small\color{gray}\mbox{}\hspace{-9mm}list of rexps}; - \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1); - - \coordinate (s1) at (0.86,-0.5); - \draw (1.5,0.3) node (s2) {\small\color{gray} string}; - \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1); - \end{tikzpicture} - \end{textblock} - - \begin{textblock}{9}(2.8,11.8) - \bl{matches$_1$ r s $\;=\;$ match [r] s} - \end{textblock} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1->[c] - \frametitle{Testing} - - \small - Every good programmer should do thourough tests: - - \begin{center} - \begin{tabular}{@ {\hspace{-20mm}}lcl} - \bl{matches$_1$ (a$\cdot$b)$^*\;$ []} & \bl{$\mapsto$} & \bl{true}\\ - \bl{matches$_1$ (a$\cdot$b)$^*\;$ ab} & \bl{$\mapsto$} & \bl{true}\\ - \bl{matches$_1$ (a$\cdot$b)$^*\;$ aba} & \bl{$\mapsto$} & \bl{false}\\ - \bl{matches$_1$ (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\ - \bl{matches$_1$ (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\ - \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x} & \bl{$\mapsto$} & \bl{true}}\\ - \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x0} & \bl{$\mapsto$} & \bl{true}}\\ - \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x3} & \bl{$\mapsto$} & \bl{false}} - \end{tabular} - \end{center} - - \onslide<3-> - {Looks OK \ldots let's ship it to customers\hspace{5mm} - \raisebox{-5mm}{\includegraphics[scale=0.05]{sun.png}}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1->[c] - \frametitle{Version 1} - - \only<1->{Several hours later\ldots}\pause - - - \begin{center} - \begin{tabular}{@ {\hspace{0mm}}lcl} - \bl{matches$_1$ []$^*$ s} & \bl{$\mapsto$} & loops\\ - \onslide<4->{\bl{matches$_1$ ([] + \ldots)$^*$ s} & \bl{$\mapsto$} & loops\\} - \end{tabular} - \end{center} - - \small - \onslide<3->{ - \begin{center} - \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}} - \ldots\\ - \bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\ - \ldots\\ - \bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\ - \end{tabular} - \end{center}} - - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1->[t] - \frametitle{Testing} - - \begin{itemize} - \item While testing is an important part in the process of programming development\pause\ldots - - \item we can only test a {\bf finite} amount of examples.\bigskip\pause - - \begin{center} - \colorbox{cream} - {\gr{\begin{minipage}{10cm} - ``Testing can only show the presence of errors, never their - absence.'' (Edsger W.~Dijkstra) - \end{minipage}}} - \end{center}\bigskip\pause - - \item In a theorem prover we can establish properties that apply to - {\bf all} input and {\bf all} output. - - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1->[t] - \frametitle{Version 2} - \mbox{}\\[-14mm]\mbox{} - - \small - \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}} - \bl{nullable ($\varnothing$)} & \bl{$=$} & \bl{false} &\\ - \bl{nullable ([])} & \bl{$=$} & \bl{true} &\\ - \bl{nullable (c)} & \bl{$=$} & \bl{false} &\\ - \bl{nullable (r$_1$ + r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\vee$ nullable r$_2$} & \\ - \bl{nullable (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\wedge$ nullable r$_2$} & \\ - \bl{nullable (r$^*$)} & \bl{$=$} & \bl{true} & \\ - \end{tabular}\medskip - - \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} - \bl{der c ($\varnothing$)} & \bl{$=$} & \bl{$\varnothing$} & \\ - \bl{der c ([])} & \bl{$=$} & \bl{$\varnothing$} & \\ - \bl{der c (d)} & \bl{$=$} & \bl{if c = d then [] else $\varnothing$} & \\ - \bl{der c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\ - \bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{((der c r$_1$) $\cdot$ r$_2$)} & \\ - & & \bl{\;\;\;\;+ (if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\ - \bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ r$^*$} &\smallskip\\ - - \bl{derivative r []} & \bl{$=$} & \bl{r} & \\ - \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\ - \end{tabular}\medskip - - \bl{matches$_2$ r s $=$ nullable (derivative r s)} - - \begin{textblock}{6}(9.5,0.9) - \begin{flushright} - \color{gray}``if r matches []'' - \end{flushright} - \end{textblock} - - \begin{textblock}{6}(9.5,6.18) - \begin{flushright} - \color{gray}``derivative w.r.t.~a char'' - \end{flushright} - \end{textblock} - - \begin{textblock}{6}(9.5,12.1) - \begin{flushright} - \color{gray}``deriv.~w.r.t.~a string'' - \end{flushright} - \end{textblock} - - \begin{textblock}{6}(9.5,13.98) - \begin{flushright} - \color{gray}``main'' - \end{flushright} - \end{textblock} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1->[t] - \frametitle{Is the Matcher Error-Free?} - - We expect that - - \begin{center} - \begin{tabular}{lcl} - \bl{matches$_2$ r s = true} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}% - \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\in$ \LL(r)}\\ - \bl{matches$_2$ r s = false} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}% - \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\notin$ \LL(r)}\\ - \end{tabular} - \end{center} - \pause\pause\bigskip - By \alert<4->{induction}, we can {\bf prove} these properties.\bigskip - - \begin{tabular}{lrcl} - Lemmas: & \bl{nullable (r)} & \bl{$\Longleftrightarrow$} & \bl{[] $\in$ \LL (r)}\\ - & \bl{s $\in$ \LL (der c r)} & \bl{$\Longleftrightarrow$} & \bl{(c::s) $\in$ \LL (r)}\\ - \end{tabular} - - \only<4->{ - \begin{textblock}{3}(0.9,4.5) - \rd{\huge$\forall$\large{}r s.} - \end{textblock}} - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[c] - \frametitle{ - \begin{tabular}{c} - \mbox{}\\[23mm] - \LARGE Demo - \end{tabular}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1->[t] - - \mbox{}\\[-2mm] - - \small - \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}} - \bl{nullable (NULL)} & \bl{$=$} & \bl{false} &\\ - \bl{nullable (EMPTY)} & \bl{$=$} & \bl{true} &\\ - \bl{nullable (CHR c)} & \bl{$=$} & \bl{false} &\\ - \bl{nullable (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) orelse (nullable r$_2$)} & \\ - \bl{nullable (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) andalso (nullable r$_2$)} & \\ - \bl{nullable (STAR r)} & \bl{$=$} & \bl{true} & \\ - \end{tabular}\medskip - - \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} - \bl{der c (NULL)} & \bl{$=$} & \bl{NULL} & \\ - \bl{der c (EMPTY)} & \bl{$=$} & \bl{NULL} & \\ - \bl{der c (CHR d)} & \bl{$=$} & \bl{if c=d then EMPTY else NULL} & \\ - \bl{der c (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (der c r$_1$) (der c r$_2$)} & \\ - \bl{der c (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (SEQ (der c r$_1$) r$_2$)} & \\ - & & \bl{\phantom{ALT} (if nullable r$_1$ then der c r$_2$ else NULL)}\\ - \bl{der c (STAR r)} & \bl{$=$} & \bl{SEQ (der c r) (STAR r)} &\smallskip\\ - - \bl{derivative r []} & \bl{$=$} & \bl{r} & \\ - \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\ - \end{tabular}\medskip - - \bl{matches r s $=$ nullable (derivative r s)} - - \only<2>{ - \begin{textblock}{8}(1.5,4) - \includegraphics[scale=0.3]{approved.png} - \end{textblock}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{No Automata?} - - You might be wondering why I did not use any automata? - - \begin{itemize} - \item {\bf Def.:} A \alert{regular language} is one where there is a DFA that - recognises it.\bigskip\pause - \end{itemize} - - - There are many reasons why this is a good definition:\medskip - \begin{itemize} - \item pumping lemma - \item closure properties of regular languages\\ (e.g.~closure under complement) - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[t] - \frametitle{Really Bad News!} - - DFAs are bad news for formalisations in theorem provers. They might - be represented as: - - \begin{itemize} - \item graphs - \item matrices - \item partial functions - \end{itemize} - - All constructions are messy to reason about.\bigskip\bigskip - \pause - - \small - \only<2>{ - Constable et al needed (on and off) 18 months for a 3-person team - to formalise automata theory in Nuprl including Myhill-Nerode. There is - only very little other formalised work on regular languages I know of - in Coq, Isabelle and HOL.} - \only<3>{Typical textbook reasoning goes like: ``\ldots if \smath{M} and \smath{N} are any two - automata with no inaccessible states \ldots'' - } - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{} - \large - \begin{center} - \begin{tabular}{p{9cm}} - My point:\bigskip\\ - - The theory about regular languages can be reformulated - to be more\\ suitable for theorem proving. - \end{tabular} - \end{center} - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE The Myhill-Nerode Theorem} - - \begin{itemize} - \item provides necessary and suf\!ficient conditions for a language - being regular (pumping lemma only necessary)\medskip - - \item will help with closure properties of regular languages\bigskip\pause - - \item key is the equivalence relation:\smallskip - \begin{center} - \smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L} - \end{center} - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE The Myhill-Nerode Theorem} - - \mbox{}\\[5cm] - - \begin{itemize} - \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}} - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE Equivalence Classes} - - \begin{itemize} - \item \smath{L = []} - \begin{center} - \smath{\Big\{\{[]\},\; U\!N\!IV - \{[]\}\Big\}} - \end{center}\bigskip\bigskip - - \item \smath{L = [c]} - \begin{center} - \smath{\Big\{\{[]\},\; \{[c]\},\; U\!N\!IV - \{[], [c]\}\Big\}} - \end{center}\bigskip\bigskip - - \item \smath{L = \varnothing} - \begin{center} - \smath{\Big\{U\!N\!IV\Big\}} - \end{center} - - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE Regular Languages} - - \begin{itemize} - \item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M} - such that \smath{\mathbb{L}(M) = L}\\[1.5cm] - - \item Myhill-Nerode: - - \begin{center} - \begin{tabular}{l} - finite $\Rightarrow$ regular\\ - \;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r.\; L = \mathbb{L}(r)}\\[3mm] - regular $\Rightarrow$ finite\\ - \;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})} - \end{tabular} - \end{center} - - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE Final Equiv.~Classes} - - \mbox{}\\[3cm] - - \begin{itemize} - \item \smath{\text{finals}\,L \dn - \{{\lbrack\mkern-2mu\lbrack{s}\rbrack\mkern-2mu\rbrack}_\approx\;|\; s \in L\}}\\ - \medskip - - \item we can prove: \smath{L = \bigcup (\text{finals}\,L)} - - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE Transitions between ECs} - - \smath{L = \{[c]\}} - - \begin{tabular}{@ {\hspace{-7mm}}cc} - \begin{tabular}{c} - \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick] - \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm] - - %\draw[help lines] (0,0) grid (3,2); - - \node[state,initial] (q_0) {$R_1$}; - \node[state,accepting] (q_1) [above right of=q_0] {$R_2$}; - \node[state] (q_2) [below right of=q_0] {$R_3$}; - - \path[->] (q_0) edge node {c} (q_1) - edge node [swap] {$\Sigma-{c}$} (q_2) - (q_2) edge [loop below] node {$\Sigma$} () - (q_1) edge node {$\Sigma$} (q_2); - \end{tikzpicture} - \end{tabular} - & - \begin{tabular}[t]{ll} - \\[-20mm] - \multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm] - - \smath{R_1}: & \smath{\{[]\}}\\ - \smath{R_2}: & \smath{\{[c]\}}\\ - \smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm] - \multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ;; [c] \subseteq Y}}} - \end{tabular} - - \end{tabular} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE Systems of Equations} - - Inspired by a method of Brzozowski\;'64, we can build an equational system - characterising the equivalence classes: - - \begin{center} - \begin{tabular}{@ {\hspace{-20mm}}c} - \\[-13mm] - \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick] - \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm] - - %\draw[help lines] (0,0) grid (3,2); - - \node[state,initial] (p_0) {$R_1$}; - \node[state,accepting] (p_1) [right of=q_0] {$R_2$}; - - \path[->] (p_0) edge [bend left] node {a} (p_1) - edge [loop above] node {b} () - (p_1) edge [loop above] node {a} () - edge [bend left] node {b} (p_0); - \end{tikzpicture}\\ - \\[-13mm] - \end{tabular} - \end{center} - - \begin{center} - \begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l} - & \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\ - & \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\ - \onslide<3->{we can prove} - & \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}} - & \onslide<3->{\smath{R_1;; \mathbb{L}(b) \,\cup\, R_2;;\mathbb{L}(b) \,\cup\, \{[]\}}}\\ - & \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}} - & \onslide<3->{\smath{R_1;; \mathbb{L}(a) \,\cup\, R_2;;\mathbb{L}(a)}}\\ - \end{tabular} - \end{center} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[t] - \small - - \begin{center} - \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll} - \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}} - & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\ - \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}} - & \onslide<1->{\smath{R_1; a + R_2; a}}\\ - - & & & \onslide<2->{by Arden}\\ - - \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}} - & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\ - \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}} - & \only<2>{\smath{R_1; a + R_2; a}}% - \only<3->{\smath{R_1; a\cdot a^\star}}\\ - - & & & \onslide<4->{by Arden}\\ - - \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}} - & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\ - \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}} - & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\ - - & & & \onslide<5->{by substitution}\\ - - \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}} - & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\ - \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}} - & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\ - - & & & \onslide<6->{by Arden}\\ - - \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}} - & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ - \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}} - & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\ - - & & & \onslide<7->{by substitution}\\ - - \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}} - & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ - \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}} - & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star - \cdot a\cdot a^\star}}\\ - \end{tabular} - \end{center} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE A Variant of Arden's Lemma} - - {\bf Arden's Lemma:}\smallskip - - If \smath{[] \not\in A} then - \begin{center} - \smath{X = X; A + \text{something}} - \end{center} - has the (unique) solution - \begin{center} - \smath{X = \text{something} ; A^\star} - \end{center} - - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1->[t] - \small - - \begin{center} - \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll} - \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}} - & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\ - \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}} - & \onslide<1->{\smath{R_1; a + R_2; a}}\\ - - & & & \onslide<2->{by Arden}\\ - - \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}} - & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\ - \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}} - & \only<2>{\smath{R_1; a + R_2; a}}% - \only<3->{\smath{R_1; a\cdot a^\star}}\\ - - & & & \onslide<4->{by Arden}\\ - - \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}} - & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\ - \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}} - & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\ - - & & & \onslide<5->{by substitution}\\ - - \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}} - & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\ - \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}} - & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\ - - & & & \onslide<6->{by Arden}\\ - - \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}} - & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ - \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}} - & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\ - - & & & \onslide<7->{by substitution}\\ - - \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}} - & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ - \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}} - & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star - \cdot a\cdot a^\star}}\\ - \end{tabular} - \end{center} - - \only<8->{ - \begin{textblock}{6}(2.5,4) - \begin{block}{} - \begin{minipage}{8cm}\raggedright - - \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm] - \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm] - - %\draw[help lines] (0,0) grid (3,2); - - \node[state,initial] (p_0) {$R_1$}; - \node[state,accepting] (p_1) [right of=q_0] {$R_2$}; - - \path[->] (p_0) edge [bend left] node {a} (p_1) - edge [loop above] node {b} () - (p_1) edge [loop above] node {a} () - edge [bend left] node {b} (p_0); - \end{tikzpicture} - - \end{minipage} - \end{block} - \end{textblock}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE The Equ's Solving Algorithm} - - \begin{itemize} - \item The algorithm must terminate: Arden makes one equation smaller; - substitution deletes one variable from the right-hand sides.\bigskip - - \item We need to maintain the invariant that Arden is applicable - (if \smath{[] \not\in A} then \ldots):\medskip - - \begin{center}\small - \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll} - \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\ - \smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\ - - & & & by Arden\\ - - \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\ - \smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\ - \end{tabular} - \end{center} - - \end{itemize} - - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE The Other Direction} - - One has to prove - - \begin{center} - \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})} - \end{center} - - by induction on \smath{r}. This is straightforward for \\the base cases:\small - - \begin{center} - \begin{tabular}{l@ {\hspace{1mm}}l} - \smath{U\!N\!IV /\!/ \!\approx_{\emptyset}} & \smath{= \{U\!N\!IV\}}\smallskip\\ - \smath{U\!N\!IV /\!/ \!\approx_{\{[]\}}} & \smath{\subseteq \{\{[]\}, U\!N\!IV - \{[]\}\}}\smallskip\\ - \smath{U\!N\!IV /\!/ \!\approx_{\{[c]\}}} & \smath{\subseteq \{\{[]\}, \{[c]\}, U\!N\!IV - \{[], [c]\}\}} - \end{tabular} - \end{center} - - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[t] - \frametitle{\LARGE The Other Direction} - - More complicated are the inductive cases:\\ one needs to prove that if - - \begin{center} - \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{3mm} - \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})} - \end{center} - - then - - \begin{center} - \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})} - \end{center} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[t] - \frametitle{\LARGE Helper Lemma} - - \begin{center} - \begin{tabular}{p{10cm}} - %If \smath{\text{finite} (f\;' A)} and \smath{f} is injective - %(on \smath{A}),\\ then \smath{\text{finite}\,A}. - Given two equivalence relations \smath{R_1} and \smath{R_2} with - \smath{R_1} refining \smath{R_2} (\smath{R_1 \subseteq R_2}).\\ - Then\medskip\\ - \smath{\;\;\text{finite} (U\!N\!IV /\!/ R_1) \Rightarrow \text{finite} (U\!N\!IV /\!/ R_2)} - \end{tabular} - \end{center} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\Large Derivatives and Left-Quotients} - \small - Work by Brozowski ('64) and Antimirov ('96):\pause\smallskip - - - \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} - \multicolumn{4}{@ {}l}{Left-Quotient:}\\ - \multicolumn{4}{@ {}l}{\bl{$\text{Ders}\;\text{s}\,A \dn \{\text{s'} \;|\; \text{s @ s'} \in A\}$}}\bigskip\\ - - \multicolumn{4}{@ {}l}{Derivative:}\\ - \bl{der c ($\varnothing$)} & \bl{$=$} & \bl{$\varnothing$} & \\ - \bl{der c ([])} & \bl{$=$} & \bl{$\varnothing$} & \\ - \bl{der c (d)} & \bl{$=$} & \bl{if c = d then [] else $\varnothing$} & \\ - \bl{der c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\ - \bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{((der c r$_1$) $\cdot$ r$_2$)} & \\ - & & \bl{\;\;\;\;+ (if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\ - \bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ r$^*$} &\smallskip\\ - - \bl{ders [] r} & \bl{$=$} & \bl{r} & \\ - \bl{ders (s @ [c]) r} & \bl{$=$} & \bl{der c (ders s r)} & \\ - \end{tabular}\pause - - \begin{center} - \alert{$\Rightarrow$}\smath{\;\;\text{Ders}\,\text{s}\,(\mathbb{L}(\text{r})) = \mathbb{L} (\text{ders s r})} - \end{center} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE Left-Quotients and MN-Rels} - - \begin{itemize} - \item \smath{x \approx_{A} y \,\dn\, \forall z.\; x @ z \in A \Leftrightarrow y @ z \in A}\medskip - \item \bl{$\text{Ders}\;s\,A \dn \{s' \;|\; s @ s' \in A\}$} - \end{itemize}\bigskip - - \begin{center} - \smath{x \approx_A y \Longleftrightarrow \text{Ders}\;x\;A = \text{Ders}\;y\;A} - \end{center}\bigskip\pause\small - - which means - - \begin{center} - \smath{x \approx_{\mathbb{L}(r)} y \Longleftrightarrow - \mathbb{L}(\text{ders}\;x\;r) = \mathbb{L}(\text{ders}\;y\;r)} - \end{center}\pause - - \hspace{8.8mm}or - \smath{\;x \approx_{\mathbb{L}(r)} y \Longleftarrow - \text{ders}\;x\;r = \text{ders}\;y\;r} - - - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE Partial Derivatives} - - Antimirov: \bl{pder : rexp $\Rightarrow$ rexp set}\bigskip - - \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} - \bl{pder c ($\varnothing$)} & \bl{$=$} & \bl{\{$\varnothing$\}} & \\ - \bl{pder c ([])} & \bl{$=$} & \bl{\{$\varnothing$\}} & \\ - \bl{pder c (d)} & \bl{$=$} & \bl{if c = d then \{[]\} else \{$\varnothing$\}} & \\ - \bl{pder c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(pder c r$_1$) $\cup$ (pder c r$_2$)} & \\ - \bl{pder c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{(pder c r$_1$) $\odot$ r$_2$} & \\ - & & \bl{\hspace{-10mm}$\cup$ (if nullable r$_1$ then pder c r$_2$ else $\varnothing$)}\\ - \bl{pder c (r$^*$)} & \bl{$=$} & \bl{(pder c r) $\odot$ r$^*$} &\smallskip\\ - \end{tabular} - - \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} - \bl{pders [] r} & \bl{$=$} & \bl{r} & \\ - \bl{pders (s @ [c]) r} & \bl{$=$} & \bl{pder c (pders s r)} & \\ - \end{tabular}\pause - - \begin{center} - \alert{$\Rightarrow$}\smath{\;\;\text{Ders}\,\text{s}\,(\mathbb{L}(\text{r})) = \bigcup (\mathbb{L}\;`\; (\text{pders s r}))} - \end{center} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[t] - \frametitle{\LARGE Final Result} - - \mbox{}\\[7mm] - - \begin{itemize} - \item \alt<1>{\smath{\text{pders x r \mbox{$=$} pders y r}}} - {\smath{\underbrace{\text{pders x r \mbox{$=$} pders y r}}_{R_1}}} - refines \bl{x $\approx_{\mathbb{L}(\text{r})}$ y}\pause - \item \smath{\text{finite} (U\!N\!IV /\!/ R_1)} \bigskip\pause - \item Therefore \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}. Qed. - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE What Have We Achieved?} - - \begin{itemize} - \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}} - \bigskip\pause - \item regular languages are closed under complementation; this is now easy\medskip - \begin{center} - \smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}} - \end{center} - \end{itemize} - - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE Examples} - - \begin{itemize} - \item \smath{L \equiv \Sigma^\star 0 \Sigma} is regular - \begin{quote}\small - \begin{tabular}{lcl} - \smath{A_1} & \smath{=} & \smath{\Sigma^\star 00}\\ - \smath{A_2} & \smath{=} & \smath{\Sigma^\star 01}\\ - \smath{A_3} & \smath{=} & \smath{\Sigma^\star 10 \cup \{0\}}\\ - \smath{A_4} & \smath{=} & \smath{\Sigma^\star 11 \cup \{1\} \cup \{[]\}}\\ - \end{tabular} - \end{quote} - - \item \smath{L \equiv \{ 0^n 1^n \,|\, n \ge 0\}} is not regular - \begin{quote}\small - \begin{tabular}{lcl} - \smath{B_0} & \smath{=} & \smath{\{0^n 1^n \,|\, n \ge 0\}}\\ - \smath{B_1} & \smath{=} & \smath{\{0^n 1^{(n-1)} \,|\, n \ge 1\}}\\ - \smath{B_2} & \smath{=} & \smath{\{0^n 1^{(n-2)} \,|\, n \ge 2\}}\\ - \smath{B_3} & \smath{=} & \smath{\{0^n 1^{(n-3)} \,|\, n \ge 3\}}\\ - & \smath{\vdots} &\\ - \end{tabular} - \end{quote} - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE What We Have Not Achieved} - - \begin{itemize} - \item regular expressions are not good if you look for a minimal - one for a language (DFAs have this notion)\pause\bigskip - - \item Is there anything to be said about context free languages:\medskip - - \begin{quote} - A context free language is where every string can be recognised by - a pushdown automaton.\bigskip - \end{quote} - \end{itemize} - - \textcolor{gray}{\footnotesize Yes. Derivatives also work for c-f grammars. Ongoing work.} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}[c] - \frametitle{\LARGE Conclusion} - - \begin{itemize} - \item We formalised the Myhill-Nerode theorem based on - regular expressions only (DFAs are difficult to deal with in a theorem prover).\smallskip - - \item Seems to be a common theme: algorithms need to be reformulated - to better suit formal treatment.\smallskip - - \item The most interesting aspect is that we are able to - implement the matcher directly inside the theorem prover - (ongoing work).\smallskip - - \item Parsing is a vast field which seem to offer new results. - \end{itemize} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - -text_raw {* - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \mode{ - \begin{frame}<1>[b] - \frametitle{ - \begin{tabular}{c} - \mbox{}\\[13mm] - \alert{\LARGE Thank you very much!}\\ - \alert{\Large Questions?} - \end{tabular}} - - \end{frame}} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -*} - - - -(*<*) -end -(*>*) \ No newline at end of file