diff -r 66ef2a2c64fb -r c3ff26204d2a Slides/Slides7.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Slides/Slides7.thy Thu Apr 28 11:44:36 2011 +0800 @@ -0,0 +1,1086 @@ +(*<*) +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}{Hefei, 15.~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 do not 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 States} + + \mbox{}\\[3cm] + + \begin{itemize} + \item ??? \smath{\text{final}_L\,X \dn \{[|s|]_\approx\;|\; s \in X\}}\\ + \medskip + + \item we can prove: \smath{L = \bigcup \{X\;|\;\text{final}_L\,X\}} + + \end{itemize} + + \end{frame}} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +*} + +text_raw {* + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \mode{ + \begin{frame}[c] + \frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes} + + \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 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}. Not trivial, but after a bit + of thinking, one can prove that if + + \begin{center} + \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm} + \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}[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