| author | Christian Urban <urbanc@in.tum.de> |
| Thu, 22 Dec 2011 04:35:01 +0000 | |
| changeset 3090 | 19f5e7afad89 |
| parent 2771 | 66ef2a2c64fb |
| permissions | -rw-r--r-- |
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(*<*) |
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theory Slides6 |
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imports "~~/src/HOL/Library/LaTeXsugar" "Nominal" |
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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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2771
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more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
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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}<1->[c]
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\frametitle{My Background}
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\mbox{}\\[-10mm]
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\begin{itemize}
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\item My background is in theory and programming languages.\bigskip |
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\pause |
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\item But I am also a programmer with a \alert<2>{software system} of around 800 kloc
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(though I am responsible for only appr.~35 kloc), |
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\item and I write papers. |
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\end{itemize}
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\only<2>{
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\begin{textblock}{6}(6.5,11.5)
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\begin{tikzpicture}
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\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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{\color{darkgray}
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\begin{minipage}{6.5cm}\raggedright
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\begin{tabular}[b]{@ {}p{4.5cm}c@ {}}
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\raggedright |
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The software is a theorem prover, called {\bf Isabelle}.
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& \mbox{}\hspace{-5mm}\raisebox{-14mm}{\includegraphics[scale=0.28]{isabelle1.png}}
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\end{tabular}%
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\end{minipage}};
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\end{tikzpicture}
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\end{textblock}}
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\only<4>{
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\begin{textblock}{6}(3,11.5)
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\begin{tikzpicture}
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\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] |
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{\color{darkgray}
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\begin{minipage}{9.6cm}\raggedright
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So I can experience every day that writing error-free code is {\bf very, very hard}
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and that papers are also {\bf hard} to get correct.
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\end{minipage}};
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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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2771
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
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text_raw {*
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66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
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\mode<presentation>{
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66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
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\begin{frame}[c]
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66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
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\frametitle{3 Points}
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66ef2a2c64fb
more to the pearl paper
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parents:
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\large |
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66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
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\begin{itemize}
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66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
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\item It is easy to make mistakes.\bigskip |
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66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
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changeset
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\item Theorem provers can prevent mistakes, {\bf if} the problem
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66ef2a2c64fb
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parents:
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is formulated so that it is suitable for theorem provers.\bigskip |
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Christian Urban <urbanc@in.tum.de>
parents:
2764
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changeset
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\item This re-formulation can be done, even in domains where |
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66ef2a2c64fb
more to the pearl paper
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parents:
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we do not expect it. |
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66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
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\end{itemize}
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66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
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more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
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changeset
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\end{frame}}
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66ef2a2c64fb
more to the pearl paper
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parents:
2764
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changeset
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
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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}
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\frametitle{Getting Papers Correct}
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\begin{minipage}{1.1\textwidth}
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My work over the last 5 years.\\ |
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{\small (in the fields of programming languages, logic and lambda-calculi)}
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\end{minipage}\bigskip
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\only<1>{
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\mbox{}\\[15mm]
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\begin{center}
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\begin{tikzpicture}[node distance=0.5mm]
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\node at (-1.0,-0.3) (proof) [double arrow, fill=gray,text=white, minimum height=2cm]{\bf Proof};
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\node [left=of proof]{\Large\bf Specification};
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\node [right=of proof]{\Large\bf Code};
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\end{tikzpicture}
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\end{center}
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} |
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\pause |
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\begin{tabular}{c@ {\hspace{2mm}}c}
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\begin{tabular}{c}
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\includegraphics[scale=0.09]{harper.jpg}\\[-2mm]
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{\footnotesize Bob Harper}\\[-2.5mm]
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{\footnotesize (CMU)}
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\end{tabular}
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\begin{tabular}{c}
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\includegraphics[scale=0.31]{pfenning.jpg}\\[-2mm]
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{\footnotesize Frank Pfenning}\\[-2.5mm]
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{\footnotesize (CMU)}
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\end{tabular} &
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\begin{tabular}{p{6cm}}
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\raggedright\small |
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\color{gray}{published a proof in ACM Transactions on Computational Logic (2005),
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$\sim$31pp} |
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\end{tabular}\\
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\\[-4mm] |
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\begin{tabular}{c}
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\includegraphics[scale=0.3]{appel.jpg}\\[-2mm]
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{\footnotesize Andrew Appel}\\[-2.5mm]
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{\footnotesize (Princeton)}
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\end{tabular} &
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\begin{tabular}{p{6cm}}
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\raggedright\small |
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\color{gray}{relied on their proof in a safety critical system (proof carrying code)}
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\end{tabular}
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\end{tabular}\medskip
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\end{frame}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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*} |
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more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
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text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\mode<presentation>{
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\begin{frame}
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\frametitle{Proof-Carrying Code}
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\begin{textblock}{10}(2.5,2.2)
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\begin{block}{Idea:}
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\begin{center}
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\begin{tikzpicture}
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\draw[help lines,cream] (0,0.2) grid (8,4); |
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\draw[line width=1mm, red] (5.5,0.6) rectangle (7.5,4); |
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\node[anchor=base] at (6.5,2.8) |
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{\small\begin{tabular}{@ {}p{1.9cm}@ {}}\centering user needs to run untrusted code\end{tabular}};
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\draw[line width=1mm, red] (0.5,0.6) rectangle (2.5,4); |
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\node[anchor=base] at (1.5,2.3) |
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{\small\begin{tabular}{@ {}p{1.9cm}@ {}}\centering code developer/ web server/ Apple
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Store\end{tabular}};
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\onslide<4->{
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\draw[line width=1mm, red, fill=red] (5.5,0.6) rectangle (7.5,1.8); |
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\node[anchor=base,white] at (6.5,1.1) |
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{\small\begin{tabular}{@ {}p{1.9cm}@ {}}\bf\centering proof- checker\end{tabular}};}
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\node at (3.8,3.0) [single arrow, fill=red,text=white, minimum height=3cm]{\bf code};
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\onslide<3->{
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\node at (3.8,1.3) [single arrow, fill=red,text=white, minimum height=3cm]{\bf LF proof};
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\node at (3.8,1.9) {\small certificate};
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} |
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\onslide<2>{\node at (4.0,1.3) [text=red]{\begin{tabular}{c}\bf Highly\\\bf Dangerous!\end{tabular}};}
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\end{tikzpicture}
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\end{center}
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\end{block}
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\end{textblock}
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\begin{textblock}{15}(2,12)
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\small |
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\begin{itemize}
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\item<4-> Appel's checker is $\sim$2700 lines of code (1865 loc of\\ LF definitions; |
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803 loc in C including 2 library functions)\\[-3mm] |
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\item<5-> 167 loc in C implement a type-checker |
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\end{itemize}
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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 {*
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\tikzstyle{every node}=[node distance=25mm,text height=1.5ex, text depth=.25ex]
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\tikzstyle{node1}=[rectangle, minimum size=10mm, rounded corners=3mm, very thick,
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draw=black!50, top color=white, bottom color=black!20] |
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\tikzstyle{node2}=[rectangle, minimum size=12mm, rounded corners=3mm, very thick,
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draw=red!70, top color=white, bottom color=red!50!black!20] |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\mode<presentation>{
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\begin{frame}<2->[squeeze]
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\frametitle{Type-Checking in LF}
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\begin{columns}
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\begin{column}{0.8\textwidth}
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\begin{textblock}{0}(1,2)
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\begin{tikzpicture}
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\matrix[ampersand replacement=\&,column sep=7mm, row sep=5mm] |
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{ \&[-10mm]
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\node (def1) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}}; \&
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\node (proof1) [node1] {\large Proof}; \&
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\node (alg1) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}}; \\
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\onslide<4->{\node {\begin{tabular}{c}\small 1st\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
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\onslide<4->{\node (def2) [node2] {\large Spec$^\text{+ex}$};} \&
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\onslide<4->{\node (proof2) [node1] {\large Proof};} \&
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\onslide<4->{\node (alg2) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\
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\onslide<5->{\node {\begin{tabular}{c}\small 2nd\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
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\onslide<5->{\node (def3) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \&
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\onslide<5->{\node (proof3) [node1] {\large Proof};} \&
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\onslide<5->{\node (alg3) [node2] {\large Alg$^\text{-ex}$};} \\
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\onslide<6->{\node {\begin{tabular}{c}\small 3rd\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
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\onslide<6->{\node (def4) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \&
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\onslide<6->{\node (proof4) [node2] {\large\hspace{1mm}Proof\hspace{1mm}};} \&
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\onslide<6->{\node (alg4) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\
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}; |
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\draw[->,black!50,line width=2mm] (proof1) -- (def1); |
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\draw[->,black!50,line width=2mm] (proof1) -- (alg1); |
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\onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (def2);}
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\onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (alg2);}
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\onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (def3);}
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\onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (alg3);}
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\onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (def4);}
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\onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (alg4);}
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\onslide<3->{\draw[white,line width=1mm] (1.1,3.2) -- (0.9,2.85) -- (1.1,2.35) -- (0.9,2.0);}
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\end{tikzpicture}
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\end{textblock}
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\end{column}
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\end{columns}
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\begin{textblock}{3}(12,3.6)
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\onslide<4->{
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\node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{2h};
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\end{tikzpicture}}
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\end{textblock}
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\only<7->{
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\begin{textblock}{14}(0.6,12.8)
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|
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\begin{block}{}
|
|
335 |
\small Each time one needs to check $\sim$31pp~of informal paper proofs. |
|
336 |
You have to be able to keep definitions and proofs consistent. |
|
337 |
\end{block}
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|
338 |
\end{textblock}}
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|
339 |
||
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\end{frame}}
|
|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
342 |
||
343 |
*} |
|
344 |
||
| 2764 | 345 |
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| 2762 | 346 |
text_raw {*
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
348 |
\mode<presentation>{
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| 2764 | 349 |
\begin{frame}<1->[c]
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2771
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
350 |
\frametitle{Lessons Learned}
|
| 2762 | 351 |
|
352 |
\begin{itemize}
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|
353 |
\item Theorem provers help with keeping large proofs consistent; |
|
354 |
make them modifiable.\medskip |
|
355 |
||
356 |
\item They can ensure that all cases are covered.\medskip |
|
357 |
||
| 2764 | 358 |
\item Some reasoning can be automated. |
359 |
\end{itemize}\bigskip\pause
|
|
360 |
||
361 |
\begin{minipage}{1.1\textwidth}
|
|
362 |
Formal reasoning needs to be ``smooth''.\\ |
|
363 |
{\small (ideally as close as possible to reasoning with ``pen-and-paper'')}
|
|
364 |
\end{minipage}
|
|
365 |
||
366 |
\only<2->{
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|
367 |
\begin{textblock}{3}(0.1,9.9)
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|
368 |
\begin{tikzpicture}
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|
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\node at (0,0) [single arrow, shape border rotate=0, fill=red,text=red]{a};
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|
370 |
\end{tikzpicture}
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|
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\end{textblock}}
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372 |
||
373 |
||
374 |
\end{frame}}
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|
375 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
376 |
*} |
|
377 |
||
378 |
||
379 |
(*<*) |
|
380 |
atom_decl name |
|
381 |
||
382 |
nominal_datatype lam = |
|
383 |
Var "name" |
|
384 |
| App "lam" "lam" |
|
385 |
| Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
|
|
386 |
||
387 |
nominal_primrec |
|
388 |
subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_[_::=_]")
|
|
389 |
where |
|
390 |
"(Var x)[y::=s] = (if x=y then s else (Var x))" |
|
391 |
| "(App t\<^isub>1 t\<^isub>2)[y::=s] = App (t\<^isub>1[y::=s]) (t\<^isub>2[y::=s])" |
|
392 |
| "x\<sharp>(y,s) \<Longrightarrow> (Lam [x].t)[y::=s] = Lam [x].(t[y::=s])" |
|
393 |
apply(finite_guess)+ |
|
394 |
apply(rule TrueI)+ |
|
395 |
apply(simp add: abs_fresh) |
|
396 |
apply(fresh_guess)+ |
|
397 |
done |
|
398 |
||
399 |
lemma subst_eqvt[eqvt]: |
|
400 |
fixes pi::"name prm" |
|
401 |
shows "pi\<bullet>(t1[x::=t2]) = (pi\<bullet>t1)[(pi\<bullet>x)::=(pi\<bullet>t2)]" |
|
402 |
by (nominal_induct t1 avoiding: x t2 rule: lam.strong_induct) |
|
403 |
(auto simp add: perm_bij fresh_atm fresh_bij) |
|
404 |
||
405 |
lemma fresh_fact: |
|
406 |
fixes z::"name" |
|
407 |
shows "\<lbrakk>z\<sharp>s; (z=y \<or> z\<sharp>t)\<rbrakk> \<Longrightarrow> z\<sharp>t[y::=s]" |
|
408 |
by (nominal_induct t avoiding: z y s rule: lam.strong_induct) |
|
409 |
(auto simp add: abs_fresh fresh_prod fresh_atm) |
|
410 |
||
411 |
lemma forget: |
|
412 |
assumes asm: "x\<sharp>L" |
|
413 |
shows "L[x::=P] = L" |
|
414 |
using asm |
|
415 |
by (nominal_induct L avoiding: x P rule: lam.strong_induct) |
|
416 |
(auto simp add: abs_fresh fresh_atm) |
|
417 |
(*>*) |
|
| 2762 | 418 |
|
| 2764 | 419 |
text_raw {*
|
420 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
421 |
\mode<presentation>{
|
|
422 |
\begin{frame}
|
|
423 |
||
424 |
\begin{textblock}{16}(1,1)
|
|
425 |
\renewcommand{\isasymbullet}{$\cdot$}
|
|
426 |
\tiny\color{black}
|
|
427 |
*} |
|
428 |
lemma substitution_lemma_not_to_be_tried_at_home: |
|
429 |
assumes asm: "x\<noteq>y" "x\<sharp>L" |
|
430 |
shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]" |
|
431 |
using asm |
|
432 |
proof (induct M arbitrary: x y N L rule: lam.induct) |
|
433 |
case (Lam z M1) |
|
434 |
have ih: "\<And>x y N L. \<lbrakk>x\<noteq>y; x\<sharp>L\<rbrakk> \<Longrightarrow> M1[x::=N][y::=L] = M1[y::=L][x::=N[y::=L]]" by fact |
|
435 |
have "x\<noteq>y" by fact |
|
436 |
have "x\<sharp>L" by fact |
|
437 |
obtain z'::"name" where fc: "z'\<sharp>(x,y,z,M1,N,L)" by (rule exists_fresh) (auto simp add: fs_name1) |
|
438 |
have eq: "Lam [z'].([(z',z)]\<bullet>M1) = Lam [z].M1" using fc |
|
439 |
by (auto simp add: lam.inject alpha fresh_prod fresh_atm) |
|
440 |
have fc': "z'\<sharp>N[y::=L]" using fc by (simp add: fresh_fact fresh_prod) |
|
441 |
have "([(z',z)]\<bullet>x) \<noteq> ([(z',z)]\<bullet>y)" using `x\<noteq>y` by (auto simp add: calc_atm) |
|
442 |
moreover |
|
443 |
have "([(z',z)]\<bullet>x)\<sharp>([(z',z)]\<bullet>L)" using `x\<sharp>L` by (simp add: fresh_bij) |
|
444 |
ultimately |
|
445 |
have "M1[([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)][([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)] |
|
446 |
= M1[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)][([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]]" |
|
447 |
using ih by simp |
|
448 |
then have "[(z',z)]\<bullet>(M1[([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)][([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)] |
|
449 |
= M1[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)][([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]])" |
|
450 |
by (simp add: perm_bool) |
|
451 |
then have ih': "([(z',z)]\<bullet>M1)[x::=N][y::=L] = ([(z',z)]\<bullet>M1)[y::=L][x::=N[y::=L]]" |
|
452 |
by (simp add: eqvts perm_swap) |
|
453 |
show "(Lam [z].M1)[x::=N][y::=L] = (Lam [z].M1)[y::=L][x::=N[y::=L]]" (is "?LHS=?RHS") |
|
454 |
proof - |
|
455 |
have "?LHS = (Lam [z'].([(z',z)]\<bullet>M1))[x::=N][y::=L]" using eq by simp |
|
456 |
also have "\<dots> = Lam [z'].(([(z',z)]\<bullet>M1)[x::=N][y::=L])" using fc by (simp add: fresh_prod) |
|
457 |
also from ih have "\<dots> = Lam [z'].(([(z',z)]\<bullet>M1)[y::=L][x::=N[y::=L]])" sorry |
|
458 |
also have "\<dots> = (Lam [z'].([(z',z)]\<bullet>M1))[y::=L][x::=N[y::=L]]" using fc fc' by (simp add: fresh_prod) |
|
459 |
also have "\<dots> = ?RHS" using eq by simp |
|
460 |
finally show "?LHS = ?RHS" . |
|
461 |
qed |
|
462 |
qed (auto simp add: forget) |
|
463 |
text_raw {*
|
|
464 |
\end{textblock}
|
|
465 |
\mbox{}
|
|
466 |
||
467 |
\only<2->{
|
|
468 |
\begin{textblock}{11.5}(4,2.3)
|
|
469 |
\begin{minipage}{9.3cm}
|
|
470 |
\begin{block}{}\footnotesize
|
|
471 |
*} |
|
472 |
lemma substitution_lemma\<iota>: |
|
473 |
assumes asm: "x \<noteq> y" "x \<sharp> L" |
|
474 |
shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]" |
|
475 |
using asm |
|
476 |
by (nominal_induct M avoiding: x y N L rule: lam.strong_induct) |
|
477 |
(auto simp add: forget fresh_fact) |
|
478 |
text_raw {*
|
|
479 |
\end{block}
|
|
480 |
\end{minipage}
|
|
481 |
\end{textblock}}
|
|
| 2762 | 482 |
\end{frame}}
|
483 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
484 |
*} |
|
485 |
||
486 |
text_raw {*
|
|
487 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
488 |
\mode<presentation>{
|
|
489 |
\begin{frame}<1>[c]
|
|
| 2764 | 490 |
\frametitle{Getting Programs Correct}
|
491 |
||
492 |
\begin{center}
|
|
493 |
\begin{tikzpicture}[node distance=0.5mm]
|
|
494 |
\node at (-1.0,-0.3) (proof) [double arrow, fill=gray,text=white, minimum height=2cm]{\bf Proof};
|
|
495 |
\node [left=of proof]{\Large\bf Specification};
|
|
496 |
\node [right=of proof]{\Large\bf Code};
|
|
497 |
\end{tikzpicture}
|
|
498 |
\end{center}
|
|
499 |
||
500 |
||
501 |
||
502 |
\end{frame}}
|
|
503 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
504 |
*} |
|
505 |
||
506 |
text_raw {*
|
|
507 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
508 |
\mode<presentation>{
|
|
509 |
\begin{frame}<1->[t]
|
|
510 |
\frametitle{Regular Expressions}
|
|
| 2762 | 511 |
|
| 2764 | 512 |
\begin{textblock}{6}(2,4)
|
513 |
\begin{tabular}{@ {}rrl}
|
|
514 |
\bl{r} & \bl{$::=$} & \bl{$\varnothing$}\\
|
|
515 |
& \bl{$\mid$} & \bl{[]}\\
|
|
516 |
& \bl{$\mid$} & \bl{c}\\
|
|
517 |
& \bl{$\mid$} & \bl{r$_1$ + r$_2$}\\
|
|
518 |
& \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$}\\
|
|
519 |
& \bl{$\mid$} & \bl{r$^*$}\\
|
|
520 |
\end{tabular}
|
|
521 |
\end{textblock}
|
|
522 |
||
523 |
\begin{textblock}{6}(8,3.5)
|
|
524 |
\includegraphics[scale=0.35]{Screen1.png}
|
|
525 |
\end{textblock}
|
|
| 2762 | 526 |
|
| 2764 | 527 |
\begin{textblock}{6}(10.2,2.8)
|
528 |
\footnotesize Isabelle: |
|
529 |
\end{textblock}
|
|
530 |
||
531 |
\only<2>{
|
|
532 |
\begin{textblock}{9}(3.6,11.8)
|
|
533 |
\bl{matches r s $\;\Longrightarrow\;$ true $\vee$ false}\\[3.5mm]
|
|
534 |
||
535 |
\hspace{10mm}\begin{tikzpicture}
|
|
536 |
\coordinate (m1) at (0.4,1); |
|
537 |
\draw (0,0.3) node (m2) {\small\color{gray}rexp};
|
|
538 |
\path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1); |
|
539 |
||
540 |
\coordinate (s1) at (0.81,1); |
|
541 |
\draw (1.3,0.3) node (s2) {\small\color{gray} string};
|
|
542 |
\path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1); |
|
543 |
\end{tikzpicture}
|
|
544 |
\end{textblock}}
|
|
545 |
||
546 |
||
| 2762 | 547 |
|
548 |
\end{frame}}
|
|
549 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
550 |
*} |
|
551 |
||
552 |
text_raw {*
|
|
553 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
554 |
\mode<presentation>{
|
|
| 2764 | 555 |
\begin{frame}<1->[t]
|
556 |
\frametitle{Specification}
|
|
557 |
||
558 |
\small |
|
559 |
\begin{textblock}{6}(0,3.5)
|
|
560 |
\begin{tabular}{r@ {\hspace{0.5mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l}
|
|
561 |
\multicolumn{4}{c}{rexp $\Rightarrow$ set of strings}\bigskip\\
|
|
562 |
&\bl{\LL ($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$}\\
|
|
563 |
&\bl{\LL ([])} & \bl{$\dn$} & \bl{\{[]\}}\\
|
|
564 |
&\bl{\LL (c)} & \bl{$\dn$} & \bl{\{c\}}\\
|
|
565 |
&\bl{\LL (r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) $\cup$ \LL (r$_2$)}\\
|
|
566 |
\rd{$\Rightarrow$} &\bl{\LL (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) ;; \LL (r$_2$)}\\
|
|
567 |
\rd{$\Rightarrow$} &\bl{\LL (r$^*$)} & \bl{$\dn$} & \bl{(\LL (r))$^\star$}\\
|
|
568 |
\end{tabular}
|
|
569 |
\end{textblock}
|
|
570 |
||
571 |
\begin{textblock}{9}(7.3,3)
|
|
572 |
{\mbox{}\hspace{2cm}\footnotesize Isabelle:\smallskip}
|
|
573 |
\includegraphics[scale=0.325]{Screen3.png}
|
|
574 |
\end{textblock}
|
|
575 |
||
576 |
\end{frame}}
|
|
577 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
578 |
*} |
|
579 |
||
580 |
||
581 |
text_raw {*
|
|
582 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
583 |
\mode<presentation>{
|
|
584 |
\begin{frame}<1->[t]
|
|
585 |
\frametitle{Version 1}
|
|
586 |
\small |
|
587 |
\mbox{}\\[-8mm]\mbox{}
|
|
588 |
||
589 |
\begin{center}\def\arraystretch{1.05}
|
|
590 |
\begin{tabular}{@ {\hspace{-5mm}}l@ {\hspace{2.5mm}}c@ {\hspace{2.5mm}}l@ {}}
|
|
591 |
\bl{match [] []} & \bl{$=$} & \bl{true}\\
|
|
592 |
\bl{match [] (c::s)} & \bl{$=$} & \bl{false}\\
|
|
593 |
\bl{match ($\varnothing$::rs) s} & \bl{$=$} & \bl{false}\\
|
|
594 |
\bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\
|
|
595 |
\bl{match (c::rs) []} & \bl{$=$} & \bl{false}\\
|
|
596 |
\bl{match (c::rs) (d::s)} & \bl{$=$} & \bl{if c = d then match rs s else false}\\
|
|
597 |
\bl{match (r$_1$ + r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s $\vee$ match (r$_2$::rs) s}\\
|
|
598 |
\bl{match (r$_1$ $\cdot$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\
|
|
599 |
\bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\
|
|
600 |
\end{tabular}
|
|
601 |
\end{center}
|
|
602 |
||
603 |
\begin{textblock}{9}(0.2,1.6)
|
|
604 |
\hspace{10mm}\begin{tikzpicture}
|
|
605 |
\coordinate (m1) at (0.44,-0.5); |
|
606 |
\draw (0,0.3) node (m2) {\small\color{gray}\mbox{}\hspace{-9mm}list of rexps};
|
|
607 |
\path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1); |
|
608 |
||
609 |
\coordinate (s1) at (0.86,-0.5); |
|
610 |
\draw (1.5,0.3) node (s2) {\small\color{gray} string};
|
|
611 |
\path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1); |
|
612 |
\end{tikzpicture}
|
|
613 |
\end{textblock}
|
|
614 |
||
615 |
\begin{textblock}{9}(2.8,11.8)
|
|
616 |
\bl{matches$_1$ r s $\;=\;$ match [r] s}
|
|
617 |
\end{textblock}
|
|
618 |
||
619 |
\end{frame}}
|
|
620 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
621 |
*} |
|
| 2762 | 622 |
|
| 2764 | 623 |
text_raw {*
|
624 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
625 |
\mode<presentation>{
|
|
626 |
\begin{frame}<1->[c]
|
|
627 |
\frametitle{Testing}
|
|
628 |
||
629 |
\small |
|
630 |
Every good programmer should do thourough tests: |
|
631 |
||
632 |
\begin{center}
|
|
633 |
\begin{tabular}{@ {\hspace{-20mm}}lcl}
|
|
634 |
\bl{matches (a$\cdot$b)$^*\;$ []} & \bl{$\mapsto$} & \bl{true}\\
|
|
635 |
\bl{matches (a$\cdot$b)$^*\;$ ab} & \bl{$\mapsto$} & \bl{true}\\
|
|
636 |
\bl{matches (a$\cdot$b)$^*\;$ aba} & \bl{$\mapsto$} & \bl{false}\\
|
|
637 |
\bl{matches (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\
|
|
638 |
\bl{matches (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\
|
|
639 |
\onslide<2->{\bl{matches x$\cdot$(0$|$1)$^*\;$ x} & \bl{$\mapsto$} & \bl{true}}\\
|
|
640 |
\onslide<2->{\bl{matches x$\cdot$(0$|$1)$^*\;$ x0} & \bl{$\mapsto$} & \bl{true}}\\
|
|
641 |
\onslide<2->{\bl{matches x$\cdot$(0$|$1)$^*\;$ x3} & \bl{$\mapsto$} & \bl{false}}
|
|
642 |
\end{tabular}
|
|
643 |
\end{center}
|
|
644 |
||
645 |
\onslide<3-> |
|
646 |
{looks OK \ldots let's ship it to customers\hspace{5mm}
|
|
647 |
\raisebox{-5mm}{\includegraphics[scale=0.05]{sun.png}}}
|
|
648 |
||
649 |
\end{frame}}
|
|
650 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
651 |
*} |
|
652 |
||
653 |
text_raw {*
|
|
654 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
655 |
\mode<presentation>{
|
|
656 |
\begin{frame}<1->[c]
|
|
657 |
\frametitle{Version 1}
|
|
658 |
||
659 |
\only<1->{Several hours later\ldots}\pause
|
|
660 |
||
661 |
||
662 |
\begin{center}
|
|
663 |
\begin{tabular}{@ {\hspace{0mm}}lcl}
|
|
664 |
\bl{matches$_1$ []$^*$ s} & \bl{$\mapsto$} & loops\\
|
|
665 |
\onslide<4->{\bl{matches$_1$ ([] + \ldots)$^*$ s} & \bl{$\mapsto$} & loops\\}
|
|
666 |
\end{tabular}
|
|
667 |
\end{center}
|
|
668 |
||
669 |
\small |
|
670 |
\onslide<3->{
|
|
671 |
\begin{center}
|
|
672 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
|
|
673 |
\ldots\\ |
|
674 |
\bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\
|
|
675 |
\ldots\\ |
|
676 |
\bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\
|
|
677 |
\end{tabular}
|
|
678 |
\end{center}}
|
|
679 |
||
680 |
||
681 |
\end{frame}}
|
|
682 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
683 |
*} |
|
684 |
||
685 |
||
686 |
text_raw {*
|
|
687 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
688 |
\mode<presentation>{
|
|
689 |
\begin{frame}<1->[t]
|
|
690 |
\frametitle{Testing}
|
|
| 2762 | 691 |
|
692 |
\begin{itemize}
|
|
| 2764 | 693 |
\item While testing is an important part in the process of programming development\pause |
694 |
||
695 |
\item We can only test a {\bf finite} amount of examples.\bigskip\pause
|
|
| 2762 | 696 |
|
| 2764 | 697 |
\begin{center}
|
698 |
\colorbox{cream}
|
|
699 |
{\gr{\begin{minipage}{10cm}
|
|
700 |
``Testing can only show the presence of errors, never their |
|
701 |
absence'' (Edsger W.~Dijkstra) |
|
702 |
\end{minipage}}}
|
|
703 |
\end{center}\bigskip\pause
|
|
704 |
||
705 |
\item In a theorem prover we can establish properties that apply to |
|
706 |
{\bf all} input and {\bf all} output.\pause
|
|
707 |
||
| 2762 | 708 |
\end{itemize}
|
709 |
||
710 |
\end{frame}}
|
|
711 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
712 |
*} |
|
713 |
||
714 |
||
715 |
text_raw {*
|
|
716 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
717 |
\mode<presentation>{
|
|
| 2764 | 718 |
\begin{frame}<1->[t]
|
719 |
\frametitle{Version 2}
|
|
720 |
\mbox{}\\[-14mm]\mbox{}
|
|
721 |
||
722 |
\small |
|
723 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}}
|
|
724 |
\bl{nullable ($\varnothing$)} & \bl{$=$} & \bl{false} &\\
|
|
725 |
\bl{nullable ([])} & \bl{$=$} & \bl{true} &\\
|
|
726 |
\bl{nullable (c)} & \bl{$=$} & \bl{false} &\\
|
|
727 |
\bl{nullable (r$_1$ + r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\vee$ nullable r$_2$} & \\
|
|
728 |
\bl{nullable (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\wedge$ nullable r$_2$} & \\
|
|
729 |
\bl{nullable (r$^*$)} & \bl{$=$} & \bl{true} & \\
|
|
730 |
\end{tabular}\medskip
|
|
731 |
||
732 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
|
|
733 |
\bl{der c ($\varnothing$)} & \bl{$=$} & \bl{$\varnothing$} & \\
|
|
734 |
\bl{der c ([])} & \bl{$=$} & \bl{$\varnothing$} & \\
|
|
735 |
\bl{der c (d)} & \bl{$=$} & \bl{if c = d then [] else $\varnothing$} & \\
|
|
736 |
\bl{der c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\
|
|
737 |
\bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{((der c r$_1$) $\cdot$ r$_2$)} & \\
|
|
738 |
& & \bl{\;\;+ (if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\
|
|
739 |
\bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ r$^*$} &\smallskip\\
|
|
| 2762 | 740 |
|
| 2764 | 741 |
\bl{derivative r []} & \bl{$=$} & \bl{r} & \\
|
742 |
\bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\
|
|
743 |
\end{tabular}\medskip
|
|
744 |
||
745 |
\bl{matches$_2$ r s $=$ nullable (derivative r s)}
|
|
746 |
||
747 |
\begin{textblock}{6}(9.5,0.9)
|
|
748 |
\begin{flushright}
|
|
749 |
\color{gray}``if r matches []''
|
|
750 |
\end{flushright}
|
|
751 |
\end{textblock}
|
|
752 |
||
753 |
\begin{textblock}{6}(9.5,6.18)
|
|
754 |
\begin{flushright}
|
|
|
2771
66ef2a2c64fb
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Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
755 |
\color{gray}``derivative w.r.t.~a char''
|
| 2764 | 756 |
\end{flushright}
|
757 |
\end{textblock}
|
|
758 |
||
759 |
\begin{textblock}{6}(9.5,12.1)
|
|
760 |
\begin{flushright}
|
|
|
2771
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
761 |
\color{gray}``deriv.~w.r.t.~a string''
|
| 2764 | 762 |
\end{flushright}
|
763 |
\end{textblock}
|
|
764 |
||
765 |
\begin{textblock}{6}(9.5,13.98)
|
|
766 |
\begin{flushright}
|
|
767 |
\color{gray}``main''
|
|
768 |
\end{flushright}
|
|
769 |
\end{textblock}
|
|
| 2762 | 770 |
|
771 |
\end{frame}}
|
|
772 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
773 |
*} |
|
774 |
||
775 |
text_raw {*
|
|
776 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
777 |
\mode<presentation>{
|
|
| 2764 | 778 |
\begin{frame}<1->[t]
|
779 |
\frametitle{Is the Matcher Error-Free?}
|
|
| 2762 | 780 |
|
| 2764 | 781 |
We expect that |
| 2762 | 782 |
|
| 2764 | 783 |
\begin{center}
|
784 |
\begin{tabular}{lcl}
|
|
785 |
\bl{matches$_2$ r s = true} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}%
|
|
786 |
\only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\in$ \LL(r)}\\
|
|
787 |
\bl{matches$_2$ r s = false} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}%
|
|
788 |
\only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\notin$ \LL(r)}\\
|
|
789 |
\end{tabular}
|
|
790 |
\end{center}
|
|
791 |
\pause\pause\bigskip |
|
792 |
By \alert<4->{induction}, we can {\bf prove} these properties.\bigskip
|
|
793 |
||
794 |
\begin{tabular}{lrcl}
|
|
795 |
Lemmas: & \bl{nullable (r)} & \bl{$\Longleftrightarrow$} & \bl{[] $\in$ \LL (r)}\\
|
|
796 |
& \bl{s $\in$ \LL (der c r)} & \bl{$\Longleftrightarrow$} & \bl{(c::s) $\in$ \LL (r)}\\
|
|
797 |
\end{tabular}
|
|
| 2762 | 798 |
|
| 2764 | 799 |
\only<4->{
|
800 |
\begin{textblock}{3}(0.9,4.5)
|
|
801 |
\rd{\huge$\forall$\large{}r s.}
|
|
| 2762 | 802 |
\end{textblock}}
|
803 |
\end{frame}}
|
|
804 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
805 |
*} |
|
806 |
||
807 |
text_raw {*
|
|
808 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
809 |
\mode<presentation>{
|
|
810 |
\begin{frame}<1>[c]
|
|
811 |
\frametitle{
|
|
812 |
\begin{tabular}{c}
|
|
813 |
\mbox{}\\[23mm]
|
|
814 |
\LARGE Demo |
|
815 |
\end{tabular}}
|
|
816 |
||
817 |
\end{frame}}
|
|
818 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
819 |
*} |
|
820 |
||
821 |
||
822 |
text_raw {*
|
|
823 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
824 |
\mode<presentation>{
|
|
| 2764 | 825 |
\begin{frame}<1->[t]
|
826 |
||
827 |
\mbox{}\\[-2mm]
|
|
828 |
||
829 |
\small |
|
830 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}}
|
|
831 |
\bl{nullable (NULL)} & \bl{$=$} & \bl{false} &\\
|
|
832 |
\bl{nullable (EMPTY)} & \bl{$=$} & \bl{true} &\\
|
|
833 |
\bl{nullable (CHR c)} & \bl{$=$} & \bl{false} &\\
|
|
834 |
\bl{nullable (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) orelse (nullable r$_2$)} & \\
|
|
835 |
\bl{nullable (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) andalso (nullable r$_2$)} & \\
|
|
836 |
\bl{nullable (STAR r)} & \bl{$=$} & \bl{true} & \\
|
|
837 |
\end{tabular}\medskip
|
|
838 |
||
839 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
|
|
840 |
\bl{der c (NULL)} & \bl{$=$} & \bl{NULL} & \\
|
|
841 |
\bl{der c (EMPTY)} & \bl{$=$} & \bl{NULL} & \\
|
|
842 |
\bl{der c (CHR d)} & \bl{$=$} & \bl{if c=d then EMPTY else NULL} & \\
|
|
843 |
\bl{der c (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (der c r$_1$) (der c r$_2$)} & \\
|
|
844 |
\bl{der c (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (SEQ (der c r$_1$) r$_2$)} & \\
|
|
845 |
& & \bl{\phantom{ALT} (if nullable r$_1$ then der c r$_2$ else NULL)}\\
|
|
846 |
\bl{der c (STAR r)} & \bl{$=$} & \bl{SEQ (der c r) (STAR r)} &\smallskip\\
|
|
847 |
||
848 |
\bl{derivative r []} & \bl{$=$} & \bl{r} & \\
|
|
849 |
\bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\
|
|
850 |
\end{tabular}\medskip
|
|
851 |
||
852 |
\bl{matches r s $=$ nullable (derivative r s)}
|
|
853 |
||
854 |
\only<2>{
|
|
855 |
\begin{textblock}{8}(1.5,4)
|
|
856 |
\includegraphics[scale=0.3]{approved.png}
|
|
857 |
\end{textblock}}
|
|
| 2762 | 858 |
|
| 2764 | 859 |
\end{frame}}
|
860 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
861 |
*} |
|
862 |
||
863 |
||
864 |
text_raw {*
|
|
865 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
866 |
\mode<presentation>{
|
|
867 |
\begin{frame}[c]
|
|
868 |
\frametitle{No Automata?}
|
|
869 |
||
870 |
You might be wondering why I did not use any automata: |
|
871 |
||
872 |
\begin{itemize}
|
|
873 |
\item A \alert{regular language} is one where there is a DFA that
|
|
874 |
recognises it.\bigskip\pause |
|
875 |
\end{itemize}
|
|
876 |
||
877 |
||
|
2771
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
878 |
There are many reasons why this is a good definition:\medskip |
| 2764 | 879 |
\begin{itemize}
|
880 |
\item pumping lemma |
|
|
2771
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
881 |
\item closure properties of regular languages\\ (e.g.~closure under complement) |
| 2764 | 882 |
\end{itemize}
|
883 |
||
884 |
\end{frame}}
|
|
885 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
886 |
||
887 |
*} |
|
888 |
||
889 |
text_raw {*
|
|
890 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
891 |
\mode<presentation>{
|
|
892 |
\begin{frame}[t]
|
|
893 |
\frametitle{Really Bad News!}
|
|
894 |
||
895 |
DFAs are bad news for formalisations in theorem provers. They might |
|
896 |
be represented as: |
|
897 |
||
| 2762 | 898 |
\begin{itemize}
|
| 2764 | 899 |
\item graphs |
900 |
\item matrices |
|
901 |
\item partial functions |
|
902 |
\end{itemize}
|
|
903 |
||
904 |
All constructions are messy to reason about.\bigskip\bigskip |
|
905 |
\pause |
|
906 |
||
907 |
\small |
|
908 |
\only<2>{
|
|
909 |
Constable et al needed (on and off) 18 months for a 3-person team |
|
910 |
to formalise automata theory in Nuprl including Myhill-Nerode. There is |
|
911 |
only very little other formalised work on regular languages I know of |
|
912 |
in Coq, Isabelle and HOL.} |
|
913 |
\only<3>{typical textbook reasoning goes like: ``\ldots if \smath{M} and \smath{N} are any two
|
|
914 |
automata with no inaccessible states \ldots'' |
|
915 |
} |
|
916 |
||
917 |
\end{frame}}
|
|
918 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
919 |
||
920 |
*} |
|
921 |
||
922 |
text_raw {*
|
|
923 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
924 |
\mode<presentation>{
|
|
925 |
\begin{frame}[c]
|
|
|
2771
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
926 |
\frametitle{}
|
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
927 |
\large |
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
928 |
\begin{center}
|
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
929 |
\begin{tabular}{p{9cm}}
|
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
930 |
My point:\bigskip\\ |
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
931 |
|
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
932 |
The theory about regular languages can be reformulated |
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
933 |
to be more suitable for theorem proving. |
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
934 |
\end{tabular}
|
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
935 |
\end{center}
|
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
936 |
\end{frame}}
|
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
937 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
938 |
*} |
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
939 |
|
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
940 |
text_raw {*
|
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
941 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
942 |
\mode<presentation>{
|
|
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
943 |
\begin{frame}[c]
|
| 2764 | 944 |
\frametitle{\LARGE The Myhill-Nerode Theorem}
|
945 |
||
946 |
\begin{itemize}
|
|
947 |
\item provides necessary and suf\!ficient conditions for a language |
|
948 |
being regular (pumping lemma only necessary)\medskip |
|
949 |
||
950 |
\item will help with closure properties of regular languages\bigskip\pause |
|
951 |
||
952 |
\item key is the equivalence relation:\smallskip |
|
953 |
\begin{center}
|
|
954 |
\smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
|
|
955 |
\end{center}
|
|
956 |
\end{itemize}
|
|
957 |
||
958 |
\end{frame}}
|
|
959 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
960 |
*} |
|
| 2762 | 961 |
|
| 2764 | 962 |
text_raw {*
|
963 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
964 |
\mode<presentation>{
|
|
965 |
\begin{frame}[c]
|
|
966 |
\frametitle{\LARGE The Myhill-Nerode Theorem}
|
|
967 |
||
968 |
\mbox{}\\[5cm]
|
|
969 |
||
970 |
\begin{itemize}
|
|
971 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
|
|
972 |
\end{itemize}
|
|
973 |
||
974 |
\end{frame}}
|
|
975 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
976 |
||
977 |
*} |
|
978 |
||
979 |
text_raw {*
|
|
980 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
981 |
\mode<presentation>{
|
|
982 |
\begin{frame}[c]
|
|
983 |
\frametitle{\LARGE Equivalence Classes}
|
|
984 |
||
985 |
\begin{itemize}
|
|
986 |
\item \smath{L = []}
|
|
987 |
\begin{center}
|
|
988 |
\smath{\Big\{\{[]\},\; U\!N\!IV - \{[]\}\Big\}}
|
|
989 |
\end{center}\bigskip\bigskip
|
|
990 |
||
991 |
\item \smath{L = [c]}
|
|
992 |
\begin{center}
|
|
993 |
\smath{\Big\{\{[]\},\; \{[c]\},\; U\!N\!IV - \{[], [c]\}\Big\}}
|
|
994 |
\end{center}\bigskip\bigskip
|
|
995 |
||
996 |
\item \smath{L = \varnothing}
|
|
997 |
\begin{center}
|
|
998 |
\smath{\Big\{U\!N\!IV\Big\}}
|
|
999 |
\end{center}
|
|
1000 |
||
1001 |
\end{itemize}
|
|
1002 |
||
1003 |
\end{frame}}
|
|
1004 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1005 |
||
1006 |
*} |
|
1007 |
||
1008 |
text_raw {*
|
|
1009 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1010 |
\mode<presentation>{
|
|
1011 |
\begin{frame}[c]
|
|
1012 |
\frametitle{\LARGE Regular Languages}
|
|
1013 |
||
1014 |
\begin{itemize}
|
|
1015 |
\item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M}
|
|
1016 |
such that \smath{\mathbb{L}(M) = L}\\[1.5cm]
|
|
| 2762 | 1017 |
|
| 2764 | 1018 |
\item Myhill-Nerode: |
| 2762 | 1019 |
|
1020 |
\begin{center}
|
|
| 2764 | 1021 |
\begin{tabular}{l}
|
1022 |
finite $\Rightarrow$ regular\\ |
|
1023 |
\;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r. L = \mathbb{L}(r)}\\[3mm]
|
|
1024 |
regular $\Rightarrow$ finite\\ |
|
1025 |
\;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
|
|
1026 |
\end{tabular}
|
|
1027 |
\end{center}
|
|
1028 |
||
1029 |
\end{itemize}
|
|
1030 |
||
1031 |
\end{frame}}
|
|
1032 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1033 |
||
1034 |
*} |
|
1035 |
||
1036 |
text_raw {*
|
|
1037 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1038 |
\mode<presentation>{
|
|
1039 |
\begin{frame}[c]
|
|
1040 |
\frametitle{\LARGE Final States}
|
|
1041 |
||
1042 |
\mbox{}\\[3cm]
|
|
1043 |
||
1044 |
\begin{itemize}
|
|
1045 |
\item \smath{\text{final}_L\,X \dn}\\
|
|
1046 |
\smath{\hspace{6mm}X \in (U\!N\!IV /\!/\approx_L) \;\wedge\; \forall s \in X.\; s \in L}
|
|
1047 |
\smallskip |
|
1048 |
\item we can prove: \smath{L = \bigcup \{X.\;\text{final}_L\,X\}}
|
|
1049 |
||
1050 |
\end{itemize}
|
|
1051 |
||
1052 |
\end{frame}}
|
|
1053 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1054 |
*} |
|
1055 |
||
1056 |
text_raw {*
|
|
1057 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1058 |
\mode<presentation>{
|
|
1059 |
\begin{frame}[c]
|
|
1060 |
\frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes}
|
|
1061 |
||
1062 |
\smath{L = \{[c]\}}
|
|
1063 |
||
1064 |
\begin{tabular}{@ {\hspace{-7mm}}cc}
|
|
1065 |
\begin{tabular}{c}
|
|
1066 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
|
|
1067 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
1068 |
||
1069 |
%\draw[help lines] (0,0) grid (3,2); |
|
1070 |
||
1071 |
\node[state,initial] (q_0) {$R_1$};
|
|
1072 |
\node[state,accepting] (q_1) [above right of=q_0] {$R_2$};
|
|
1073 |
\node[state] (q_2) [below right of=q_0] {$R_3$};
|
|
1074 |
||
1075 |
\path[->] (q_0) edge node {c} (q_1)
|
|
1076 |
edge node [swap] {$\Sigma-{c}$} (q_2)
|
|
1077 |
(q_2) edge [loop below] node {$\Sigma$} ()
|
|
1078 |
(q_1) edge node {$\Sigma$} (q_2);
|
|
1079 |
\end{tikzpicture}
|
|
1080 |
\end{tabular}
|
|
1081 |
& |
|
1082 |
\begin{tabular}[t]{ll}
|
|
1083 |
\\[-20mm] |
|
1084 |
\multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm]
|
|
1085 |
||
1086 |
\smath{R_1}: & \smath{\{[]\}}\\
|
|
1087 |
\smath{R_2}: & \smath{\{[c]\}}\\
|
|
1088 |
\smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm]
|
|
|
2771
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
1089 |
\multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ;; [c] \subseteq Y}}}
|
| 2764 | 1090 |
\end{tabular}
|
1091 |
||
1092 |
\end{tabular}
|
|
1093 |
||
1094 |
\end{frame}}
|
|
1095 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1096 |
*} |
|
1097 |
||
1098 |
||
1099 |
text_raw {*
|
|
1100 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1101 |
\mode<presentation>{
|
|
1102 |
\begin{frame}[c]
|
|
1103 |
\frametitle{\LARGE Systems of Equations}
|
|
1104 |
||
1105 |
Inspired by a method of Brzozowski\;'64, we can build an equational system |
|
1106 |
characterising the equivalence classes: |
|
1107 |
||
1108 |
\begin{center}
|
|
1109 |
\begin{tabular}{@ {\hspace{-20mm}}c}
|
|
1110 |
\\[-13mm] |
|
1111 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
|
|
1112 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
1113 |
||
1114 |
%\draw[help lines] (0,0) grid (3,2); |
|
1115 |
||
1116 |
\node[state,initial] (p_0) {$R_1$};
|
|
1117 |
\node[state,accepting] (p_1) [right of=q_0] {$R_2$};
|
|
1118 |
||
1119 |
\path[->] (p_0) edge [bend left] node {a} (p_1)
|
|
1120 |
edge [loop above] node {b} ()
|
|
1121 |
(p_1) edge [loop above] node {a} ()
|
|
1122 |
edge [bend left] node {b} (p_0);
|
|
1123 |
\end{tikzpicture}\\
|
|
1124 |
\\[-13mm] |
|
1125 |
\end{tabular}
|
|
1126 |
\end{center}
|
|
1127 |
||
1128 |
\begin{center}
|
|
1129 |
\begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
|
|
1130 |
& \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
|
|
1131 |
& \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\
|
|
1132 |
\onslide<3->{we can prove}
|
|
1133 |
& \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}}
|
|
1134 |
& \onslide<3->{\smath{R_1; \mathbb{L}(b) \,\cup\, R_2;\mathbb{L}(b) \,\cup\, \{[]\};\{[]\}}}\\
|
|
1135 |
& \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}}
|
|
1136 |
& \onslide<3->{\smath{R_1; \mathbb{L}(a) \,\cup\, R_2;\mathbb{L}(a)}}\\
|
|
1137 |
\end{tabular}
|
|
1138 |
\end{center}
|
|
1139 |
||
1140 |
\end{frame}}
|
|
1141 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1142 |
*} |
|
1143 |
||
1144 |
||
1145 |
text_raw {*
|
|
1146 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1147 |
\mode<presentation>{
|
|
1148 |
\begin{frame}<1>[t]
|
|
| 2762 | 1149 |
\small |
| 2764 | 1150 |
|
1151 |
\begin{center}
|
|
1152 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
1153 |
\onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
|
|
1154 |
& \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
1155 |
\onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
|
|
1156 |
& \onslide<1->{\smath{R_1; a + R_2; a}}\\
|
|
1157 |
||
1158 |
& & & \onslide<2->{by Arden}\\
|
|
1159 |
||
1160 |
\onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
|
|
1161 |
& \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
1162 |
\onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
|
|
1163 |
& \only<2>{\smath{R_1; a + R_2; a}}%
|
|
1164 |
\only<3->{\smath{R_1; a\cdot a^\star}}\\
|
|
1165 |
||
1166 |
& & & \onslide<4->{by Arden}\\
|
|
1167 |
||
1168 |
\onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
|
|
1169 |
& \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
1170 |
\onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
|
|
1171 |
& \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
|
|
1172 |
||
1173 |
& & & \onslide<5->{by substitution}\\
|
|
1174 |
||
1175 |
\onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
|
|
1176 |
& \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
1177 |
\onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
|
|
1178 |
& \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
|
|
1179 |
||
1180 |
& & & \onslide<6->{by Arden}\\
|
|
1181 |
||
1182 |
\onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
|
|
1183 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
1184 |
\onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
|
|
1185 |
& \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
|
|
1186 |
||
1187 |
& & & \onslide<7->{by substitution}\\
|
|
1188 |
||
1189 |
\onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
|
|
1190 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
1191 |
\onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
|
|
1192 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
|
1193 |
\cdot a\cdot a^\star}}\\ |
|
1194 |
\end{tabular}
|
|
| 2762 | 1195 |
\end{center}
|
1196 |
||
1197 |
\end{frame}}
|
|
1198 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1199 |
*} |
|
1200 |
||
1201 |
text_raw {*
|
|
1202 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1203 |
\mode<presentation>{
|
|
| 2764 | 1204 |
\begin{frame}[c]
|
1205 |
\frametitle{\LARGE A Variant of Arden's Lemma}
|
|
1206 |
||
1207 |
{\bf Arden's Lemma:}\smallskip
|
|
1208 |
||
1209 |
If \smath{[] \not\in A} then
|
|
1210 |
\begin{center}
|
|
1211 |
\smath{X = X; A + \text{something}}
|
|
1212 |
\end{center}
|
|
1213 |
has the (unique) solution |
|
1214 |
\begin{center}
|
|
1215 |
\smath{X = \text{something} ; A^\star}
|
|
1216 |
\end{center}
|
|
1217 |
||
1218 |
||
1219 |
\end{frame}}
|
|
1220 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1221 |
*} |
|
1222 |
||
1223 |
||
1224 |
text_raw {*
|
|
1225 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1226 |
\mode<presentation>{
|
|
1227 |
\begin{frame}<1->[t]
|
|
1228 |
\small |
|
1229 |
||
1230 |
\begin{center}
|
|
1231 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
1232 |
\onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
|
|
1233 |
& \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
1234 |
\onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
|
|
1235 |
& \onslide<1->{\smath{R_1; a + R_2; a}}\\
|
|
1236 |
||
1237 |
& & & \onslide<2->{by Arden}\\
|
|
1238 |
||
1239 |
\onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
|
|
1240 |
& \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
|
|
1241 |
\onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
|
|
1242 |
& \only<2>{\smath{R_1; a + R_2; a}}%
|
|
1243 |
\only<3->{\smath{R_1; a\cdot a^\star}}\\
|
|
1244 |
||
1245 |
& & & \onslide<4->{by Arden}\\
|
|
1246 |
||
1247 |
\onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
|
|
1248 |
& \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
|
|
1249 |
\onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
|
|
1250 |
& \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
|
|
1251 |
||
1252 |
& & & \onslide<5->{by substitution}\\
|
|
1253 |
||
1254 |
\onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
|
|
1255 |
& \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
|
|
1256 |
\onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
|
|
1257 |
& \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
|
|
1258 |
||
1259 |
& & & \onslide<6->{by Arden}\\
|
|
1260 |
||
1261 |
\onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
|
|
1262 |
& \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
1263 |
\onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
|
|
1264 |
& \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
|
|
1265 |
||
1266 |
& & & \onslide<7->{by substitution}\\
|
|
1267 |
||
1268 |
\onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
|
|
1269 |
& \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
|
|
1270 |
\onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
|
|
1271 |
& \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
|
|
1272 |
\cdot a\cdot a^\star}}\\ |
|
1273 |
\end{tabular}
|
|
1274 |
\end{center}
|
|
1275 |
||
1276 |
\only<8->{
|
|
1277 |
\begin{textblock}{6}(2.5,4)
|
|
1278 |
\begin{block}{}
|
|
1279 |
\begin{minipage}{8cm}\raggedright
|
|
| 2762 | 1280 |
|
| 2764 | 1281 |
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
|
1282 |
\tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
|
|
1283 |
||
1284 |
%\draw[help lines] (0,0) grid (3,2); |
|
1285 |
||
1286 |
\node[state,initial] (p_0) {$R_1$};
|
|
1287 |
\node[state,accepting] (p_1) [right of=q_0] {$R_2$};
|
|
1288 |
||
1289 |
\path[->] (p_0) edge [bend left] node {a} (p_1)
|
|
1290 |
edge [loop above] node {b} ()
|
|
1291 |
(p_1) edge [loop above] node {a} ()
|
|
1292 |
edge [bend left] node {b} (p_0);
|
|
1293 |
\end{tikzpicture}
|
|
1294 |
||
1295 |
\end{minipage}
|
|
1296 |
\end{block}
|
|
1297 |
\end{textblock}}
|
|
1298 |
||
1299 |
\end{frame}}
|
|
1300 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1301 |
*} |
|
1302 |
||
1303 |
||
1304 |
text_raw {*
|
|
1305 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1306 |
\mode<presentation>{
|
|
1307 |
\begin{frame}[c]
|
|
1308 |
\frametitle{\LARGE The Equ's Solving Algorithm}
|
|
1309 |
||
| 2762 | 1310 |
\begin{itemize}
|
| 2764 | 1311 |
\item The algorithm must terminate: Arden makes one equation smaller; |
1312 |
substitution deletes one variable from the right-hand sides.\bigskip |
|
1313 |
||
1314 |
\item We need to maintain the invariant that Arden is applicable |
|
1315 |
(if \smath{[] \not\in A} then \ldots):\medskip
|
|
1316 |
||
1317 |
\begin{center}\small
|
|
1318 |
\begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
|
|
1319 |
\smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
|
|
1320 |
\smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\
|
|
1321 |
||
1322 |
& & & by Arden\\ |
|
1323 |
||
1324 |
\smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
|
|
1325 |
\smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\
|
|
1326 |
\end{tabular}
|
|
1327 |
\end{center}
|
|
1328 |
||
1329 |
\end{itemize}
|
|
1330 |
||
1331 |
||
1332 |
\end{frame}}
|
|
1333 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1334 |
*} |
|
1335 |
||
1336 |
||
1337 |
text_raw {*
|
|
1338 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1339 |
\mode<presentation>{
|
|
1340 |
\begin{frame}[c]
|
|
1341 |
\frametitle{\LARGE Other Direction}
|
|
1342 |
||
1343 |
One has to prove |
|
1344 |
||
1345 |
\begin{center}
|
|
1346 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
|
|
1347 |
\end{center}
|
|
1348 |
||
1349 |
by induction on \smath{r}. Not trivial, but after a bit
|
|
1350 |
of thinking, one can prove that if |
|
1351 |
||
1352 |
\begin{center}
|
|
1353 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm}
|
|
1354 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})}
|
|
1355 |
\end{center}
|
|
1356 |
||
1357 |
then |
|
| 2762 | 1358 |
|
| 2764 | 1359 |
\begin{center}
|
1360 |
\smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})}
|
|
1361 |
\end{center}
|
|
1362 |
||
1363 |
||
1364 |
||
1365 |
\end{frame}}
|
|
1366 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1367 |
*} |
|
1368 |
||
1369 |
||
1370 |
text_raw {*
|
|
1371 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1372 |
\mode<presentation>{
|
|
1373 |
\begin{frame}[c]
|
|
1374 |
\frametitle{\LARGE What Have We Achieved?}
|
|
1375 |
||
1376 |
\begin{itemize}
|
|
1377 |
\item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
|
|
1378 |
\bigskip\pause |
|
|
2771
66ef2a2c64fb
more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents:
2764
diff
changeset
|
1379 |
\item regular languages are closed under complementation; this is now easy\medskip |
| 2764 | 1380 |
\begin{center}
|
1381 |
\smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}}
|
|
1382 |
\end{center}
|
|
1383 |
\end{itemize}
|
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| 2762 | 1384 |
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| 2764 | 1385 |
|
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|
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|
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*} |
|
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text_raw {*
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|
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|
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\mode<presentation>{
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|
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\begin{frame}[c]
|
|
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\frametitle{\LARGE Examples}
|
|
1395 |
||
1396 |
\begin{itemize}
|
|
1397 |
\item \smath{L \equiv \Sigma^\star 0 \Sigma} is regular
|
|
1398 |
\begin{quote}\small
|
|
1399 |
\begin{tabular}{lcl}
|
|
1400 |
\smath{A_1} & \smath{=} & \smath{\Sigma^\star 00}\\
|
|
1401 |
\smath{A_2} & \smath{=} & \smath{\Sigma^\star 01}\\
|
|
1402 |
\smath{A_3} & \smath{=} & \smath{\Sigma^\star 10 \cup \{0\}}\\
|
|
1403 |
\smath{A_4} & \smath{=} & \smath{\Sigma^\star 11 \cup \{1\} \cup \{[]\}}\\
|
|
1404 |
\end{tabular}
|
|
1405 |
\end{quote}
|
|
1406 |
||
1407 |
\item \smath{L \equiv \{ 0^n 1^n \,|\, n \ge 0\}} is not regular
|
|
1408 |
\begin{quote}\small
|
|
1409 |
\begin{tabular}{lcl}
|
|
1410 |
\smath{B_0} & \smath{=} & \smath{\{0^n 1^n \,|\, n \ge 0\}}\\
|
|
1411 |
\smath{B_1} & \smath{=} & \smath{\{0^n 1^{(n-1)} \,|\, n \ge 1\}}\\
|
|
1412 |
\smath{B_2} & \smath{=} & \smath{\{0^n 1^{(n-2)} \,|\, n \ge 2\}}\\
|
|
1413 |
\smath{B_3} & \smath{=} & \smath{\{0^n 1^{(n-3)} \,|\, n \ge 3\}}\\
|
|
1414 |
& \smath{\vdots} &\\
|
|
1415 |
\end{tabular}
|
|
1416 |
\end{quote}
|
|
| 2762 | 1417 |
\end{itemize}
|
1418 |
||
1419 |
\end{frame}}
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|
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|
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*} |
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text_raw {*
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\mode<presentation>{
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| 2764 | 1427 |
\begin{frame}[c]
|
1428 |
\frametitle{\LARGE What We Have Not Achieved}
|
|
1429 |
||
1430 |
\begin{itemize}
|
|
1431 |
\item regular expressions are not good if you look for a minimal |
|
1432 |
one for a language (DFAs have this notion)\pause\bigskip |
|
1433 |
||
1434 |
\item Is there anything to be said about context free languages:\medskip |
|
1435 |
||
1436 |
\begin{quote}
|
|
1437 |
A context free language is where every string can be recognised by |
|
1438 |
a pushdown automaton.\bigskip |
|
1439 |
\end{quote}
|
|
1440 |
\end{itemize}
|
|
1441 |
||
1442 |
\textcolor{gray}{\footnotesize Yes. Derivatives also work for c-f grammars. Ongoing work.}
|
|
1443 |
||
1444 |
\end{frame}}
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|
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*} |
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text_raw {*
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\mode<presentation>{
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|
1452 |
\begin{frame}[c]
|
|
1453 |
\frametitle{\LARGE Conclusion}
|
|
1454 |
||
1455 |
\begin{itemize}
|
|
1456 |
\item We formalised the Myhill-Nerode theorem based on |
|
|
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|
1457 |
regular expressions only (DFAs are difficult to deal with in a theorem prover).\smallskip |
| 2764 | 1458 |
|
1459 |
\item Seems to be a common theme: algorithms need to be reformulated |
|
1460 |
to better suit formal treatment.\smallskip |
|
1461 |
||
1462 |
\item The most interesting aspect is that we are able to |
|
1463 |
implement the matcher directly inside the theorem prover |
|
1464 |
(ongoing work).\smallskip |
|
1465 |
||
|
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|
1466 |
\item Parsing is a vast field which seem to offer new results. |
| 2764 | 1467 |
\end{itemize}
|
1468 |
||
1469 |
\end{frame}}
|
|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
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*} |
|
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text_raw {*
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\mode<presentation>{
|
|
1476 |
\begin{frame}<1>[b]
|
|
| 2762 | 1477 |
\frametitle{
|
1478 |
\begin{tabular}{c}
|
|
| 2764 | 1479 |
\mbox{}\\[13mm]
|
| 2762 | 1480 |
\alert{\LARGE Thank you very much!}\\
|
1481 |
\alert{\Large Questions?}
|
|
1482 |
\end{tabular}}
|
|
| 2764 | 1483 |
|
|
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|
1484 |
\begin{center}
|
|
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|
1485 |
\bf \underline{Short Bio:}
|
|
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|
1486 |
\end{center}
|
|
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|
1487 |
\mbox{}\\[-17mm]\mbox{}\small
|
|
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|
1488 |
\begin{itemize}
|
|
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|
1489 |
\item PhD in Cambridge |
|
66ef2a2c64fb
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|
1490 |
\item Emmy-Noether Research Fellowship at the TU Munich |
|
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|
1491 |
\item talks at: CMU, Yale, Princeton, MIT,$\ldots$ |
|
66ef2a2c64fb
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|
1492 |
\end{itemize}
|
|
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|
1493 |
|
|
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|
1494 |
\end{frame}}
|
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|
1495 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
1496 |
*} |
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|
1497 |
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|
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text_raw {*
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|
1499 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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|
1500 |
\mode<presentation>{
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|
1501 |
\begin{frame}[c]
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|
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|
1502 |
\frametitle{Future Research}
|
|
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|
1503 |
|
|
66ef2a2c64fb
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|
1504 |
My existing strengths:\bigskip |
|
66ef2a2c64fb
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|
1505 |
|
|
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|
1506 |
\begin{itemize}
|
|
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|
1507 |
\item Isabelle (implementation)\bigskip |
|
66ef2a2c64fb
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|
1508 |
\item background in logic, programming languages, formal methods |
|
66ef2a2c64fb
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|
1509 |
\end{itemize}
|
|
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|
1510 |
|
|
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|
1511 |
\end{frame}}
|
|
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|
1512 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
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|
1513 |
*} |
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|
1514 |
|
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|
1515 |
text_raw {*
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|
1516 |
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|
1517 |
\mode<presentation>{
|
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|
1518 |
\begin{frame}[c]
|
|
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|
1519 |
\frametitle{Future Research}
|
|
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|
1520 |
|
|
66ef2a2c64fb
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|
1521 |
I want to have a single logic framework in which I can |
|
66ef2a2c64fb
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|
1522 |
write programs and prove their correctness.\bigskip |
|
66ef2a2c64fb
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|
1523 |
|
|
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|
1524 |
\begin{itemize}
|
|
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|
1525 |
\item extensions of HOL (IO, modules, advanced types) |
|
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|
1526 |
\item high-level programming languages |
|
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|
1527 |
\end{itemize}
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|
1528 |
|
|
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|
1529 |
\end{frame}}
|
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|
1530 |
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|
1531 |
*} |
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|
1532 |
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text_raw {*
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1535 |
\mode<presentation>{
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1536 |
\begin{frame}[c]
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1537 |
\frametitle{Future Research}
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|
1538 |
|
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|
1539 |
Compilers\bigskip |
|
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|
1540 |
|
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|
1541 |
\begin{itemize}
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|
1542 |
\item the high-level language needs to be compiled to correct machine |
|
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|
1543 |
code |
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|
1544 |
\item compiler verification, machine code verification |
|
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|
1545 |
\end{itemize}
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|
1546 |
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|
1547 |
\end{frame}}
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1548 |
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1549 |
*} |
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|
1550 |
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text_raw {*
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\mode<presentation>{
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\frametitle{Future Research}
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1556 |
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|
1557 |
Stronger type-systems\bigskip |
|
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|
1558 |
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|
1559 |
\begin{itemize}
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1560 |
\item ``correct by construction'' |
|
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|
1561 |
\item GADTs, dependent types |
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|
1562 |
\end{itemize}
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1563 |
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|
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\end{frame}}
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1565 |
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*} |
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1567 |
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\mode<presentation>{
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\frametitle{Future Research}
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|
1574 |
Proof automation\bigskip |
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Christian Urban <urbanc@in.tum.de>
parents:
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|
1575 |
|
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|
1576 |
\begin{itemize}
|
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|
1577 |
\item external tools generate ``proof-certificates'' |
|
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parents:
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|
1578 |
\item certificates are imported into Isabelle |
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|
1579 |
\item GPU based external provers |
|
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parents:
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|
1580 |
\end{itemize}
|
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|
1581 |
|
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|
1582 |
\end{frame}}
|
|
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|
1583 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
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|
1584 |
*} |
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|
1585 |
|
|
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|
1586 |
text_raw {*
|
|
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|
1587 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
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|
1588 |
\mode<presentation>{
|
|
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|
1589 |
\begin{frame}[c]
|
|
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|
1590 |
\frametitle{Future Research}
|
|
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|
1591 |
|
|
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|
1592 |
Large-scale applications\bigskip |
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|
1593 |
|
|
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|
1594 |
\begin{itemize}
|
|
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changeset
|
1595 |
\item verification of Java-Script, Scala,$\ldots$ |
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parents:
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changeset
|
1596 |
\item interesting code (INTEL in Shanghai) |
|
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|
1597 |
\end{itemize}
|
| 2764 | 1598 |
|
| 2762 | 1599 |
\end{frame}}
|
| 2764 | 1600 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 2762 | 1601 |
*} |
1602 |
||
1603 |
||
1604 |
(*<*) |
|
1605 |
end |
|
1606 |
(*>*) |