(*<*)+
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theory Slides4+
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imports "LaTeXsugar" "Nominal"+
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begin+
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notation (latex output)+
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  set ("_") and+
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  Cons  ("_::/_" [66,65] 65) +
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(*>*)+
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text_raw {*+
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  \renewcommand{\slidecaption}{Nanjing, 31.~August 2010}+
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  \newcommand{\abst}[2]{#1.#2}% atom-abstraction+
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  \newcommand{\susp}{{\boldsymbol{\cdot}}}% for suspensions+
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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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  +
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  \newcommand{\bl}[1]{\textcolor{blue}{#1}}+
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  \newcommand{\LL}{$\mathbb{L}\,$}+
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  \pgfdeclareradialshading{smallbluesphere}{\pgfpoint{0.5mm}{0.5mm}}%+
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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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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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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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  \huge Error-Free Programming\\[-1mm] +
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  \huge with Theorem Provers\\[5mm]+
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  \end{tabular}}+
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  \begin{center}+
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  Christian Urban+
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  \end{center}+
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  \begin{center}+
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  \small Technical University of Munich, Germany\\[7mm]+
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+
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  \small in Nanjing on the kind invitation of\\ Professor Xingyuan Zhang+
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  and his group+
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  \end{center}+
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  \end{frame}}+
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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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  \mode<presentation>{+
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  \begin{frame}<1->[c]+
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  \frametitle{My Background}+
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+
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  \begin{itemize}+
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  \item researcher in Theoretical Computer Science\medskip+
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+
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  \item programmer on a \alert<2->{software system} with 800 kloc (though I am +
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  responsible only for 35 kloc)+
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  \end{itemize}+
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+
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  \only<2->{+
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  \begin{textblock}{6}(2,11)+
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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}{4cm}\raggedright+
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  A theorem prover called {\bf Isabelle}.+
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  \end{minipage}};+
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  \end{tikzpicture}+
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  \end{textblock}}+
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+
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  +
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  \only<3>{+
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  \begin{textblock}{6}(9,11)+
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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}{4cm}\raggedright+
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  Like every other code, this code is very hard to +
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  get correct.+
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  \end{minipage}};+
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  \end{textblock}}+
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+
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  \end{frame}}+
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*}+
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text_raw {*+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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  \mode<presentation>{+
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  \begin{frame}<1->[t]+
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  \frametitle{Regular Expressions}+
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+
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  An example many (should) know about:\\ +
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  \rd{\bf Regular Expressions:} +
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+
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  \only<2>{+
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  \begin{center}+
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  \bl{[] $\;\;\;|\;\;\;$ c $\;\;\;|\;\;\;$  r$_1$$|$r$_2$  $\;\;\;|\;\;\;$  +
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   r$_1$$\cdot$r$_2$ $\;\;\;|\;\;\;$ r$^*$}+
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  \end{center}}+
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  \only<3->{+
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  \begin{center}+
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  \begin{tabular}{@ {}rrll@ {}}+
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  \bl{r} & \bl{$::=$}  & \bl{NULL}            & \gr{(matches no string)}\\ +
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         & \bl{$\mid$} & \bl{EMPTY}           & \gr{(matches the empty string, [])}\\ +
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         & \bl{$\mid$} & \bl{CHR c}           & \gr{(matches the character c)}\\ +
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         & \bl{$\mid$} & \bl{ALT r$_1$ r$_2$} & \gr{(alternative, r$_1 |\,$r$_2$)}\\ +
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         & \bl{$\mid$} & \bl{SEQ r$_1$ r$_2$} & \gr{(sequential, r$_1\cdot\,$r$_2$)}\\ +
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         & \bl{$\mid$} & \bl{STAR r}          & \gr{(repeat, r$^*$)}\\+
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  \end{tabular}+
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  \end{center}\medskip}+
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+
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  \small+
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  \begin{textblock}{14.5}(1,12.5)+
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  \only<2->{\gr{(a$\cdot$b)$^*$ \hspace{3mm}$\mapsto$\hspace{3mm} \{[], ab, abab, ababab, \ldots\}}\\}+
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  \only<2->{\gr{x$\cdot$(0 $|$ 1 $|$ 2  \ldots  8 $|$ 9)$^*$ \hspace{3mm}$\mapsto$\hspace{3mm} +
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  \{x, x0, x1, \ldots, x00, \ldots, x123, \ldots\}}}+
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  \end{textblock}+
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  \end{frame}}+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
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*}+
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text_raw {*+
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  \mode<presentation>{+
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  \begin{frame}<1>[c]+
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  \frametitle{RegExp Matcher}+
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+
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  Let's implement a regular expression matcher:\bigskip+
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+
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  \begin{center}+
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  \begin{tikzpicture}+
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  %%\draw[help lines, black] (-3,0) grid (6,3);+
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  +
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  \draw[line width=1mm, red] (0.0,0.0) rectangle (4,2.3);+
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  \node[anchor=base] at (2,1) +
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     {\small\begin{tabular}{@ {}c@ {}}\Large\bf Regular\\ +
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                                      \Large\bf Expression \\ +
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                                      \Large\bf Matcher\end{tabular}};+
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  +
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  \coordinate (m1) at (0,1.5);+
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  \draw (-2,2) node (m2) {\small\begin{tabular}{c}\bl{regular}\\[-1mm] \bl{expression}\end{tabular}};+
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  \path[overlay, ->, line width = 1mm, shorten <=-3mm] (m2) edge (m1);+
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  +
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  \coordinate (s1) at (0,0.5);+
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  \draw (-1.8,-0) node (s2) {\small\begin{tabular}{c}\bl{string}\end{tabular}};+
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  \path[overlay, ->, line width = 1mm, shorten <=-3mm] (s2) edge (s1);+
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+
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  \coordinate (r1) at (4,1.2);+
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  \draw (6,1.2) node (r2) {\small\begin{tabular}{c}\bl{true}, \bl{false}\end{tabular}};+
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  \path[overlay, ->, line width = 1mm, shorten >=-3mm] (r1) edge (r2);+
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+
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  \end{tikzpicture}+
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  \end{center}+
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+
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+
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  \end{frame}}+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
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*}+
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text_raw {*+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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  \mode<presentation>{+
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  \begin{frame}<1->[t]+
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  \frametitle{RegExp Matcher}+
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  \small+
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+
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  {\bf input:} a \underline{list} of RegExps and a string \hspace{6mm}{\bf output:} true or false+
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+
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  \only<2->{+
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  \begin{center}+
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  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}+
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  \bl{match [] []}                   & \bl{$=$} & \bl{true}\\+
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  \bl{match [] \_}                   & \bl{$=$} & \bl{false}\\+
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  \bl{match (NULL::rs) s}            & \bl{$=$} & \bl{false}\\+
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  \bl{match (EMPTY::rs) s}           & \bl{$=$} & \bl{match rs s}\\+
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  \bl{match (CHR c::rs) (c::s)}      & \bl{$=$} & \bl{match rs s}\\         +
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  \bl{match (CHR c::rs) \_}          & \bl{$=$} & \bl{false}\\    +
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  \bl{match (ALT r$_1$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s}\\+
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                                     &          & \bl{\;\;\;\;orelse match (r$_2$::rs) s}\\ +
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  \bl{match (SEQ r$_1$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\+
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  \bl{match (STAR r::rs) s}          & \bl{$=$} & \bl{match rs s}\\+
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                                     &          & \bl{\;\;\;\;orelse match (r::STAR r::rs) s}\\+
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  \end{tabular}+
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  \end{center}}+
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+
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  \onslide<3->{we start the program with\\+
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  \hspace{6mm}\bl{matches r s $=$ match [r] s}}+
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+
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  \end{frame}}+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
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text_raw {*+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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  \mode<presentation>{+
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  \frametitle{Program in Scala}+
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+
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  \bl{\footnotesize+
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  \begin{tabular}{l}+
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  sealed abstract class Rexp\\ +
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  sealed case class Null extends Rexp\\+
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  sealed case class Empty extends Rexp\\+
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  sealed case class Chr(c: Char) extends Rexp\\+
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  sealed case class Alt(r1 : Rexp, r2 : Rexp) extends Rexp\\+
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  sealed case class Seq(r1 : Rexp, r2 : Rexp) extends Rexp\\+
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  sealed case class Star(r : Rexp) extends Rexp\medskip\\+
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  def match1 (rs : List[Rexp], s : List[Char]) : Boolean = rs match \{\\+
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  \hspace{3mm}case Nil @{text "\<Rightarrow>"} if (s == Nil) true else false\\+
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  \hspace{3mm}case (Null()::rs) @{text "\<Rightarrow>"} false\\+
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  \hspace{3mm}case (Empty()::rs) @{text "\<Rightarrow>"} match1 (rs, s)\\+
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  \hspace{3mm}case (Chr(c)::rs) @{text "\<Rightarrow>"} s match \\+
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  \hspace{6mm}\{ case Nil @{text "\<Rightarrow>"} false\\+
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  \hspace{8mm}case (d::s) @{text "\<Rightarrow>"} if (c==d) match1 (rs,s) else false \}\\+
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  \hspace{3mm}case (Alt (r1, r2)::rs) @{text "\<Rightarrow>"} match1 (r1::rs, s) || match1 (r2::rs, s)\\+
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  \hspace{3mm}case (Seq (r1, r2)::rs) @{text "\<Rightarrow>"} match1 (r1::r2::rs, s) \\+
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  \hspace{3mm}case (Star (r)::rs) @{text "\<Rightarrow>"} match1 (r::rs, s) || match1 (r::Star (r)::rs, s)\\+
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  \}+
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  \end{tabular}}+
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+
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  \end{frame}}+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
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text_raw {*+
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  \mode<presentation>{+
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  \frametitle{Testing}+
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  +
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  \small+
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  Every good programmer should do thourough tests: +
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+
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+
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  \begin{center}+
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  \begin{tabular}{@ {\hspace{-20mm}}lcl}+
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  \bl{matches (a$\cdot$b)$^*\;$ []}     & \bl{$\mapsto$} & \bl{true}\\+
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  \bl{matches (a$\cdot$b)$^*\;$ ab}   & \bl{$\mapsto$} & \bl{true}\\ +
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  \bl{matches (a$\cdot$b)$^*\;$ aba}  & \bl{$\mapsto$} & \bl{false}\\+
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  \bl{matches (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\ +
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  \bl{matches (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\+
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  \onslide<2->{\bl{matches x$\cdot$(0$|$1)$^*\;$ x}   & \bl{$\mapsto$} & \bl{true}}\\+
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  \onslide<2->{\bl{matches x$\cdot$(0$|$1)$^*\;$ x0}  & \bl{$\mapsto$} & \bl{true}}\\+
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  \onslide<2->{\bl{matches x$\cdot$(0$|$1)$^*\;$ x3}  & \bl{$\mapsto$} & \bl{false}}+
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  \end{tabular}+
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  \end{center}+
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 +
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  \onslide<3->+
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  {looks OK \ldots let's ship it to customers\hspace{5mm} +
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   \raisebox{-5mm}{\includegraphics[scale=0.05]{sun.png}}}+
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  +
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  \end{frame}}+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
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text_raw {*+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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  \mode<presentation>{+
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  \begin{frame}<1->[t]+
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  \frametitle{Testing}+
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+
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  \begin{itemize}+
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  \item While testing is an important part in the process of programming development\pause+
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+
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  \item we can only test a {\bf finite} amount of examples\bigskip\pause+
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+
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  \begin{center}+
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  \colorbox{cream}+
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  {\gr{\begin{minipage}{10cm}+
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  ``Testing can only show the presence of errors, never their+
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  absence'' (Edsger W.~Dijkstra)+
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  \end{minipage}}}+
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  \end{center}\bigskip\pause+
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+
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  \item In a theorem prover we can establish properties that apply to +
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  {\bf all} input and {\bf all} output.\pause +
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+
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  \item For example we can establish that the matcher terminates +
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  on all input.+
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  \end{itemize}+
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  \end{frame}}+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
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text_raw {*+
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  \mode<presentation>{+
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  \begin{frame}<1->[t]+
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  \frametitle{RegExp Matcher}+
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+
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  \small+
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  We need to find a measure that gets smaller in each recursive call.\bigskip+
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+
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  \onslide<1->{+
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  \begin{center}+
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  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-9mm}}l@ {}}+
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  \bl{match [] []}                   & \bl{$=$} & \bl{true} & \onslide<2->{\ok}\\+
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  \bl{match [] \_}                   & \bl{$=$} & \bl{false} & \onslide<2->{\ok}\\+
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  \bl{match (NULL::rs) s}            & \bl{$=$} & \bl{false} & \onslide<2->{\ok}\\+
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  \bl{match (EMPTY::rs) s}           & \bl{$=$} & \bl{match rs s} & \onslide<3->{\ok}\\+
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  \bl{match (CHR c::rs) (c::s)}      & \bl{$=$} & \bl{match rs s} & \onslide<4->{\ok}\\         +
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  \bl{match (CHR c::rs) \_}          & \bl{$=$} & \bl{false} & \onslide<2->{\ok}\\    +
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  \bl{match (ALT r$_1$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s} & \onslide<5->{\ok}\\+
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                                     &          & \bl{\;\;\;\;orelse match (r$_2$::rs) s}\\ +
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  \bl{match (SEQ r$_1$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s} & \onslide<6->{\ok}\\+
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  \bl{match (STAR r::rs) s}          & \bl{$=$} & \bl{match rs s} & \onslide<7->{\notok}\\+
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                                     &          & \bl{\;\;\;\;orelse match (r::STAR r::rs) s}\\+
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  \end{tabular}+
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  \end{center}}+
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+
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  \begin{textblock}{5}(4,4)+
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  \begin{tikzpicture}+
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  %%\draw[help lines, black] (-3,0) grid (6,3);+
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  +
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  \coordinate (m1) at (-2,0);+
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  \coordinate (m2) at ( 2,0);+
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  \path[overlay, ->, line width = 0.6mm, color = red] (m1) edge (m2);+
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  \draw (0,0) node[above=-1mm] {\footnotesize\rd{needs to get smaller}};+
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text_raw {*+
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  \mode<presentation>{+
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  \begin{frame}<1->[c]+
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  \frametitle{Bug Hunting}+
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+
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  \only<1>{Several hours later\ldots}\pause+
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+
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+
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  \begin{center}+
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  \begin{tabular}{@ {\hspace{-20mm}}lcl}+
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  \bl{matches (STAR (EMPTY)) s}     & \bl{$\mapsto$} & loops\\+
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  \onslide<4->{\bl{matches (STAR (EMPTY $|$ \ldots)) s}   & \bl{$\mapsto$} & loops\\} +
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  \end{tabular}+
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  \end{center}+
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+
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  \small+
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  \onslide<3->{+
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  \begin{center}+
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  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}+
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  \ldots\\+
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  \bl{match (EMPTY::rs) s}           & \bl{$=$} & \bl{match rs s}\\+
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  \ldots\\+
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  \bl{match (STAR r::rs) s}          & \bl{$=$} & \bl{match rs s}\\+
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                                     &          & \bl{\;\;\;\;orelse match (r::STAR r::rs) s}\\+
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  \end{tabular}+
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  \end{center}}+
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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{RegExp Matcher}+
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  \small+
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+
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  \begin{center}+
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  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}+
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  \bl{match [] []}                   & \bl{$=$} & \bl{true}\\+
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  \bl{match [] \_}                   & \bl{$=$} & \bl{false}\\+
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  \bl{match (NULL::rs) s}            & \bl{$=$} & \bl{false}\\+
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  \bl{match (EMPTY::rs) s}           & \bl{$=$} & \bl{match rs s}\\+
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  \bl{match (CHR c::rs) (c::s)}      & \bl{$=$} & \bl{match rs s}\\         +
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  \bl{match (CHR c::rs) \_}          & \bl{$=$} & \bl{false}\\    +
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  \bl{match (ALT r$_1$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s}\\+
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                                     &          & \bl{\;\;\;\;orelse match (r$_2$::rs) s}\\ +
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  \bl{match (SEQ r$_1$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\+
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  \bl{match (STAR r::rs) s}          & \bl{$=$} & \bl{match rs s}\\+
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                                     &          & \bl{\;\;\;\;orelse match (r::STAR r::rs) s}\\+
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  \end{tabular}+
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  \end{center}+
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+
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  \only<1>{+
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  \begin{textblock}{5}(4,4)+
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  \largenotok+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
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text_raw {*+
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  \mode<presentation>{+
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  \begin{frame}<1->[t]+
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  \frametitle{Second Attempt}+
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+
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  Can a regular expression match the empty string?+
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+
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  \small+
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  \begin{center}+
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  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}}+
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  \bl{nullable (NULL)}            & \bl{$=$} & \bl{false} & \onslide<2->{\ok}\\+
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  \bl{nullable (EMPTY)}           & \bl{$=$} & \bl{true}  & \onslide<2->{\ok}\\+
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  \bl{nullable (CHR c)}           & \bl{$=$} & \bl{false} & \onslide<2->{\ok}\\         +
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  \bl{nullable (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) orelse (nullable r$_2$)}+
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    & \onslide<2->{\ok}\\ +
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  \bl{nullable (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) andalso (nullable r$_2$)}+
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    & \onslide<2->{\ok}\\ +
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  \bl{nullable (STAR r)}          & \bl{$=$} & \bl{true} & \onslide<2->{\ok}\\+
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  \end{tabular}+
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  \end{center}+
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+
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  \end{frame}}+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
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text_raw {*+
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  \begin{frame}<1->[t]+
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  \frametitle{RegExp Matcher 2}+
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+
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  If \bl{r} matches \bl{c::s}, which regular expression can match the string \bl{s}?+
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+
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  \small+
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  \begin{center}+
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  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}+
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  \bl{der c (NULL)}            & \bl{$=$} & \bl{NULL} & \onslide<3->{\ok}\\+
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  \bl{der c (EMPTY)}           & \bl{$=$} & \bl{NULL} & \onslide<3->{\ok}\\+
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  \bl{der c (CHR d)}           & \bl{$=$} & \bl{if c=d then EMPTY else NULL} & \onslide<3->{\ok}\\+
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  \bl{der c (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (der c r$_1$) (der c r$_2$)} & \onslide<3->{\ok}\\ +
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  \bl{der c (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (SEQ (der c r$_1$) r$_2$)} & \onslide<3->{\ok}\\+
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       &          & \bl{\phantom{ALT} (if nullable r$_1$ then der c r$_2$ else NULL)}\\+
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  \bl{der c (STAR r)}          & \bl{$=$} & \bl{SEQ (der c r) (STAR r)} & \onslide<3->{\ok}\medskip\\+
−
  \pause+
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+
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  \bl{derivative r []}     & \bl{$=$} & \bl{r} & \onslide<3->{\ok}\\+
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  \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \onslide<3->{\ok}\\+
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  \end{tabular}+
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  \end{center}+
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+
−
  we call the program with\\+
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  \bl{matches r s $=$ nullable (derivative r s)}+
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  +
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+
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  \end{frame}}+
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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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  \mode<presentation>{+
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  \begin{frame}<1->[c]+
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  \frametitle{But Now What?}+
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+
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  \begin{center}+
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  {\usefont{T1}{ptm}{b}{N}\VERYHuge{\rd{?}}}+
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  \end{center}+
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+
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  \end{frame}}+
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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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  \begin{frame}<1->[c]+
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  \frametitle{Testing}+
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  +
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  \small+
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+
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  \begin{center}+
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  \begin{tabular}{@ {\hspace{-20mm}}lcl}+
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  \bl{matches []$^*$ []} & \bl{$\mapsto$} & \bl{true}\\+
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  \bl{matches ([]$|$a)$^*$ a}  & \bl{$\mapsto$} & \bl{true}\medskip\\+
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+
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  \bl{matches (a$\cdot$b)$^*\;$ []}     & \bl{$\mapsto$} & \bl{true}\\+
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  \bl{matches (a$\cdot$b)$^*\;$ ab}   & \bl{$\mapsto$} & \bl{true}\\ +
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  \bl{matches (a$\cdot$b)$^*\;$ aba}  & \bl{$\mapsto$} & \bl{false}\\+
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  \bl{matches (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\ +
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  \bl{matches (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\+
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  +
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  \bl{matches x$\cdot$(0$|$1)$^*\;$ x}   & \bl{$\mapsto$} & \bl{true}\\+
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  \bl{matches x$\cdot$(0$|$1)$^*\;$ x0}  & \bl{$\mapsto$} & \bl{true}\\+
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  \bl{matches x$\cdot$(0$|$1)$^*\;$ x3}  & \bl{$\mapsto$} & \bl{false}+
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  \end{tabular}+
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  \end{center}+
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 +
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  +
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  \end{frame}}+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
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*}+
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text_raw {*+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%+
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  \mode<presentation>{+
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  \begin{frame}<1->[t]+
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  \frametitle{Specification}+
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+
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  We have to specify what it means for a regular expression to match+
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  a string. +
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  %+
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  \only<2>{+
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  \mbox{}\\[8mm]+
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  \bl{(a$\cdot$b)$^*$}\\ +
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  \hspace{7mm}\bl{$\mapsto$\hspace{3mm}\{[], ab, abab, ababab, \ldots\}}\bigskip\\+
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  \bl{x$\cdot$(0 $|$ 1 $|$ 2  \ldots  8 $|$ 9 )$^*$}\\ +
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  \hspace{7mm}\bl{$\mapsto$\hspace{3mm} +
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  \{x, x0, x1, \ldots, x00, \ldots, x123, \ldots\}}}+
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  %+
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  \only<3->{+
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  \begin{center}+
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  \begin{tabular}{rcl}+
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  \bl{\LL (NULL)}            & \bl{$\dn$} & \bl{\{\}}\\+
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  \bl{\LL (EMPTY)}           & \bl{$\dn$} & \bl{\{[]\}}\\+
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  \bl{\LL (CHR c)}           & \bl{$\dn$} & \bl{\{c\}}\\+
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  \bl{\LL (ALT r$_1$ r$_2$)} & \bl{$\dn$} & \onslide<4->{\bl{\LL (r$_1$) $\cup$ \LL (r$_2$)}}\\+
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  \bl{\LL (SEQ r$_1$ r$_2$)} & \bl{$\dn$} & \onslide<6->{\bl{\LL (r$_1$) ; \LL (r$_2$)}}\\+
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  \bl{\LL (STAR r)}          & \bl{$\dn$} & \onslide<8->{\bl{(\LL (r))$^\star$}}\\+
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  \end{tabular}+
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  \end{center}}+
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+
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  \only<5-6>{+
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  \begin{textblock}{6}(2.9,13.3)+
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  \colorbox{cream}{\bl{S$_1$ ; S$_2$ $\;\dn\;$ \{ s$_1$@s$_2$ $|$ s$_1$$\in$S$_1$ $\wedge$+
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                                                          s$_2$$\in$S$_2$ \}}}+
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  \end{textblock}}+
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+
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  \only<7->{+
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  \begin{textblock}{9}(4,13)+
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  \colorbox{cream}{\bl{$\infer{\mbox{[]} \in \mbox{S}^\star}{}$}}\hspace{3mm}+
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  \colorbox{cream}{\bl{$\infer{\mbox{s}_1\mbox{@}\mbox{s}_2 \in \mbox{S}^\star}+
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                      {\mbox{s}_1 \in \mbox{S} & \mbox{s}_2 \in \mbox{S}^\star}$}}+
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  \end{textblock}}+
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  +
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  \end{frame}}+
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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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  \mode<presentation>{+
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  \begin{frame}<1->[t]+
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  \frametitle{Is the Matcher Error-Free?}+
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+
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  We expect that+
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+
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  \begin{center}+
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  \begin{tabular}{lcl}+
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  \bl{matches r s = true}  & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}% +
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  \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\in$ \LL(r)}\\+
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  \bl{matches r s = false} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}%+
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  \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\notin$ \LL(r)}\\+
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  \end{tabular}+
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  \end{center}+
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  \pause\pause\bigskip+
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  By \alert<4->{induction}, we can {\bf prove} these properties.\bigskip+
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+
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  \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)}\\+
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  \end{tabular}+
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  +
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  \only<4->{+
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  \begin{textblock}{3}(0.9,4.5)+
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  \rd{\huge$\forall$\large{}r s.}+
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  \end{textblock}}+
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  \end{frame}}+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
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*}+
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+
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text_raw {*+
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  \mode<presentation>{+
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  \begin{frame}<1->[t]+
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+
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  \mbox{}\\[-2mm]+
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+
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  \small+
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  \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$_2$)} & \\ +
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  \bl{nullable (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) andalso (nullable r$_2$)} & \\+
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  \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+
−
+
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  \bl{matches r s $=$ nullable (derivative r s)}+
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  +
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  \only<2>{+
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  \begin{textblock}{8}(1.5,4)+
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  \includegraphics[scale=0.3]{approved.png}+
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  \end{textblock}}+
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  +
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+
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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{Interlude: TCB}+
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+
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  \begin{itemize}+
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  \item The \alert{\bf Trusted Code Base} (TCB) is the code that can make your +
−
  program behave in unintended ways (i.e.~cause bugs).\medskip+
−
+
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  \item Typically the TCB includes: CPUs, operating systems, C-libraries,+
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  device drivers, (J)VMs\ldots\bigskip+
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  \pause+
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+
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  \item It also includes the compiler.+
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  \end{itemize}+
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+
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  \end{frame}}+
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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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  \begin{frame}<1-3>+
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  \frametitle{\LARGE\begin{tabular}{c}Hacking Compilers +
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  \end{tabular}}+
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  +
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  %Why is it so paramount to have a small trusted code base (TCB)?+
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  \bigskip\bigskip+
−
+
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  \begin{columns}+
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  \begin{column}{2.7cm}+
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  \begin{minipage}{2.5cm}%+
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  \begin{tabular}{c@ {}}+
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  \includegraphics[scale=0.2]{ken-thompson.jpg}\\[-1.8mm]+
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  \footnotesize Ken Thompson\\[-1.8mm]+
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  \footnotesize Turing Award, 1983\\+
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  \end{tabular}+
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  \end{minipage}+
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  \end{column}+
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  \begin{column}{9cm}+
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  \begin{tabular}{l@ {\hspace{1mm}}p{8cm}}+
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  \myitemi+
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  & Ken Thompson showed how to hide a Trojan Horse in a +
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  compiler \textcolor{red}{without} leaving any traces in the source code.\\[2mm]+
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  \myitemi+
−
  & No amount of source level verification will protect +
−
  you from such Thompson-hacks.\\[2mm]+
−
+
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  \myitemi+
−
  & Therefore in safety-critical systems it is important to rely +
−
  on only a very small TCB.+
−
  \end{tabular}+
−
  \end{column}+
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  \end{columns}+
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+
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  \only<2>{+
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  \begin{textblock}{6}(4,2)+
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  \begin{tikzpicture}+
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  \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] +
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  {\normalsize+
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  \begin{minipage}{8cm}+
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  \begin{quote}+
−
  \includegraphics[scale=0.05]{evil.png}+
−
  \begin{enumerate}+
−
  \item[1)] Assume you ship the compiler as binary and also with sources.+
−
  \item[2)] Make the compiler aware when it compiles itself.+
−
  \item[3)] Add the Trojan horse.+
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  \item[4)] Compile.+
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  \item[5)] Delete Trojan horse from the sources of the compiler.+
−
  \item[6)] Go on holiday for the rest of your life. ;o)\\[-7mm]\mbox{}+
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  \end{enumerate}+
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  \end{quote}+
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  \end{minipage}};+
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  \end{tikzpicture}+
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text_raw {*+
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  \frametitle{\LARGE\begin{tabular}{c}An Example: Small TCB for\\[-2mm] +
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  A Critical Infrastructure\end{tabular}}+
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  \mbox{}\\[-14mm]\mbox{}+
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+
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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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  \end{column}+
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  +
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  \begin{column}{0.8\textwidth}+
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  \begin{textblock}{10}(4.5,3.95)+
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  \begin{block}{Proof-Carrying Code}+
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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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  +
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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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+
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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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  +
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     {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\bf\centering proof- checker\end{tabular}};}+
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+
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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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+
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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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  % Code Developer+
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  % User (runs untrusted code)+
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  % transmits a proof that the code is safe+
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  % +
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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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  \end{columns}+
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  +
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  \small\mbox{}\\[2.5cm]+
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  \begin{itemize}+
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  \item<4-> TCB of the 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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+
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  \end{frame}}+
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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     +
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  \tikzstyle{every node}=[node distance=25mm,text height=1.5ex, text depth=.25ex]+
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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}[squeeze]+
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  \frametitle{Type-Checking in LF} +
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  +
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  \begin{columns}+
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  \begin{column}{0.2\textwidth}+
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  \begin{tabular}{@ {\hspace{-4mm}}c@ {}}+
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  \\[-4mm]+
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  \includegraphics[scale=0.1]{harper.jpg}\\[-2mm] +
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  {\footnotesize Bob Harper}\\[-2.5mm]+
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  {\footnotesize (CMU)}\\[2mm]+
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+
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  \includegraphics[scale=0.3]{pfenning.jpg}\\[-2mm] +
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  {\footnotesize Frank Pfenning}\\[-2.5mm]+
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  {\footnotesize (CMU)}\\[2mm]+
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+
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  \onslide<-6>{+
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  {\footnotesize 31 pages in }\\[-2.5mm]+
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  {\footnotesize ACM Transact.~on}\\[-2.5mm]+
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  {\footnotesize Comp.~Logic.,~2005}\\}+
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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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    +
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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 (alg2)   [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\+
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     +
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+
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  \begin{block}{}+
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  \small Each time one needs to check $\sim$31pp~of informal paper proofs.+
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  You have to be able to keep definitions and proofs consistent.+
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+
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  \begin{itemize}+
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  \item Theorem provers help with keeping large proofs consistent;+
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  make them modifiable.\medskip+
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  +
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  \item They can ensure that all cases are covered.\medskip+
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+
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  \item Sometimes, tedious reasoning can be automated.+
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  \frametitle{Theorem Provers}+
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+
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  \begin{itemize}+
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  \item You also pay a (sometimes heavy) price: reasoning can be much, much harder.\medskip+
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+
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  \item Depending on your domain, suitable reasoning infrastructure+
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  might not yet be available.+
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  \frametitle{Theorem Provers}+
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+
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  Recently impressive work has been accomplished proving the correctness+
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+
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  \begin{itemize}+
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  \item of a compiler for C-light (compiled code has the same observable+
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  behaviour as the original source code),\medskip+
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+
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  \item a mirco-kernel operating system (absence of certain +
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  bugs\ldots no nil pointers, no buffer overflows).+
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  \frametitle{Trust in Theorem Provers}+
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+
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  Why should we trust theorem provers? +
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  +
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   \begin{itemize}+
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  \item Theorem provers are a \textcolor{red}{special kind} of software.+
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  +
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  \item We do \textcolor{red}{\bf not} need to trust them; we only need to trust:+
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  \end{itemize}+
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+
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  \begin{quote}+
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  \begin{itemize}+
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  \item The logic they are based on \textcolor{gray}{(e.g.~HOL)}, and\smallskip+
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  \item a proof checker that checks the proofs+
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  \textcolor{gray}{(this can be a very small program)}.\smallskip\pause+
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  \item To a little extend, we also need to trust our definitions+
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  \textcolor{gray}{(this can be mitigated)}.+
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  \end{itemize}+
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  \end{quote}+
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  +
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  \begin{itemize}+
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  \item I am using the Isabelle theorem prover (development since 1990).\bigskip\bigskip\bigskip+
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  +
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  \item It follows the LCF-approach:+
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  +
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  \begin{itemize}+
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  \item Have a special abstract type \alert{\bf thm}.+
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  \item Make the constructors of this abstract type the inference rules +
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  of your logic.+
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  \item Implement the theorem prover in a strongly-typed language (e.g.~ML).+
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  \end{itemize}+
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+
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  $\Rightarrow$ everything of type {\bf thm} has been proved (even if we do not+
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  have to explicitly generate proofs).+
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  \end{itemize}+
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  +
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  \only<1>{+
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  \begin{textblock}{5}(11,2.3)+
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  \begin{center}+
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  \includegraphics[scale=0.18]{robin-milner.jpg}\\[-0.8mm]+
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  \footnotesize Robin Milner\\[-0.8mm]+
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  \footnotesize Turing Award, 1991\\+
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  \end{center}+
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  \end{textblock}}+
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  \mbox{}\\[23mm]+
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  \LARGE Demo+
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  \end{tabular}}+
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  \frametitle{Future Research}+
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  +
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  \begin{itemize}+
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  \item Make theorem provers more like a programming environment.\medskip\pause +
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+
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  \item Use all the computational power we get from the hardware to+
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  automate reasoning (GPUs).\medskip\pause+
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+
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  \item Provide a comprehensive reasoning infrastructure for many domains and +
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  design automated decision procedures. +
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  \end{itemize}\pause+
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+
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  +
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  \begin{center}+
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  \colorbox{cream}{+
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  \begin{minipage}{10cm} +
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  \color{gray}+
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  \small+
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  ``Formal methods will never have a significant impact until+
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   they can be used by people that don't understand them.''\smallskip\\+
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  \mbox{}\footnotesize\hfill attributed to Tom Melham+
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  \end{minipage}}+
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  \begin{frame}<1->[c]+
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  \frametitle{Conclusion}+
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  +
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  \begin{itemize}+
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  \item The plan is to make this kind of programming the ``future''.\medskip\pause+
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+
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  \item Though the technology is already there\\ (compiler + micro-kernel os).\medskip\pause+
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+
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  \item Logic and reasoning (especially induction) are important skills for +
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  Computer Scientists.+
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+
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  \begin{tabular}{c}+
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  \mbox{}\\[23mm]+
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  \alert{\LARGE Thank you very much!}\\+
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  \alert{\Large Questions?}+
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  \end{tabular}}+
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  \end{frame}}+
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*}+
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+
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+
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(*<*)+
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end+
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(*>*)+
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