the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
(*<*)theory Slides4imports "LaTeXsugar" "Nominal"beginnotation (latex output) set ("_") and Cons ("_::/_" [66,65] 65) (*>*)text_raw {* \renewcommand{\slidecaption}{Nanjing, 31.~August 2010} \newcommand{\abst}[2]{#1.#2}% atom-abstraction \newcommand{\pair}[2]{\langle #1,#2\rangle} % pairing \newcommand{\susp}{{\boldsymbol{\cdot}}}% for suspensions \newcommand{\unit}{\langle\rangle}% unit \newcommand{\app}[2]{#1\,#2}% application \newcommand{\eqprob}{\mathrel{{\approx}?}} \newcommand{\freshprob}{\mathrel{\#?}} \newcommand{\redu}[1]{\stackrel{#1}{\Longrightarrow}}% reduction \newcommand{\id}{\varepsilon}% identity substitution \newcommand{\bl}[1]{\textcolor{blue}{#1}} \newcommand{\gr}[1]{\textcolor{gray}{#1}} \newcommand{\rd}[1]{\textcolor{red}{#1}} \newcommand{\ok}{\includegraphics[scale=0.07]{ok.png}} \newcommand{\notok}{\includegraphics[scale=0.07]{notok.png}} \newcommand{\largenotok}{\includegraphics[scale=1]{notok.png}} \renewcommand{\Huge}{\fontsize{61.92}{77}\selectfont} \newcommand{\veryHuge}{\fontsize{74.3}{93}\selectfont} \newcommand{\VeryHuge}{\fontsize{89.16}{112}\selectfont} \newcommand{\VERYHuge}{\fontsize{107}{134}\selectfont} \newcommand{\LL}{$\mathbb{L}\,$} \pgfdeclareradialshading{smallbluesphere}{\pgfpoint{0.5mm}{0.5mm}}% {rgb(0mm)=(0,0,0.9); rgb(0.9mm)=(0,0,0.7); rgb(1.3mm)=(0,0,0.5); rgb(1.4mm)=(1,1,1)} \def\myitemi{\begin{pgfpicture}{-1ex}{-0.55ex}{1ex}{1ex} \usebeamercolor[fg]{subitem projected} {\pgftransformscale{0.8}\pgftext{\normalsize\pgfuseshading{bigsphere}}} \pgftext{% \usebeamerfont*{subitem projected}} \end{pgfpicture}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1>[t] \frametitle{% \begin{tabular}{@ {\hspace{-3mm}}c@ {}} \\ \huge Error-Free Programming\\[-1mm] \huge with Theorem Provers\\[5mm] \end{tabular}} \begin{center} Christian Urban \end{center} \begin{center} \small Technical University of Munich, Germany\\[7mm] \small in Nanjing on the kind invitation of\\ Professor Xingyuan Zhang and his group \end{center} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[c] \frametitle{My Background} \begin{itemize} \item researcher in Theoretical Computer Science\medskip \item programmer on a \alert<2->{software system} with 800 kloc (though I am responsible only for 35 kloc) \end{itemize} \only<2->{ \begin{textblock}{6}(2,11) \begin{tikzpicture} \draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] {\color{darkgray} \begin{minipage}{4cm}\raggedright A theorem prover called {\bf Isabelle}. \end{minipage}}; \end{tikzpicture} \end{textblock}} \only<3>{ \begin{textblock}{6}(9,11) \begin{tikzpicture} \draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] {\color{darkgray} \begin{minipage}{4cm}\raggedright Like every other code, this code is very hard to get correct. \end{minipage}}; \end{tikzpicture} \end{textblock}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[t] \frametitle{Regular Expressions} An example many (should) know about:\\ \rd{\bf Regular Expressions:} \only<2>{ \begin{center} \bl{[] $\;\;\;|\;\;\;$ c $\;\;\;|\;\;\;$ r$_1$$|$r$_2$ $\;\;\;|\;\;\;$ r$_1$$\cdot$r$_2$ $\;\;\;|\;\;\;$ r$^*$} \end{center}} \only<3->{ \begin{center} \begin{tabular}{@ {}rrll@ {}} \bl{r} & \bl{$::=$} & \bl{NULL} & \gr{(matches no string)}\\ & \bl{$\mid$} & \bl{EMPTY} & \gr{(matches the empty string, [])}\\ & \bl{$\mid$} & \bl{CHR c} & \gr{(matches the character c)}\\ & \bl{$\mid$} & \bl{ALT r$_1$ r$_2$} & \gr{(alternative, r$_1 |\,$r$_2$)}\\ & \bl{$\mid$} & \bl{SEQ r$_1$ r$_2$} & \gr{(sequential, r$_1\cdot\,$r$_2$)}\\ & \bl{$\mid$} & \bl{STAR r} & \gr{(repeat, r$^*$)}\\ \end{tabular} \end{center}\medskip} \small \begin{textblock}{14.5}(1,12.5) \only<2->{\gr{(a$\cdot$b)$^*$ \hspace{3mm}$\mapsto$\hspace{3mm} \{[], ab, abab, ababab, \ldots\}}\\} \only<2->{\gr{x$\cdot$(0 $|$ 1 $|$ 2 \ldots 8 $|$ 9)$^*$ \hspace{3mm}$\mapsto$\hspace{3mm} \{x, x0, x1, \ldots, x00, \ldots, x123, \ldots\}}} \end{textblock} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1>[c] \frametitle{RegExp Matcher} Let's implement a regular expression matcher:\bigskip \begin{center} \begin{tikzpicture} %%\draw[help lines, black] (-3,0) grid (6,3); \draw[line width=1mm, red] (0.0,0.0) rectangle (4,2.3); \node[anchor=base] at (2,1) {\small\begin{tabular}{@ {}c@ {}}\Large\bf Regular\\ \Large\bf Expression \\ \Large\bf Matcher\end{tabular}}; \coordinate (m1) at (0,1.5); \draw (-2,2) node (m2) {\small\begin{tabular}{c}\bl{regular}\\[-1mm] \bl{expression}\end{tabular}}; \path[overlay, ->, line width = 1mm, shorten <=-3mm] (m2) edge (m1); \coordinate (s1) at (0,0.5); \draw (-1.8,-0) node (s2) {\small\begin{tabular}{c}\bl{string}\end{tabular}}; \path[overlay, ->, line width = 1mm, shorten <=-3mm] (s2) edge (s1); \coordinate (r1) at (4,1.2); \draw (6,1.2) node (r2) {\small\begin{tabular}{c}\bl{true}, \bl{false}\end{tabular}}; \path[overlay, ->, line width = 1mm, shorten >=-3mm] (r1) edge (r2); \end{tikzpicture} \end{center} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[t] \frametitle{RegExp Matcher} \small {\bf input:} a \underline{list} of RegExps and a string \hspace{6mm}{\bf output:} true or false \only<2->{ \begin{center} \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}} \bl{match [] []} & \bl{$=$} & \bl{true}\\ \bl{match [] \_} & \bl{$=$} & \bl{false}\\ \bl{match (NULL::rs) s} & \bl{$=$} & \bl{false}\\ \bl{match (EMPTY::rs) s} & \bl{$=$} & \bl{match rs s}\\ \bl{match (CHR c::rs) (c::s)} & \bl{$=$} & \bl{match rs s}\\ \bl{match (CHR c::rs) \_} & \bl{$=$} & \bl{false}\\ \bl{match (ALT r$_1$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s}\\ & & \bl{\;\;\;\;orelse match (r$_2$::rs) s}\\ \bl{match (SEQ r$_1$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\ \bl{match (STAR r::rs) s} & \bl{$=$} & \bl{match rs s}\\ & & \bl{\;\;\;\;orelse match (r::STAR r::rs) s}\\ \end{tabular} \end{center}} \onslide<3->{we start the program with\\ \hspace{6mm}\bl{matches r s $=$ match [r] s}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1>[c] \frametitle{Program in Scala} \bl{\footnotesize \begin{tabular}{l} sealed abstract class Rexp\\ sealed case class Null extends Rexp\\ sealed case class Empty extends Rexp\\ sealed case class Chr(c: Char) extends Rexp\\ sealed case class Alt(r1 : Rexp, r2 : Rexp) extends Rexp\\ sealed case class Seq(r1 : Rexp, r2 : Rexp) extends Rexp\\ sealed case class Star(r : Rexp) extends Rexp\medskip\\ def match1 (rs : List[Rexp], s : List[Char]) : Boolean = rs match \{\\ \hspace{3mm}case Nil @{text "\<Rightarrow>"} if (s == Nil) true else false\\ \hspace{3mm}case (Null()::rs) @{text "\<Rightarrow>"} false\\ \hspace{3mm}case (Empty()::rs) @{text "\<Rightarrow>"} match1 (rs, s)\\ \hspace{3mm}case (Chr(c)::rs) @{text "\<Rightarrow>"} s match \\ \hspace{6mm}\{ case Nil @{text "\<Rightarrow>"} false\\ \hspace{8mm}case (d::s) @{text "\<Rightarrow>"} if (c==d) match1 (rs,s) else false \}\\ \hspace{3mm}case (Alt (r1, r2)::rs) @{text "\<Rightarrow>"} match1 (r1::rs, s) || match1 (r2::rs, s)\\ \hspace{3mm}case (Seq (r1, r2)::rs) @{text "\<Rightarrow>"} match1 (r1::r2::rs, s) \\ \hspace{3mm}case (Star (r)::rs) @{text "\<Rightarrow>"} match1 (r::rs, s) || match1 (r::Star (r)::rs, s)\\ \} \end{tabular}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[c] \frametitle{Testing} \small Every good programmer should do thourough tests: \begin{center} \begin{tabular}{@ {\hspace{-20mm}}lcl} \bl{matches (a$\cdot$b)$^*\;$ []} & \bl{$\mapsto$} & \bl{true}\\ \bl{matches (a$\cdot$b)$^*\;$ ab} & \bl{$\mapsto$} & \bl{true}\\ \bl{matches (a$\cdot$b)$^*\;$ aba} & \bl{$\mapsto$} & \bl{false}\\ \bl{matches (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\ \bl{matches (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\ \onslide<2->{\bl{matches x$\cdot$(0$|$1)$^*\;$ x} & \bl{$\mapsto$} & \bl{true}}\\ \onslide<2->{\bl{matches x$\cdot$(0$|$1)$^*\;$ x0} & \bl{$\mapsto$} & \bl{true}}\\ \onslide<2->{\bl{matches x$\cdot$(0$|$1)$^*\;$ x3} & \bl{$\mapsto$} & \bl{false}} \end{tabular} \end{center} \onslide<3-> {looks OK \ldots let's ship it to customers\hspace{5mm} \raisebox{-5mm}{\includegraphics[scale=0.05]{sun.png}}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[t] \frametitle{Testing} \begin{itemize} \item While testing is an important part in the process of programming development\pause \item we can only test a {\bf finite} amount of examples\bigskip\pause \begin{center} \colorbox{cream} {\gr{\begin{minipage}{10cm} ``Testing can only show the presence of errors, never their absence'' (Edsger W.~Dijkstra) \end{minipage}}} \end{center}\bigskip\pause \item In a theorem prover we can establish properties that apply to {\bf all} input and {\bf all} output.\pause \item For example we can establish that the matcher terminates on all input. \end{itemize} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[t] \frametitle{RegExp Matcher} \small We need to find a measure that gets smaller in each recursive call.\bigskip \onslide<1->{ \begin{center} \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-9mm}}l@ {}} \bl{match [] []} & \bl{$=$} & \bl{true} & \onslide<2->{\ok}\\ \bl{match [] \_} & \bl{$=$} & \bl{false} & \onslide<2->{\ok}\\ \bl{match (NULL::rs) s} & \bl{$=$} & \bl{false} & \onslide<2->{\ok}\\ \bl{match (EMPTY::rs) s} & \bl{$=$} & \bl{match rs s} & \onslide<3->{\ok}\\ \bl{match (CHR c::rs) (c::s)} & \bl{$=$} & \bl{match rs s} & \onslide<4->{\ok}\\ \bl{match (CHR c::rs) \_} & \bl{$=$} & \bl{false} & \onslide<2->{\ok}\\ \bl{match (ALT r$_1$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s} & \onslide<5->{\ok}\\ & & \bl{\;\;\;\;orelse match (r$_2$::rs) s}\\ \bl{match (SEQ r$_1$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s} & \onslide<6->{\ok}\\ \bl{match (STAR r::rs) s} & \bl{$=$} & \bl{match rs s} & \onslide<7->{\notok}\\ & & \bl{\;\;\;\;orelse match (r::STAR r::rs) s}\\ \end{tabular} \end{center}} \begin{textblock}{5}(4,4) \begin{tikzpicture} %%\draw[help lines, black] (-3,0) grid (6,3); \coordinate (m1) at (-2,0); \coordinate (m2) at ( 2,0); \path[overlay, ->, line width = 0.6mm, color = red] (m1) edge (m2); \draw (0,0) node[above=-1mm] {\footnotesize\rd{needs to get smaller}}; \end{tikzpicture} \end{textblock} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[c] \frametitle{Bug Hunting} \only<1>{Several hours later\ldots}\pause \begin{center} \begin{tabular}{@ {\hspace{-20mm}}lcl} \bl{matches (STAR (EMPTY)) s} & \bl{$\mapsto$} & loops\\ \onslide<4->{\bl{matches (STAR (EMPTY $|$ \ldots)) s} & \bl{$\mapsto$} & loops\\} \end{tabular} \end{center} \small \onslide<3->{ \begin{center} \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}} \ldots\\ \bl{match (EMPTY::rs) s} & \bl{$=$} & \bl{match rs s}\\ \ldots\\ \bl{match (STAR r::rs) s} & \bl{$=$} & \bl{match rs s}\\ & & \bl{\;\;\;\;orelse match (r::STAR r::rs) s}\\ \end{tabular} \end{center}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[c] \frametitle{RegExp Matcher} \small \begin{center} \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}} \bl{match [] []} & \bl{$=$} & \bl{true}\\ \bl{match [] \_} & \bl{$=$} & \bl{false}\\ \bl{match (NULL::rs) s} & \bl{$=$} & \bl{false}\\ \bl{match (EMPTY::rs) s} & \bl{$=$} & \bl{match rs s}\\ \bl{match (CHR c::rs) (c::s)} & \bl{$=$} & \bl{match rs s}\\ \bl{match (CHR c::rs) \_} & \bl{$=$} & \bl{false}\\ \bl{match (ALT r$_1$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s}\\ & & \bl{\;\;\;\;orelse match (r$_2$::rs) s}\\ \bl{match (SEQ r$_1$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\ \bl{match (STAR r::rs) s} & \bl{$=$} & \bl{match rs s}\\ & & \bl{\;\;\;\;orelse match (r::STAR r::rs) s}\\ \end{tabular} \end{center} \only<1>{ \begin{textblock}{5}(4,4) \largenotok \end{textblock}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[t] \frametitle{Second Attempt} Can a regular expression match the empty string? \small \begin{center} \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}} \bl{nullable (NULL)} & \bl{$=$} & \bl{false} & \onslide<2->{\ok}\\ \bl{nullable (EMPTY)} & \bl{$=$} & \bl{true} & \onslide<2->{\ok}\\ \bl{nullable (CHR c)} & \bl{$=$} & \bl{false} & \onslide<2->{\ok}\\ \bl{nullable (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) orelse (nullable r$_2$)} & \onslide<2->{\ok}\\ \bl{nullable (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) andalso (nullable r$_2$)} & \onslide<2->{\ok}\\ \bl{nullable (STAR r)} & \bl{$=$} & \bl{true} & \onslide<2->{\ok}\\ \end{tabular} \end{center} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[t] \frametitle{RegExp Matcher 2} If \bl{r} matches \bl{c::s}, which regular expression can match the string \bl{s}? \small \begin{center} \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} \bl{der c (NULL)} & \bl{$=$} & \bl{NULL} & \onslide<3->{\ok}\\ \bl{der c (EMPTY)} & \bl{$=$} & \bl{NULL} & \onslide<3->{\ok}\\ \bl{der c (CHR d)} & \bl{$=$} & \bl{if c=d then EMPTY else NULL} & \onslide<3->{\ok}\\ \bl{der c (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (der c r$_1$) (der c r$_2$)} & \onslide<3->{\ok}\\ \bl{der c (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (SEQ (der c r$_1$) r$_2$)} & \onslide<3->{\ok}\\ & & \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)} & \onslide<3->{\ok}\medskip\\ \pause \bl{derivative r []} & \bl{$=$} & \bl{r} & \onslide<3->{\ok}\\ \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \onslide<3->{\ok}\\ \end{tabular} \end{center} we call the program with\\ \bl{matches r s $=$ nullable (derivative r s)} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[c] \frametitle{But Now What?} \begin{center} {\usefont{T1}{ptm}{b}{N}\VERYHuge{\rd{?}}} \end{center} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[c] \frametitle{Testing} \small \begin{center} \begin{tabular}{@ {\hspace{-20mm}}lcl} \bl{matches []$^*$ []} & \bl{$\mapsto$} & \bl{true}\\ \bl{matches ([]$|$a)$^*$ a} & \bl{$\mapsto$} & \bl{true}\medskip\\ \bl{matches (a$\cdot$b)$^*\;$ []} & \bl{$\mapsto$} & \bl{true}\\ \bl{matches (a$\cdot$b)$^*\;$ ab} & \bl{$\mapsto$} & \bl{true}\\ \bl{matches (a$\cdot$b)$^*\;$ aba} & \bl{$\mapsto$} & \bl{false}\\ \bl{matches (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\ \bl{matches (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\ \bl{matches x$\cdot$(0$|$1)$^*\;$ x} & \bl{$\mapsto$} & \bl{true}\\ \bl{matches x$\cdot$(0$|$1)$^*\;$ x0} & \bl{$\mapsto$} & \bl{true}\\ \bl{matches x$\cdot$(0$|$1)$^*\;$ x3} & \bl{$\mapsto$} & \bl{false} \end{tabular} \end{center} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[t] \frametitle{Specification} We have to specify what it means for a regular expression to match a string. % \only<2>{ \mbox{}\\[8mm] \bl{(a$\cdot$b)$^*$}\\ \hspace{7mm}\bl{$\mapsto$\hspace{3mm}\{[], ab, abab, ababab, \ldots\}}\bigskip\\ \bl{x$\cdot$(0 $|$ 1 $|$ 2 \ldots 8 $|$ 9 )$^*$}\\ \hspace{7mm}\bl{$\mapsto$\hspace{3mm} \{x, x0, x1, \ldots, x00, \ldots, x123, \ldots\}}} % \only<3->{ \begin{center} \begin{tabular}{rcl} \bl{\LL (NULL)} & \bl{$\dn$} & \bl{\{\}}\\ \bl{\LL (EMPTY)} & \bl{$\dn$} & \bl{\{[]\}}\\ \bl{\LL (CHR c)} & \bl{$\dn$} & \bl{\{c\}}\\ \bl{\LL (ALT r$_1$ r$_2$)} & \bl{$\dn$} & \onslide<4->{\bl{\LL (r$_1$) $\cup$ \LL (r$_2$)}}\\ \bl{\LL (SEQ r$_1$ r$_2$)} & \bl{$\dn$} & \onslide<6->{\bl{\LL (r$_1$) ; \LL (r$_2$)}}\\ \bl{\LL (STAR r)} & \bl{$\dn$} & \onslide<8->{\bl{(\LL (r))$^\star$}}\\ \end{tabular} \end{center}} \only<5-6>{ \begin{textblock}{6}(2.9,13.3) \colorbox{cream}{\bl{S$_1$ ; S$_2$ $\;\dn\;$ \{ s$_1$@s$_2$ $|$ s$_1$$\in$S$_1$ $\wedge$ s$_2$$\in$S$_2$ \}}} \end{textblock}} \only<7->{ \begin{textblock}{9}(4,13) \colorbox{cream}{\bl{$\infer{\mbox{[]} \in \mbox{S}^\star}{}$}}\hspace{3mm} \colorbox{cream}{\bl{$\infer{\mbox{s}_1\mbox{@}\mbox{s}_2 \in \mbox{S}^\star} {\mbox{s}_1 \in \mbox{S} & \mbox{s}_2 \in \mbox{S}^\star}$}} \end{textblock}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[t] \frametitle{Is the Matcher Error-Free?} We expect that \begin{center} \begin{tabular}{lcl} \bl{matches r s = true} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}% \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\in$ \LL(r)}\\ \bl{matches r s = false} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}% \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\notin$ \LL(r)}\\ \end{tabular} \end{center} \pause\pause\bigskip By \alert<4->{induction}, we can {\bf prove} these properties.\bigskip \begin{tabular}{lrcl} Lemmas: & \bl{nullable (r)} & \bl{$\Longleftrightarrow$} & \bl{[] $\in$ \LL (r)}\\ & \bl{s $\in$ \LL (der c r)} & \bl{$\Longleftrightarrow$} & \bl{(c::s) $\in$ \LL (r)}\\ \end{tabular} \only<4->{ \begin{textblock}{3}(0.9,4.5) \rd{\huge$\forall$\large{}r s.} \end{textblock}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[t] \mbox{}\\[-2mm] \small \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}} \bl{nullable (NULL)} & \bl{$=$} & \bl{false} &\\ \bl{nullable (EMPTY)} & \bl{$=$} & \bl{true} &\\ \bl{nullable (CHR c)} & \bl{$=$} & \bl{false} &\\ \bl{nullable (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) orelse (nullable r$_2$)} & \\ \bl{nullable (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) andalso (nullable r$_2$)} & \\ \bl{nullable (STAR r)} & \bl{$=$} & \bl{true} & \\ \end{tabular}\medskip \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} \bl{der c (NULL)} & \bl{$=$} & \bl{NULL} & \\ \bl{der c (EMPTY)} & \bl{$=$} & \bl{NULL} & \\ \bl{der c (CHR d)} & \bl{$=$} & \bl{if c=d then EMPTY else NULL} & \\ \bl{der c (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (der c r$_1$) (der c r$_2$)} & \\ \bl{der c (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (SEQ (der c r$_1$) r$_2$)} & \\ & & \bl{\phantom{ALT} (if nullable r$_1$ then der c r$_2$ else NULL)}\\ \bl{der c (STAR r)} & \bl{$=$} & \bl{SEQ (der c r) (STAR r)} &\smallskip\\ \bl{derivative r []} & \bl{$=$} & \bl{r} & \\ \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\ \end{tabular}\medskip \bl{matches r s $=$ nullable (derivative r s)} \only<2>{ \begin{textblock}{8}(1.5,4) \includegraphics[scale=0.3]{approved.png} \end{textblock}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[c] \frametitle{Interlude: TCB} \begin{itemize} \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 \item Typically the TCB includes: CPUs, operating systems, C-libraries, device drivers, (J)VMs\ldots\bigskip \pause \item It also includes the compiler. \end{itemize} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1-3> \frametitle{\LARGE\begin{tabular}{c}Hacking Compilers \end{tabular}} %Why is it so paramount to have a small trusted code base (TCB)? \bigskip\bigskip \begin{columns} \begin{column}{2.7cm} \begin{minipage}{2.5cm}% \begin{tabular}{c@ {}} \includegraphics[scale=0.2]{ken-thompson.jpg}\\[-1.8mm] \footnotesize Ken Thompson\\[-1.8mm] \footnotesize Turing Award, 1983\\ \end{tabular} \end{minipage} \end{column} \begin{column}{9cm} \begin{tabular}{l@ {\hspace{1mm}}p{8cm}} \myitemi & Ken Thompson showed how to hide a Trojan Horse in a compiler \textcolor{red}{without} leaving any traces in the source code.\\[2mm] \myitemi & No amount of source level verification will protect you from such Thompson-hacks.\\[2mm] \myitemi & Therefore in safety-critical systems it is important to rely on only a very small TCB. \end{tabular} \end{column} \end{columns} \only<2>{ \begin{textblock}{6}(4,2) \begin{tikzpicture} \draw (0,0) node[inner sep=3mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] {\normalsize \begin{minipage}{8cm} \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. \item[4)] Compile. \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{} \end{enumerate} \end{quote} \end{minipage}}; \end{tikzpicture} \end{textblock}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame} \frametitle{\LARGE\begin{tabular}{c}An Example: Small TCB for\\[-2mm] A Critical Infrastructure\end{tabular}} \mbox{}\\[-14mm]\mbox{} \begin{columns} \begin{column}{0.2\textwidth} \begin{tabular}{@ {} c@ {}} \includegraphics[scale=0.3]{appel.jpg}\\[-2mm] {\footnotesize Andrew Appel}\\[-2.5mm] {\footnotesize (Princeton)} \end{tabular} \end{column} \begin{column}{0.8\textwidth} \begin{textblock}{10}(4.5,3.95) \begin{block}{Proof-Carrying Code} \begin{center} \begin{tikzpicture} \draw[help lines,cream] (0,0.2) grid (8,4); \draw[line width=1mm, red] (5.5,0.6) rectangle (7.5,4); \node[anchor=base] at (6.5,2.8) {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\centering user needs to run untrusted code\end{tabular}}; \draw[line width=1mm, red] (0.5,0.6) rectangle (2.5,4); \node[anchor=base] at (1.5,2.3) {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\centering code developer/ web server/ Apple Store\end{tabular}}; \onslide<4->{ \draw[line width=1mm, red, fill=red] (5.5,0.6) rectangle (7.5,1.8); \node[anchor=base,white] at (6.5,1.1) {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\bf\centering proof- checker\end{tabular}};} \node at (3.8,3.0) [single arrow, fill=red,text=white, minimum height=3cm]{\bf code}; \onslide<3->{ \node at (3.8,1.3) [single arrow, fill=red,text=white, minimum height=3cm]{\bf LF proof}; \node at (3.8,1.9) {\small certificate}; } \onslide<2>{\node at (4.0,1.3) [text=red]{\begin{tabular}{c}\bf Highly\\\bf Dangerous!\end{tabular}};} % Code Developer % User (runs untrusted code) % transmits a proof that the code is safe % \end{tikzpicture} \end{center} \end{block} \end{textblock} \end{column} \end{columns} \small\mbox{}\\[2.5cm] \begin{itemize} \item<4-> TCB of the checker is $\sim$2700 lines of code (1865 loc of\\ LF definitions; 803 loc in C including 2 library functions)\\[-3mm] \item<5-> 167 loc in C implement a type-checker \end{itemize} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text {* \tikzstyle{every node}=[node distance=25mm,text height=1.5ex, text depth=.25ex] \tikzstyle{node1}=[rectangle, minimum size=10mm, rounded corners=3mm, very thick, draw=black!50, top color=white, bottom color=black!20] \tikzstyle{node2}=[rectangle, minimum size=12mm, rounded corners=3mm, very thick, draw=red!70, top color=white, bottom color=red!50!black!20] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}[squeeze] \frametitle{Type-Checking in LF} \begin{columns} \begin{column}{0.2\textwidth} \begin{tabular}{@ {\hspace{-4mm}}c@ {}} \\[-4mm] \includegraphics[scale=0.1]{harper.jpg}\\[-2mm] {\footnotesize Bob Harper}\\[-2.5mm] {\footnotesize (CMU)}\\[2mm] \includegraphics[scale=0.3]{pfenning.jpg}\\[-2mm] {\footnotesize Frank Pfenning}\\[-2.5mm] {\footnotesize (CMU)}\\[2mm] \onslide<-6>{ {\footnotesize 31 pages in }\\[-2.5mm] {\footnotesize ACM Transact.~on}\\[-2.5mm] {\footnotesize Comp.~Logic.,~2005}\\} \end{tabular} \end{column} \begin{column}{0.8\textwidth} \begin{textblock}{0}(3.1,2) \begin{tikzpicture} \matrix[ampersand replacement=\&,column sep=7mm, row sep=5mm] { \&[-10mm] \node (def1) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}}; \& \node (proof1) [node1] {\large Proof}; \& \node (alg1) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}}; \\ \onslide<4->{\node {\begin{tabular}{c}\small 1st\\[-2.5mm] \footnotesize solution\end{tabular}};} \& \onslide<4->{\node (def2) [node2] {\large Spec$^\text{+ex}$};} \& \onslide<4->{\node (proof2) [node1] {\large Proof};} \& \onslide<4->{\node (alg2) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\ \onslide<5->{\node {\begin{tabular}{c}\small 2nd\\[-2.5mm] \footnotesize solution\end{tabular}};} \& \onslide<5->{\node (def3) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \& \onslide<5->{\node (proof3) [node1] {\large Proof};} \& \onslide<5->{\node (alg3) [node2] {\large Alg$^\text{-ex}$};} \\ \onslide<6->{\node {\begin{tabular}{c}\small 3rd\\[-2.5mm] \footnotesize solution\end{tabular}};} \& \onslide<6->{\node (def4) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \& \onslide<6->{\node (proof4) [node2] {\large\hspace{1mm}Proof\hspace{1mm}};} \& \onslide<6->{\node (alg4) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\ }; \draw[->,black!50,line width=2mm] (proof1) -- (def1); \draw[->,black!50,line width=2mm] (proof1) -- (alg1); \onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (def2);} \onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (alg2);} \onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (def3);} \onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (alg3);} \onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (def4);} \onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (alg4);} \onslide<3->{\draw[white,line width=1mm] (1.1,3.2) -- (0.9,2.85) -- (1.1,2.35) -- (0.9,2.0);} \end{tikzpicture} \end{textblock} \end{column} \end{columns} \only<2>{% \begin{textblock}{2}(.1,12.85) \begin{tikzpicture} \draw[line width=1mm, red] (0,0) ellipse (1.5cm and 0.88cm); \end{tikzpicture} \end{textblock} } \begin{textblock}{3}(14,3.6) \onslide<4->{ \begin{tikzpicture} \node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{2h}; \end{tikzpicture}} \end{textblock} \only<7->{ \begin{textblock}{14}(0.6,12.8) \begin{block}{} \small Each time one needs to check $\sim$31pp~of informal paper proofs. You have to be able to keep definitions and proofs consistent. \end{block} \end{textblock}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1>[c] \frametitle{Theorem Provers} \begin{itemize} \item Theorem provers help with keeping large proofs consistent; make them modifiable.\medskip \item They can ensure that all cases are covered.\medskip \item Sometimes, tedious reasoning can be automated. \end{itemize} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1>[c] \frametitle{Theorem Provers} \begin{itemize} \item You also pay a (sometimes heavy) price: reasoning can be much, much harder.\medskip \item Depending on your domain, suitable reasoning infrastructure might not yet be available. \end{itemize} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1>[c] \frametitle{Theorem Provers} Recently impressive work has been accomplished proving the correctness \begin{itemize} \item of a compiler for C-light (compiled code has the same observable behaviour as the original source code),\medskip \item a mirco-kernel operating system (absence of certain bugs\ldots no nil pointers, no buffer overflows). \end{itemize} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1>[c] \frametitle{Trust in Theorem Provers} \begin{center} Why should we trust theorem provers? \end{center} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame} \frametitle{Theorem Provers} \begin{itemize} \item Theorem provers are a \textcolor{red}{special kind} of software. \item We do \textcolor{red}{\bf not} need to trust them; we only need to trust: \end{itemize} \begin{quote} \begin{itemize} \item The logic they are based on \textcolor{gray}{(e.g.~HOL)}, and\smallskip \item a proof checker that checks the proofs \textcolor{gray}{(this can be a very small program)}.\smallskip\pause \item To a little extend, we also need to trust our definitions \textcolor{gray}{(this can be mitigated)}. \end{itemize} \end{quote} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame} \frametitle{Isabelle} \begin{itemize} \item I am using the Isabelle theorem prover (development since 1990).\bigskip\bigskip\bigskip \item It follows the LCF-approach: \begin{itemize} \item Have a special abstract type \alert{\bf thm}. \item Make the constructors of this abstract type the inference rules of your logic. \item Implement the theorem prover in a strongly-typed language (e.g.~ML). \end{itemize} $\Rightarrow$ everything of type {\bf thm} has been proved (even if we do not have to explicitly generate proofs). \end{itemize} \only<1>{ \begin{textblock}{5}(11,2.3) \begin{center} \includegraphics[scale=0.18]{robin-milner.jpg}\\[-0.8mm] \footnotesize Robin Milner\\[-0.8mm] \footnotesize Turing Award, 1991\\ \end{center} \end{textblock}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1>[c] \frametitle{ \begin{tabular}{c} \mbox{}\\[23mm] \LARGE Demo \end{tabular}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[c] \frametitle{Future Research} \begin{itemize} \item Make theorem provers more like a programming environment.\medskip\pause \item Use all the computational power we get from the hardware to automate reasoning (GPUs).\medskip\pause \item Provide a comprehensive reasoning infrastructure for many domains and design automated decision procedures. \end{itemize}\pause \begin{center} \colorbox{cream}{ \begin{minipage}{10cm} \color{gray} \small ``Formal methods will never have a significant impact until they can be used by people that don't understand them.''\smallskip\\ \mbox{}\footnotesize\hfill attributed to Tom Melham \end{minipage}} \end{center} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1->[c] \frametitle{Conclusion} \begin{itemize} \item The plan is to make this kind of programming the ``future''.\medskip\pause \item Though the technology is already there\\ (compiler + micro-kernel os).\medskip\pause \item Logic and reasoning (especially induction) are important skills for Computer Scientists. \end{itemize} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}text_raw {* %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode<presentation>{ \begin{frame}<1>[c] \frametitle{ \begin{tabular}{c} \mbox{}\\[23mm] \alert{\LARGE Thank you very much!}\\ \alert{\Large Questions?} \end{tabular}} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *}(*<*)end(*>*)