(*<*)
theory Slides4
imports "LaTeXsugar" "Nominal"
begin
notation (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}}
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\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}}
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*}
text_raw {*
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\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 {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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 {*
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\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 {*
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\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}}
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*}
text_raw {*
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\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 {*
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\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 {*
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\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}}
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*}
text_raw {*
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\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 {*
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\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}}
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*}
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}}
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*}
text_raw {*
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\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}}
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*}
text_raw {*
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\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}}
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*}
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}}
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*}
text_raw {*
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\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);
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{\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 {*
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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)
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\begin{tikzpicture}
\node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{2h};
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\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
(*>*)