--- a/Slides/Slides6.thy Mon Apr 11 02:25:25 2011 +0100
+++ b/Slides/Slides6.thy Tue Apr 12 15:46:35 2011 +0800
@@ -77,351 +77,50 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
+
+
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1->[c]
\frametitle{My Background}
+ \mbox{}\\[-10mm]
\begin{itemize}
- \item researcher in Theoretical Computer Science\medskip
+ \item My background is in theory and programming languages.\bigskip
+ \pause
- \item programmer on a \alert<2->{software system} with 800 kloc (though I am
- responsible only for 35 kloc)
+ \item But I am also a programmer with a \alert<2>{software system} of around 800 kloc
+ (though I am responsible for only appr.~35 kloc),
+
+ \item and I write papers.
\end{itemize}
-
- \only<2->{
- \begin{textblock}{6}(2,11)
+
+ \only<2>{
+ \begin{textblock}{6}(6.5,11.5)
\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.
+ \begin{minipage}{6.5cm}\raggedright
+ \begin{tabular}[b]{@ {}p{4.5cm}c@ {}}
+ \raggedright
+ The software is a theorem prover, called {\bf Isabelle}.
+ & \mbox{}\hspace{-5mm}\raisebox{-14mm}{\includegraphics[scale=0.28]{isabelle1.png}}
+ \end{tabular}%
\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}
+
+ \only<4>{
+ \begin{textblock}{6}(3,11.5)
\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);
-
+ \draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm]
+ {\color{darkgray}
+ \begin{minipage}{9.6cm}\raggedright
+ So I can experience every day that writing error-free code is {\bf very, very hard}
+ and that papers are also {\bf hard} to get correct.
+ \end{minipage}};
\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}}
@@ -431,308 +130,65 @@
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{frame}
+ \frametitle{Getting Papers Correct}
+
+ \begin{minipage}{1.1\textwidth}
+ My work over the last 5 years.\\
+ {\small (in the fields of programming languages, logic and lambda-calculi)}
+ \end{minipage}\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]
+ \only<1>{
+ \mbox{}\\[15mm]
+ \begin{center}
+ \begin{tikzpicture}[node distance=0.5mm]
+ \node at (-1.0,-0.3) (proof) [double arrow, fill=gray,text=white, minimum height=2cm]{\bf Proof};
+ \node [left=of proof]{\Large\bf Specification};
+ \node [right=of proof]{\Large\bf Code};
+ \end{tikzpicture}
+ \end{center}
+ }
+ \pause
- \myitemi
- & Therefore in safety-critical systems it is important to rely
- on only a very small TCB.
+ \begin{tabular}{c@ {\hspace{2mm}}c}
+ \begin{tabular}{c}
+ \includegraphics[scale=0.09]{harper.jpg}\\[-2mm]
+ {\footnotesize Bob Harper}\\[-2.5mm]
+ {\footnotesize (CMU)}
\end{tabular}
- \end{column}
- \end{columns}
+ \begin{tabular}{c}
+ \includegraphics[scale=0.31]{pfenning.jpg}\\[-2mm]
+ {\footnotesize Frank Pfenning}\\[-2.5mm]
+ {\footnotesize (CMU)}
+ \end{tabular} &
+
+ \begin{tabular}{p{6cm}}
+ \raggedright\small
+ \color{gray}{published a proof in ACM Transactions on Computational Logic (2005),
+ $\sim$31pp}
+ \end{tabular}\\
- \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}}
+ \\[-4mm]
+
+ \begin{tabular}{c}
+ \includegraphics[scale=0.3]{appel.jpg}\\[-2mm]
+ {\footnotesize Andrew Appel}\\[-2.5mm]
+ {\footnotesize (Princeton)}
+ \end{tabular} &
+
+ \begin{tabular}{p{6cm}}
+ \raggedright\small
+ \color{gray}{relied on their proof in a safety critical system (proof carrying code)}
+ \end{tabular}
+
+ \end{tabular}\medskip
+
+
+
+
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -740,25 +196,14 @@
*}
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{}
+ \frametitle{Proof-Carrying Code}
- \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{textblock}{10}(2.5,2.2)
+ \begin{block}{Idea:}
\begin{center}
\begin{tikzpicture}
\draw[help lines,cream] (0,0.2) grid (8,4);
@@ -784,31 +229,26 @@
}
\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{textblock}{15}(2,12)
+ \small
\begin{itemize}
- \item<4-> TCB of the checker is $\sim$2700 lines of code (1865 loc of\\ LF definitions;
+ \item<4-> Appel's 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{textblock}
\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,
@@ -817,30 +257,13 @@
draw=red!70, top color=white, bottom color=red!50!black!20]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}[squeeze]
+ \begin{frame}<2->[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{textblock}{0}(1,2)
\begin{tikzpicture}
\matrix[ampersand replacement=\&,column sep=7mm, row sep=5mm]
@@ -884,15 +307,8 @@
\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)
+ \begin{textblock}{3}(12,3.6)
\onslide<4->{
\begin{tikzpicture}
\node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{2h};
@@ -912,10 +328,11 @@
*}
+
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1>[c]
+ \begin{frame}<1->[c]
\frametitle{Theorem Provers}
\begin{itemize}
@@ -924,9 +341,130 @@
\item They can ensure that all cases are covered.\medskip
- \item Sometimes, tedious reasoning can be automated.
- \end{itemize}
+ \item Some reasoning can be automated.
+ \end{itemize}\bigskip\pause
+
+ \begin{minipage}{1.1\textwidth}
+ Formal reasoning needs to be ``smooth''.\\
+ {\small (ideally as close as possible to reasoning with ``pen-and-paper'')}
+ \end{minipage}
+
+ \only<2->{
+ \begin{textblock}{3}(0.1,9.9)
+ \begin{tikzpicture}
+ \node at (0,0) [single arrow, shape border rotate=0, fill=red,text=red]{a};
+ \end{tikzpicture}
+ \end{textblock}}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+(*<*)
+atom_decl name
+
+nominal_datatype lam =
+ Var "name"
+ | App "lam" "lam"
+ | Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
+
+nominal_primrec
+ subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_[_::=_]")
+where
+ "(Var x)[y::=s] = (if x=y then s else (Var x))"
+| "(App t\<^isub>1 t\<^isub>2)[y::=s] = App (t\<^isub>1[y::=s]) (t\<^isub>2[y::=s])"
+| "x\<sharp>(y,s) \<Longrightarrow> (Lam [x].t)[y::=s] = Lam [x].(t[y::=s])"
+apply(finite_guess)+
+apply(rule TrueI)+
+apply(simp add: abs_fresh)
+apply(fresh_guess)+
+done
+
+lemma subst_eqvt[eqvt]:
+ fixes pi::"name prm"
+ shows "pi\<bullet>(t1[x::=t2]) = (pi\<bullet>t1)[(pi\<bullet>x)::=(pi\<bullet>t2)]"
+by (nominal_induct t1 avoiding: x t2 rule: lam.strong_induct)
+ (auto simp add: perm_bij fresh_atm fresh_bij)
+
+lemma fresh_fact:
+ fixes z::"name"
+ shows "\<lbrakk>z\<sharp>s; (z=y \<or> z\<sharp>t)\<rbrakk> \<Longrightarrow> z\<sharp>t[y::=s]"
+by (nominal_induct t avoiding: z y s rule: lam.strong_induct)
+ (auto simp add: abs_fresh fresh_prod fresh_atm)
+
+lemma forget:
+ assumes asm: "x\<sharp>L"
+ shows "L[x::=P] = L"
+ using asm
+by (nominal_induct L avoiding: x P rule: lam.strong_induct)
+ (auto simp add: abs_fresh fresh_atm)
+(*>*)
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}
+
+ \begin{textblock}{16}(1,1)
+ \renewcommand{\isasymbullet}{$\cdot$}
+ \tiny\color{black}
+*}
+lemma substitution_lemma_not_to_be_tried_at_home:
+ assumes asm: "x\<noteq>y" "x\<sharp>L"
+ shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"
+using asm
+proof (induct M arbitrary: x y N L rule: lam.induct)
+ case (Lam z M1)
+ have ih: "\<And>x y N L. \<lbrakk>x\<noteq>y; x\<sharp>L\<rbrakk> \<Longrightarrow> M1[x::=N][y::=L] = M1[y::=L][x::=N[y::=L]]" by fact
+ have "x\<noteq>y" by fact
+ have "x\<sharp>L" by fact
+ obtain z'::"name" where fc: "z'\<sharp>(x,y,z,M1,N,L)" by (rule exists_fresh) (auto simp add: fs_name1)
+ have eq: "Lam [z'].([(z',z)]\<bullet>M1) = Lam [z].M1" using fc
+ by (auto simp add: lam.inject alpha fresh_prod fresh_atm)
+ have fc': "z'\<sharp>N[y::=L]" using fc by (simp add: fresh_fact fresh_prod)
+ have "([(z',z)]\<bullet>x) \<noteq> ([(z',z)]\<bullet>y)" using `x\<noteq>y` by (auto simp add: calc_atm)
+ moreover
+ have "([(z',z)]\<bullet>x)\<sharp>([(z',z)]\<bullet>L)" using `x\<sharp>L` by (simp add: fresh_bij)
+ ultimately
+ have "M1[([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)][([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]
+ = M1[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)][([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]]"
+ using ih by simp
+ then have "[(z',z)]\<bullet>(M1[([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)][([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]
+ = M1[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)][([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]])"
+ by (simp add: perm_bool)
+ then have ih': "([(z',z)]\<bullet>M1)[x::=N][y::=L] = ([(z',z)]\<bullet>M1)[y::=L][x::=N[y::=L]]"
+ by (simp add: eqvts perm_swap)
+ show "(Lam [z].M1)[x::=N][y::=L] = (Lam [z].M1)[y::=L][x::=N[y::=L]]" (is "?LHS=?RHS")
+ proof -
+ have "?LHS = (Lam [z'].([(z',z)]\<bullet>M1))[x::=N][y::=L]" using eq by simp
+ also have "\<dots> = Lam [z'].(([(z',z)]\<bullet>M1)[x::=N][y::=L])" using fc by (simp add: fresh_prod)
+ also from ih have "\<dots> = Lam [z'].(([(z',z)]\<bullet>M1)[y::=L][x::=N[y::=L]])" sorry
+ also have "\<dots> = (Lam [z'].([(z',z)]\<bullet>M1))[y::=L][x::=N[y::=L]]" using fc fc' by (simp add: fresh_prod)
+ also have "\<dots> = ?RHS" using eq by simp
+ finally show "?LHS = ?RHS" .
+ qed
+qed (auto simp add: forget)
+text_raw {*
+ \end{textblock}
+ \mbox{}
+
+ \only<2->{
+ \begin{textblock}{11.5}(4,2.3)
+ \begin{minipage}{9.3cm}
+ \begin{block}{}\footnotesize
+*}
+lemma substitution_lemma\<iota>:
+ assumes asm: "x \<noteq> y" "x \<sharp> L"
+ shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"
+ using asm
+by (nominal_induct M avoiding: x y N L rule: lam.strong_induct)
+ (auto simp add: forget fresh_fact)
+text_raw {*
+ \end{block}
+ \end{minipage}
+ \end{textblock}}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
@@ -935,14 +473,63 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
\begin{frame}<1>[c]
- \frametitle{Theorem Provers}
+ \frametitle{Getting Programs Correct}
+
+ \begin{center}
+ \begin{tikzpicture}[node distance=0.5mm]
+ \node at (-1.0,-0.3) (proof) [double arrow, fill=gray,text=white, minimum height=2cm]{\bf Proof};
+ \node [left=of proof]{\Large\bf Specification};
+ \node [right=of proof]{\Large\bf Code};
+ \end{tikzpicture}
+ \end{center}
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \frametitle{Regular Expressions}
- \begin{itemize}
- \item You also pay a (sometimes heavy) price: reasoning can be much, much harder.\medskip
+ \begin{textblock}{6}(2,4)
+ \begin{tabular}{@ {}rrl}
+ \bl{r} & \bl{$::=$} & \bl{$\varnothing$}\\
+ & \bl{$\mid$} & \bl{[]}\\
+ & \bl{$\mid$} & \bl{c}\\
+ & \bl{$\mid$} & \bl{r$_1$ + r$_2$}\\
+ & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$}\\
+ & \bl{$\mid$} & \bl{r$^*$}\\
+ \end{tabular}
+ \end{textblock}
+
+ \begin{textblock}{6}(8,3.5)
+ \includegraphics[scale=0.35]{Screen1.png}
+ \end{textblock}
- \item Depending on your domain, suitable reasoning infrastructure
- might not yet be available.
- \end{itemize}
+ \begin{textblock}{6}(10.2,2.8)
+ \footnotesize Isabelle:
+ \end{textblock}
+
+ \only<2>{
+ \begin{textblock}{9}(3.6,11.8)
+ \bl{matches r s $\;\Longrightarrow\;$ true $\vee$ false}\\[3.5mm]
+
+ \hspace{10mm}\begin{tikzpicture}
+ \coordinate (m1) at (0.4,1);
+ \draw (0,0.3) node (m2) {\small\color{gray}rexp};
+ \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1);
+
+ \coordinate (s1) at (0.81,1);
+ \draw (1.3,0.3) node (s2) {\small\color{gray} string};
+ \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1);
+ \end{tikzpicture}
+ \end{textblock}}
+
+
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -951,17 +538,159 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1>[c]
- \frametitle{Theorem Provers}
+ \begin{frame}<1->[t]
+ \frametitle{Specification}
+
+ \small
+ \begin{textblock}{6}(0,3.5)
+ \begin{tabular}{r@ {\hspace{0.5mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l}
+ \multicolumn{4}{c}{rexp $\Rightarrow$ set of strings}\bigskip\\
+ &\bl{\LL ($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$}\\
+ &\bl{\LL ([])} & \bl{$\dn$} & \bl{\{[]\}}\\
+ &\bl{\LL (c)} & \bl{$\dn$} & \bl{\{c\}}\\
+ &\bl{\LL (r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) $\cup$ \LL (r$_2$)}\\
+ \rd{$\Rightarrow$} &\bl{\LL (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) ;; \LL (r$_2$)}\\
+ \rd{$\Rightarrow$} &\bl{\LL (r$^*$)} & \bl{$\dn$} & \bl{(\LL (r))$^\star$}\\
+ \end{tabular}
+ \end{textblock}
+
+ \begin{textblock}{9}(7.3,3)
+ {\mbox{}\hspace{2cm}\footnotesize Isabelle:\smallskip}
+ \includegraphics[scale=0.325]{Screen3.png}
+ \end{textblock}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \frametitle{Version 1}
+ \small
+ \mbox{}\\[-8mm]\mbox{}
+
+ \begin{center}\def\arraystretch{1.05}
+ \begin{tabular}{@ {\hspace{-5mm}}l@ {\hspace{2.5mm}}c@ {\hspace{2.5mm}}l@ {}}
+ \bl{match [] []} & \bl{$=$} & \bl{true}\\
+ \bl{match [] (c::s)} & \bl{$=$} & \bl{false}\\
+ \bl{match ($\varnothing$::rs) s} & \bl{$=$} & \bl{false}\\
+ \bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\
+ \bl{match (c::rs) []} & \bl{$=$} & \bl{false}\\
+ \bl{match (c::rs) (d::s)} & \bl{$=$} & \bl{if c = d then match rs s else false}\\
+ \bl{match (r$_1$ + r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s $\vee$ match (r$_2$::rs) s}\\
+ \bl{match (r$_1$ $\cdot$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\
+ \bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\
+ \end{tabular}
+ \end{center}
+
+ \begin{textblock}{9}(0.2,1.6)
+ \hspace{10mm}\begin{tikzpicture}
+ \coordinate (m1) at (0.44,-0.5);
+ \draw (0,0.3) node (m2) {\small\color{gray}\mbox{}\hspace{-9mm}list of rexps};
+ \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1);
+
+ \coordinate (s1) at (0.86,-0.5);
+ \draw (1.5,0.3) node (s2) {\small\color{gray} string};
+ \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1);
+ \end{tikzpicture}
+ \end{textblock}
+
+ \begin{textblock}{9}(2.8,11.8)
+ \bl{matches$_1$ r s $\;=\;$ match [r] s}
+ \end{textblock}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
- Recently impressive work has been accomplished proving the correctness
+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->[c]
+ \frametitle{Version 1}
+
+ \only<1->{Several hours later\ldots}\pause
+
+
+ \begin{center}
+ \begin{tabular}{@ {\hspace{0mm}}lcl}
+ \bl{matches$_1$ []$^*$ s} & \bl{$\mapsto$} & loops\\
+ \onslide<4->{\bl{matches$_1$ ([] + \ldots)$^*$ s} & \bl{$\mapsto$} & loops\\}
+ \end{tabular}
+ \end{center}
+
+ \small
+ \onslide<3->{
+ \begin{center}
+ \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
+ \ldots\\
+ \bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\
+ \ldots\\
+ \bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\
+ \end{tabular}
+ \end{center}}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \frametitle{Testing}
\begin{itemize}
- \item of a compiler for C-light (compiled code has the same observable
- behaviour as the original source code),\medskip
+ \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
- \item a mirco-kernel operating system (absence of certain
- bugs\ldots no nil pointers, no buffer overflows).
+ \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
+
\end{itemize}
\end{frame}}
@@ -972,78 +701,91 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1>[c]
- \frametitle{Trust in Theorem Provers}
+ \begin{frame}<1->[t]
+ \frametitle{Version 2}
+ \mbox{}\\[-14mm]\mbox{}
+
+ \small
+ \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}}
+ \bl{nullable ($\varnothing$)} & \bl{$=$} & \bl{false} &\\
+ \bl{nullable ([])} & \bl{$=$} & \bl{true} &\\
+ \bl{nullable (c)} & \bl{$=$} & \bl{false} &\\
+ \bl{nullable (r$_1$ + r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\vee$ nullable r$_2$} & \\
+ \bl{nullable (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\wedge$ nullable r$_2$} & \\
+ \bl{nullable (r$^*$)} & \bl{$=$} & \bl{true} & \\
+ \end{tabular}\medskip
+
+ \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
+ \bl{der c ($\varnothing$)} & \bl{$=$} & \bl{$\varnothing$} & \\
+ \bl{der c ([])} & \bl{$=$} & \bl{$\varnothing$} & \\
+ \bl{der c (d)} & \bl{$=$} & \bl{if c = d then [] else $\varnothing$} & \\
+ \bl{der c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\
+ \bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{((der c r$_1$) $\cdot$ r$_2$)} & \\
+ & & \bl{\;\;+ (if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\
+ \bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ r$^*$} &\smallskip\\
- \begin{center}
- Why should we trust theorem provers?
- \end{center}
+ \bl{derivative r []} & \bl{$=$} & \bl{r} & \\
+ \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\
+ \end{tabular}\medskip
+
+ \bl{matches$_2$ r s $=$ nullable (derivative r s)}
+
+ \begin{textblock}{6}(9.5,0.9)
+ \begin{flushright}
+ \color{gray}``if r matches []''
+ \end{flushright}
+ \end{textblock}
+
+ \begin{textblock}{6}(9.5,6.18)
+ \begin{flushright}
+ \color{gray}``derivative for a char''
+ \end{flushright}
+ \end{textblock}
+
+ \begin{textblock}{6}(9.5,12.1)
+ \begin{flushright}
+ \color{gray}``deriv.~for a string''
+ \end{flushright}
+ \end{textblock}
+
+ \begin{textblock}{6}(9.5,13.98)
+ \begin{flushright}
+ \color{gray}``main''
+ \end{flushright}
+ \end{textblock}
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
text_raw {*
-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<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{frame}<1->[t]
+ \frametitle{Is the Matcher Error-Free?}
- \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 {*
+ We expect that
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \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{center}
+ \begin{tabular}{lcl}
+ \bl{matches$_2$ r s = true} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}%
+ \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\in$ \LL(r)}\\
+ \bl{matches$_2$ r s = false} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}%
+ \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\notin$ \LL(r)}\\
+ \end{tabular}
+ \end{center}
+ \pause\pause\bigskip
+ By \alert<4->{induction}, we can {\bf prove} these properties.\bigskip
+
+ \begin{tabular}{lrcl}
+ Lemmas: & \bl{nullable (r)} & \bl{$\Longleftrightarrow$} & \bl{[] $\in$ \LL (r)}\\
+ & \bl{s $\in$ \LL (der c r)} & \bl{$\Longleftrightarrow$} & \bl{(c::s) $\in$ \LL (r)}\\
+ \end{tabular}
- \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}
+ \only<4->{
+ \begin{textblock}{3}(0.9,4.5)
+ \rd{\huge$\forall$\large{}r s.}
\end{textblock}}
-
-
\end{frame}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
@@ -1066,29 +808,359 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1->[c]
- \frametitle{Future Research}
+ \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}[c]
+ \frametitle{No Automata?}
+
+ You might be wondering why I did not use any automata:
+
+ \begin{itemize}
+ \item A \alert{regular language} is one where there is a DFA that
+ recognises it.\bigskip\pause
+ \end{itemize}
+
+
+ I can think of two reasons why this is a good definition:\medskip
+ \begin{itemize}
+ \item pumping lemma
+ \item closure properties of regular languages (closed under complement)
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{Really Bad News!}
+
+ DFAs are bad news for formalisations in theorem provers. They might
+ be represented as:
+
\begin{itemize}
- \item Make theorem provers more like a programming environment.\medskip\pause
+ \item graphs
+ \item matrices
+ \item partial functions
+ \end{itemize}
+
+ All constructions are messy to reason about.\bigskip\bigskip
+ \pause
+
+ \small
+ \only<2>{
+ Constable et al needed (on and off) 18 months for a 3-person team
+ to formalise automata theory in Nuprl including Myhill-Nerode. There is
+ only very little other formalised work on regular languages I know of
+ in Coq, Isabelle and HOL.}
+ \only<3>{typical textbook reasoning goes like: ``\ldots if \smath{M} and \smath{N} are any two
+ automata with no inaccessible states \ldots''
+ }
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Myhill-Nerode Theorem}
+
+ \begin{itemize}
+ \item provides necessary and suf\!ficient conditions for a language
+ being regular (pumping lemma only necessary)\medskip
+
+ \item will help with closure properties of regular languages\bigskip\pause
+
+ \item key is the equivalence relation:\smallskip
+ \begin{center}
+ \smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L}
+ \end{center}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
- \item Use all the computational power we get from the hardware to
- automate reasoning (GPUs).\medskip\pause
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Myhill-Nerode Theorem}
+
+ \mbox{}\\[5cm]
+
+ \begin{itemize}
+ \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Equivalence Classes}
+
+ \begin{itemize}
+ \item \smath{L = []}
+ \begin{center}
+ \smath{\Big\{\{[]\},\; U\!N\!IV - \{[]\}\Big\}}
+ \end{center}\bigskip\bigskip
+
+ \item \smath{L = [c]}
+ \begin{center}
+ \smath{\Big\{\{[]\},\; \{[c]\},\; U\!N\!IV - \{[], [c]\}\Big\}}
+ \end{center}\bigskip\bigskip
+
+ \item \smath{L = \varnothing}
+ \begin{center}
+ \smath{\Big\{U\!N\!IV\Big\}}
+ \end{center}
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Regular Languages}
+
+ \begin{itemize}
+ \item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M}
+ such that \smath{\mathbb{L}(M) = L}\\[1.5cm]
- \item Provide a comprehensive reasoning infrastructure for many domains and
- design automated decision procedures.
- \end{itemize}\pause
+ \item Myhill-Nerode:
-
\begin{center}
- \colorbox{cream}{
- \begin{minipage}{10cm}
- \color{gray}
+ \begin{tabular}{l}
+ finite $\Rightarrow$ regular\\
+ \;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r. L = \mathbb{L}(r)}\\[3mm]
+ regular $\Rightarrow$ finite\\
+ \;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
+ \end{tabular}
+ \end{center}
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Final States}
+
+ \mbox{}\\[3cm]
+
+ \begin{itemize}
+ \item \smath{\text{final}_L\,X \dn}\\
+ \smath{\hspace{6mm}X \in (U\!N\!IV /\!/\approx_L) \;\wedge\; \forall s \in X.\; s \in L}
+ \smallskip
+ \item we can prove: \smath{L = \bigcup \{X.\;\text{final}_L\,X\}}
+
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes}
+
+ \smath{L = \{[c]\}}
+
+ \begin{tabular}{@ {\hspace{-7mm}}cc}
+ \begin{tabular}{c}
+ \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
+ \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
+
+ %\draw[help lines] (0,0) grid (3,2);
+
+ \node[state,initial] (q_0) {$R_1$};
+ \node[state,accepting] (q_1) [above right of=q_0] {$R_2$};
+ \node[state] (q_2) [below right of=q_0] {$R_3$};
+
+ \path[->] (q_0) edge node {c} (q_1)
+ edge node [swap] {$\Sigma-{c}$} (q_2)
+ (q_2) edge [loop below] node {$\Sigma$} ()
+ (q_1) edge node {$\Sigma$} (q_2);
+ \end{tikzpicture}
+ \end{tabular}
+ &
+ \begin{tabular}[t]{ll}
+ \\[-20mm]
+ \multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm]
+
+ \smath{R_1}: & \smath{\{[]\}}\\
+ \smath{R_2}: & \smath{\{[c]\}}\\
+ \smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm]
+ \multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ; [c] \subseteq Y}}}
+ \end{tabular}
+
+ \end{tabular}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Systems of Equations}
+
+ Inspired by a method of Brzozowski\;'64, we can build an equational system
+ characterising the equivalence classes:
+
+ \begin{center}
+ \begin{tabular}{@ {\hspace{-20mm}}c}
+ \\[-13mm]
+ \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick]
+ \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
+
+ %\draw[help lines] (0,0) grid (3,2);
+
+ \node[state,initial] (p_0) {$R_1$};
+ \node[state,accepting] (p_1) [right of=q_0] {$R_2$};
+
+ \path[->] (p_0) edge [bend left] node {a} (p_1)
+ edge [loop above] node {b} ()
+ (p_1) edge [loop above] node {a} ()
+ edge [bend left] node {b} (p_0);
+ \end{tikzpicture}\\
+ \\[-13mm]
+ \end{tabular}
+ \end{center}
+
+ \begin{center}
+ \begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
+ & \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\
+ & \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\
+ \onslide<3->{we can prove}
+ & \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}}
+ & \onslide<3->{\smath{R_1; \mathbb{L}(b) \,\cup\, R_2;\mathbb{L}(b) \,\cup\, \{[]\};\{[]\}}}\\
+ & \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}}
+ & \onslide<3->{\smath{R_1; \mathbb{L}(a) \,\cup\, R_2;\mathbb{L}(a)}}\\
+ \end{tabular}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>[t]
\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}}
+
+ \begin{center}
+ \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
+ \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
+ & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+ \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
+ & \onslide<1->{\smath{R_1; a + R_2; a}}\\
+
+ & & & \onslide<2->{by Arden}\\
+
+ \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
+ & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+ \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
+ & \only<2>{\smath{R_1; a + R_2; a}}%
+ \only<3->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<4->{by Arden}\\
+
+ \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
+ & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
+ \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
+ & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<5->{by substitution}\\
+
+ \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
+ & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
+ \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
+ & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<6->{by Arden}\\
+
+ \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
+ & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+ \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
+ & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<7->{by substitution}\\
+
+ \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
+ & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+ \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
+ & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
+ \cdot a\cdot a^\star}}\\
+ \end{tabular}
\end{center}
\end{frame}}
@@ -1098,16 +1170,219 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1->[c]
- \frametitle{Conclusion}
+ \begin{frame}[c]
+ \frametitle{\LARGE A Variant of Arden's Lemma}
+
+ {\bf Arden's Lemma:}\smallskip
+
+ If \smath{[] \not\in A} then
+ \begin{center}
+ \smath{X = X; A + \text{something}}
+ \end{center}
+ has the (unique) solution
+ \begin{center}
+ \smath{X = \text{something} ; A^\star}
+ \end{center}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1->[t]
+ \small
+
+ \begin{center}
+ \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
+ \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}}
+ & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+ \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}}
+ & \onslide<1->{\smath{R_1; a + R_2; a}}\\
+
+ & & & \onslide<2->{by Arden}\\
+
+ \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}}
+ & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\
+ \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}}
+ & \only<2>{\smath{R_1; a + R_2; a}}%
+ \only<3->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<4->{by Arden}\\
+
+ \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}}
+ & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\
+ \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}}
+ & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<5->{by substitution}\\
+
+ \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}}
+ & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\
+ \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}}
+ & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<6->{by Arden}\\
+
+ \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}}
+ & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+ \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}}
+ & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\
+
+ & & & \onslide<7->{by substitution}\\
+
+ \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}}
+ & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\
+ \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}}
+ & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star
+ \cdot a\cdot a^\star}}\\
+ \end{tabular}
+ \end{center}
+
+ \only<8->{
+ \begin{textblock}{6}(2.5,4)
+ \begin{block}{}
+ \begin{minipage}{8cm}\raggedright
+ \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm]
+ \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm]
+
+ %\draw[help lines] (0,0) grid (3,2);
+
+ \node[state,initial] (p_0) {$R_1$};
+ \node[state,accepting] (p_1) [right of=q_0] {$R_2$};
+
+ \path[->] (p_0) edge [bend left] node {a} (p_1)
+ edge [loop above] node {b} ()
+ (p_1) edge [loop above] node {a} ()
+ edge [bend left] node {b} (p_0);
+ \end{tikzpicture}
+
+ \end{minipage}
+ \end{block}
+ \end{textblock}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE The Equ's Solving Algorithm}
+
\begin{itemize}
- \item The plan is to make this kind of programming the ``future''.\medskip\pause
+ \item The algorithm must terminate: Arden makes one equation smaller;
+ substitution deletes one variable from the right-hand sides.\bigskip
+
+ \item We need to maintain the invariant that Arden is applicable
+ (if \smath{[] \not\in A} then \ldots):\medskip
+
+ \begin{center}\small
+ \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll}
+ \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
+ \smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\
+
+ & & & by Arden\\
+
+ \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\
+ \smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\
+ \end{tabular}
+ \end{center}
+
+ \end{itemize}
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Other Direction}
+
+ One has to prove
+
+ \begin{center}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})}
+ \end{center}
+
+ by induction on \smath{r}. Not trivial, but after a bit
+ of thinking, one can prove that if
+
+ \begin{center}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})}
+ \end{center}
+
+ then
- \item Though the technology is already there\\ (compiler + micro-kernel os).\medskip\pause
+ \begin{center}
+ \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})}
+ \end{center}
+
+
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE What Have We Achieved?}
+
+ \begin{itemize}
+ \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}}
+ \bigskip\pause
+ \item regular languages are closed under complementation; this is easy
+ \begin{center}
+ \smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}}
+ \end{center}
+ \end{itemize}
- \item Logic and reasoning (especially induction) are important skills for
- Computer Scientists.
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Examples}
+
+ \begin{itemize}
+ \item \smath{L \equiv \Sigma^\star 0 \Sigma} is regular
+ \begin{quote}\small
+ \begin{tabular}{lcl}
+ \smath{A_1} & \smath{=} & \smath{\Sigma^\star 00}\\
+ \smath{A_2} & \smath{=} & \smath{\Sigma^\star 01}\\
+ \smath{A_3} & \smath{=} & \smath{\Sigma^\star 10 \cup \{0\}}\\
+ \smath{A_4} & \smath{=} & \smath{\Sigma^\star 11 \cup \{1\} \cup \{[]\}}\\
+ \end{tabular}
+ \end{quote}
+
+ \item \smath{L \equiv \{ 0^n 1^n \,|\, n \ge 0\}} is not regular
+ \begin{quote}\small
+ \begin{tabular}{lcl}
+ \smath{B_0} & \smath{=} & \smath{\{0^n 1^n \,|\, n \ge 0\}}\\
+ \smath{B_1} & \smath{=} & \smath{\{0^n 1^{(n-1)} \,|\, n \ge 1\}}\\
+ \smath{B_2} & \smath{=} & \smath{\{0^n 1^{(n-2)} \,|\, n \ge 2\}}\\
+ \smath{B_3} & \smath{=} & \smath{\{0^n 1^{(n-3)} \,|\, n \ge 3\}}\\
+ & \smath{\vdots} &\\
+ \end{tabular}
+ \end{quote}
\end{itemize}
\end{frame}}
@@ -1118,20 +1393,80 @@
text_raw {*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mode<presentation>{
- \begin{frame}<1>[c]
+ \begin{frame}[c]
+ \frametitle{\LARGE What We Have Not Achieved}
+
+ \begin{itemize}
+ \item regular expressions are not good if you look for a minimal
+ one for a language (DFAs have this notion)\pause\bigskip
+
+ \item Is there anything to be said about context free languages:\medskip
+
+ \begin{quote}
+ A context free language is where every string can be recognised by
+ a pushdown automaton.\bigskip
+ \end{quote}
+ \end{itemize}
+
+ \textcolor{gray}{\footnotesize Yes. Derivatives also work for c-f grammars. Ongoing work.}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{\LARGE Conclusion}
+
+ \begin{itemize}
+ \item We formalised the Myhill-Nerode theorem based on
+ regular expressions (DFA are difficult to deal with in a theorem prover).\smallskip
+
+ \item Seems to be a common theme: algorithms need to be reformulated
+ to better suit formal treatment.\smallskip
+
+ \item The most interesting aspect is that we are able to
+ implement the matcher directly inside the theorem prover
+ (ongoing work).\smallskip
+
+ \item Parsing is a vast field and seems to offer new results.
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+*}
+
+text_raw {*
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}<1>[b]
\frametitle{
\begin{tabular}{c}
- \mbox{}\\[23mm]
+ \mbox{}\\[13mm]
\alert{\LARGE Thank you very much!}\\
\alert{\Large Questions?}
\end{tabular}}
-
+
+ %\begin{center}
+ %\bf \underline{Short Bio:}
+ %\end{center}
+ %\mbox{}\\[-17mm]\mbox{}\small
+ %\begin{itemize}
+ %\item PhD in Cambridge
+ %\item Emmy-Noether Fellowship in Munich
+ %\item main results in nominal reasoning and nominal unification
+ %\end{itemize}
+
\end{frame}}
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*}
+
(*<*)
end
(*>*)
\ No newline at end of file