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\begin{document}
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\begin{frame}[t]
\frametitle{%
\begin{tabular}{@ {}c@ {}}
\\
\Large POSIX Lexing with Derivatives\\[-1.5mm]
\Large of Regular Expressions\\[-1mm]
\normalsize Or, How to Find Bugs with the\\[-5mm]
\normalsize Isabelle Theorem Prover
\end{tabular}}\bigskip\bigskip\bigskip
\normalsize
\begin{center}
\begin{tabular}{c}
\small Christian Urban\\
%\small King's College London\\
\\
\small joint work with Fahad Ausaf and Roy Dyckhoff
\end{tabular}
\end{center}
\end{frame}
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\begin{frame}[c]
\frametitle{Why Bother?}
\begin{itemize}
\item Surely regular expressions must have been studied and
implemented to death by now, no?\medskip\pause
\item \ldots{}well, take for example the ``evil'' regular
expression \bl{$({\tt a}^?)^n \cdot {\tt a}^n$} to match
strings \bl{$\underbrace{{\tt a}\ldots{\tt a}}_n$}
\end{itemize}\smallskip
\begin{center}
\begin{tikzpicture}
\begin{axis}[
xlabel={strings of {\tt a}s},
ylabel={time in secs},
enlargelimits=false,
xtick={0,5,...,30},
xmax=30,
ymax=35,
ytick={0,5,...,30},
scaled ticks=false,
axis lines=left,
width=7cm,
height=5cm,
legend entries={Python,Ruby},
legend pos=north west,
legend cell align=left
]
\addplot[blue,mark=*, mark options={fill=white}]
table {data/re-python.data};
\addplot[brown,mark=pentagon*, mark options={fill=white}]
table {data/re-ruby.data};
\end{axis}
\end{tikzpicture}
\end{center}
\end{frame}
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\begin{frame}[c]
\begin{center}
\includegraphics[scale=0.2]{pics/isabelle.png}
\end{center}
\mbox{}\\[-20mm]\mbox{}
\begin{itemize}
\item Isabelle interactive theorem prover;
some proofs are automatic -- most however need help
\item the learning curve is steep; you often have to fight the
theorem prover\ldots no different in other ITPs
\end{itemize}
\end{frame}
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\begin{frame}[c]
\frametitle{\Large Isabelle Theorem Prover}
\begin{itemize}
\item started to use Isabelle after my PhD (in 2000)
\item the thesis included a rather complicated
``pencil-and-paper'' proof for a termination argument
(SN for a sort of $\lambda$-calculus)\medskip
\item me, my supervisor, the examiners did not find any problems\medskip
\begin{center}
\begin{tabular}{@ {}c@ {}}
\includegraphics[scale=0.38]{pics/barendregt.jpg}\\[-2mm]
\footnotesize Henk Barendregt
\end{tabular}
\hspace{2mm}
\begin{tabular}{@ {}c@ {}}
\includegraphics[scale=0.20]{pics/andrewpitts.jpg}\\[-2mm]
\footnotesize Andrew Pitts
\end{tabular}
\end{center}
\item people were building their work on my result
\end{itemize}
\end{frame}
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\begin{frame}[t]
\frametitle{\Large Nominal Isabelle}
\begin{itemize}
\item implemented a package for the Isabelle prover
in order to reason conveniently about binders
\begin{center}
\large\bl{$\lambda \alert{x}.\,M$} \hspace{10mm}
\bl{$\forall \alert{x}.\,P\,x$}
\end{center}\bigskip\bigskip\bigskip\bigskip
\bigskip\bigskip\bigskip\pause\pause
\item when finally being able to formalise the proof from my PhD, I found that the main result
(termination) is correct, but a central lemma needed to
be generalised
\end{itemize}
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\includegraphics[scale=0.33]{pics/skeleton.jpg}
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\begin{frame}[c]
\frametitle{\Large Variable Convention}
\begin{center}
\begin{bubble}[10cm]
\color{gray}
\small
{\bf\mbox{}Variable Convention:}\\[1mm]
If $M_1,\ldots,M_n$ occur in a certain mathematical context
(e.g. definition, proof), then in these terms all bound variables
are chosen to be different from the free variables.\\[2mm]
\footnotesize\hfill Barendregt in ``The Lambda-Calculus: Its Syntax and Semantics''
\end{bubble}
\end{center}
\mbox{}\\[-8mm]
\begin{itemize}
\item instead of proving a property for \alert{\bf all} bound
variables, you prove it only for \alert{\bf some}\ldots?
\item this is mostly OK, but in some corner-cases you can use it
to prove \alert{\bf false}\ldots we fixed this!
\end{itemize}
\end{frame}
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\begin{frame}[c]
\frametitle{}
\begin{tabular}{c@ {\hspace{2mm}}c}
\\[6mm]
\begin{tabular}{c}
\includegraphics[scale=0.11]{pics/harper.jpg}\\[-2mm]
{\footnotesize Bob Harper}\\[-2mm]
{\footnotesize}
\end{tabular}
\begin{tabular}{c}
\includegraphics[scale=0.37]{pics/pfenning.jpg}\\[-2mm]
{\footnotesize Frank Pfenning}\\[-2mm]
{\footnotesize}
\end{tabular} &
\begin{tabular}{p{6cm}}
\raggedright
{published a proof on LF in\\ {\bf ACM Transactions on
Computational Logic}, 2005,
$\sim$31pp}
\end{tabular}\\
\\[0mm]
\begin{tabular}{c}
\includegraphics[scale=0.36]{pics/appel.jpg}\\[-2mm]
{\footnotesize Andrew Appel}\\[-2.5mm]
{\footnotesize}
\end{tabular} &
\begin{tabular}{p{6cm}}
\raggedright
{relied on their proof in a\\ {\bf security} critical application}
\end{tabular}
\end{tabular}
\end{frame}
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\begin{frame}
\frametitle{Proof-Carrying Code}
\begin{textblock}{10}(2.5,2.2)
\begin{block}{Idea:}
\begin{center}
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\begin{textblock}{14}(2,11)
\small
\begin{itemize}
\item<2-> Appel's checker is $\sim$2700 lines of code (1865 loc of\\ LF definitions;
803 loc in C including 2 library functions)\\[-3mm]
\item<2-> 167 loc in C implement a type-checker\\ (proved correct by Harper and Pfenning)
\end{itemize}
\end{textblock}
\end{frame}
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\small Each time one needs to check $\sim$31pp~of
informal paper proofs---impossible without tool support.
You have to be able to keep definitions
and proofs consistent.
\end{bubble}}
\end{textblock}
\end{frame}
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\begin{frame}[c]
\frametitle{\LARGE Lessons Learned}
\begin{itemize}
\item by using a theorem prover we were able to keep a large
proof consistent with changes in the first definitions\bigskip
\item it took us appr.~10 days to get to the error\ldots
probably the same time Harper and Pfenning needed to \LaTeX{}
their paper\bigskip
\item once there, we ran circles around them
\end{itemize}
\end{frame}
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\begin{frame}[c]
\frametitle{Real-Time Scheduling}
\begin{textblock}{11}(1,3)
\begin{tabular}{@{\hspace{-10mm}}l}
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\begin{textblock}{0}(1.5,13)%
\small
\onslide<5->{
\begin{bubble}[10.3cm]%
RT-Scheduling: You want to avoid that a
high-priority process is starved indefinitely by lower priority
processes.
\end{bubble}}
\end{textblock}
\end{frame}
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\begin{frame}[c]
\frametitle{\Large Priority Inheritance Scheduling}
\begin{itemize}
\item Idea: Let a low priority process \bl{$L$} temporarily inherit
the high priority of \bl{$H$} until \bl{$L$} leaves the critical
section unlocking the resource.\bigskip
\item Once the resource is unlocked, \bl{$L$} ``returns to its original
priority level.''\\
\mbox{}\hfill\footnotesize
\begin{tabular}{p{6cm}@{}}
L.~Sha, R.~Rajkumar, and J.~P.~Lehoczky.
{\it Priority Inheritance Protocols: An Approach to
Real-Time Synchronization}. IEEE Transactions on
Computers, 39(9):1175–1185, 1990
\end{tabular}\bigskip\normalsize\pause
\item classic, proved correct, reviewed in a respectable journal....what
could possibly be wrong?
\end{itemize}
\end{frame}
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\node at (2.0,0.9) {\small $B_L$};
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\node at (5.7,0.9) {\small $B_R$};
\draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);
\draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);
\draw[very thick,blue!100] (3.5,0) -- (3.5,0.6);
\draw[very thick,blue!100] (5.7,0) -- (5.7,0.6);
\draw[very thick,blue!100] (3,3) rectangle (4,3.6);
\node at (3.5,3.3) {\small $A$};
}
\only<4>{
\draw[fill, blue!50] (1,0) rectangle (3,0.6);
\draw[very thick, blue!50] (3,0) rectangle (6,0.6);
\node at (1.5,0.9) {\small $A_L$};
\node at (2.0,0.9) {\small $B_L$};
\node at (3.5,0.9) {\small $A_R$};
\node at (5.7,0.9) {\small $B_R$};
\draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);
\draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);
\draw[very thick,blue!100] (3.5,0) -- (3.5,0.6);
\draw[very thick,blue!100] (5.7,0) -- (5.7,0.6);
\draw[very thick,blue!100] (3,3) rectangle (4,3.6);
\node at (3.5,3.3) {\small $A$};
\draw[very thick,blue!100] (4,3) rectangle (5,3.6);
\node at (4.5,3.3) {\small $B$};
}
\only<5>{
\draw[fill, blue!50] (1,0) rectangle (3,0.6);
\draw[very thick, blue!50] (3,3) rectangle (6,3.6);
\node at (1.5,0.9) {\small $A_L$};
\node at (2.0,0.9) {\small $B_L$};
\node at (3.5,3.9) {\small $A_R$};
\node at (5.7,3.9) {\small $B_R$};
\draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);
\draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);
\draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);
\draw[very thick,blue!100] (5.7,3) -- (5.7,3.6);
\draw[very thick,blue!100] (6,3) rectangle (7,3.6);
\node at (6.5,3.3) {\small $A$};
\draw[very thick,blue!100] (7,3) rectangle (8,3.6);
\node at (7.5,3.3) {\small $B$};
\draw[blue!50, ->, line width = 2mm] (3,1) -- (3, 2.5);
}
\only<6>{
\draw[fill, blue!50] (1,0) rectangle (3,0.6);
\draw[fill, blue!50] (3,3) rectangle (3.5,3.6);
\draw[very thick, blue!50] (3.5,3) rectangle (6,3.6);
\node at (1.5,0.9) {\small $A_L$};
\node at (2.0,0.9) {\small $B_L$};
\node at (3.5,3.9) {\small $A_R$};
\node at (5.7,3.9) {\small $B_R$};
\draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);
\draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);
\draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);
\draw[very thick,blue!100] (5.7,3) -- (5.7,3.6);
\draw[very thick,blue!100] (6,3) rectangle (7,3.6);
\node at (6.5,3.3) {\small $A$};
\draw[very thick,blue!100] (7,3) rectangle (8,3.6);
\node at (7.5,3.3) {\small $B$};
}
\only<7>{
\draw[fill, blue!50] (1,0) rectangle (3,0.6);
\draw[fill, blue!50] (3,3) rectangle (3.5,3.6);
\draw[very thick, blue!50] (3.5,0) rectangle (6,0.6);
\node at (1.5,0.9) {\small $A_L$};
\node at (2.0,0.9) {\small $B_L$};
\node at (3.5,3.9) {\small $A_R$};
\node at (5.7,0.9) {\small $B_R$};
\draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);
\draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);
\draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);
\draw[very thick,blue!100] (5.7,0) -- (5.7,0.6);
\draw[very thick,blue!100] (6,3) rectangle (7,3.6);
\node at (6.5,3.3) {\small $A$};
\draw[very thick,blue!100] (7,3) rectangle (8,3.6);
\node at (7.5,3.3) {\small $B$};
\draw[blue!50, <-, line width = 2mm] (3.5,1) -- (3.5, 2.5);
\draw[blue!50, <-, line width = 2mm] (4,3.3) -- (5.5,3.3);
}
\only<8>{
\draw[fill, blue!50] (1,0) rectangle (3,0.6);
\draw[fill, blue!50] (3,3) rectangle (3.5,3.6);
\draw[very thick, blue!50] (4.5,0) rectangle (7,0.6);
\node at (1.5,0.9) {\small $A_L$};
\node at (2.0,0.9) {\small $B_L$};
\node at (3.5,3.9) {\small $A_R$};
\node at (6.7,0.9) {\small $B_R$};
\draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);
\draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);
\draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);
\draw[very thick,blue!100] (6.7,0) -- (6.7,0.6);
\draw[fill,blue!100] (3.5,3) rectangle (4.5,3.6);
\node at (4,3.3) {\small $A$};
\draw[very thick,blue!100] (7,3) rectangle (8,3.6);
\node at (7.5,3.3) {\small $B$};
}
\only<9>{
\draw[fill, blue!50] (1,0) rectangle (3,0.6);
\draw[fill, blue!50] (3,3) rectangle (3.5,3.6);
\draw[fill, blue!50] (4.5,0) rectangle (5,0.6);
\draw[very thick, blue!50] (5,0) rectangle (7,0.6);
\node at (1.5,0.9) {\small $A_L$};
\node at (2.0,0.9) {\small $B_L$};
\node at (3.5,3.9) {\small $A_R$};
\node at (6.7,0.9) {\small $B_R$};
\draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);
\draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);
\draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);
\draw[very thick,blue!100] (6.7,0) -- (6.7,0.6);
\draw[fill,blue!100] (3.5,3) rectangle (4.5,3.6);
\node at (4,3.3) {\small $A$};
\draw[very thick,blue!100] (7,3) rectangle (8,3.6);
\node at (7.5,3.3) {\small $B$};
}
\only<10>{
\draw[fill, blue!50] (1,0) rectangle (3,0.6);
\draw[fill, blue!50] (3,3) rectangle (3.5,3.6);
\draw[fill, blue!50] (4.5,0) rectangle (5,0.6);
\draw[very thick, blue!50] (5,0) rectangle (7,0.6);
\node at (1.5,0.9) {\small $A_L$};
\node at (2.0,0.9) {\small $B_L$};
\node at (3.5,3.9) {\small $A_R$};
\node at (6.7,0.9) {\small $B_R$};
\draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);
\draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);
\draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);
\draw[very thick,blue!100] (6.7,0) -- (6.7,0.6);
\draw[fill,blue!100] (3.5,3) rectangle (4.5,3.6);
\node at (4,3.3) {\small $A$};
\draw[very thick,blue!100] (7,3) rectangle (8,3.6);
\node at (7.5,3.3) {\small $B$};
\draw[red, fill] (5,1.5) rectangle (6,2.1);
\draw[red, fill] (6.05,1.5) rectangle (7,2.1);
}
\only<11>{
\draw[fill, blue!50] (1,0) rectangle (3,0.6);
\draw[fill, blue!50] (3,3) rectangle (3.5,3.6);
\draw[fill, blue!50] (4.5,0) rectangle (5,0.6);
\draw[very thick, blue!50] (5,0) rectangle (7,0.6);
\node at (1.5,0.9) {\small $A_L$};
\node at (2.0,0.9) {\small $B_L$};
\node at (3.5,3.9) {\small $A_R$};
\node at (6.7,0.9) {\small $B_R$};
\draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);
\draw[very thick,blue!100] (2.0,0) -- (2.0,0.6);
\draw[very thick,blue!100] (3.5,3) -- (3.5,3.6);
\draw[very thick,blue!100] (6.7,0) -- (6.7,0.6);
\draw[fill,blue!100] (3.5,3) rectangle (4.5,3.6);
\node at (4,3.3) {\small $A$};
\draw[very thick,blue!100] (7,3) rectangle (8,3.6);
\node at (7.5,3.3) {\small $B$};
\draw[red, fill] (5,1.5) rectangle (6,2.1);
\draw[red, fill] (6.05,1.5) rectangle (7,2.1);
\draw[blue!50, ->, line width = 2mm] (7.1,0.4) -- (8, 0.4);
\draw[blue!50, ->, line width = 2mm] (7.1,4) -- (8,4);
}
\draw[very thick,->](0,0) -- (8,0);
\node [anchor=base] at (8, -0.3) {\scriptsize time};
\node [anchor=base] at (0, -0.3) {\scriptsize 0};
\node [anchor=base] at (-1.2, 0.2) {\small low priority};
\only<2->{\node [anchor=base] at (-1.2, 3.2) {\small high priority};}
\only<10->{\node [anchor=base] at (-1.5, 1.7) {\small medium pr.};}
\end{tikzpicture}
\end{tabular}
\end{textblock}
\begin{textblock}{0}(1.5,13)%
\small
\begin{bubble}[10.3cm]%
RT-Scheduling: You want to avoid that a
high-priority process is starved indefinitely by lower priority
processes.
\end{bubble}
\end{textblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{\Large Priority Inheritance Scheduling}
\begin{itemize}
\item Idea: Let a low priority process \bl{$L$} temporarily inherit
the high priority of \bl{$H$} until \bl{$L$} leaves the critical
section unlocking the resource.\bigskip
\item Once the resource is unlocked, \bl{$L$} returns to its original
priority level. \alert{\bf BOGUS}\pause\bigskip
\item \ldots \bl{$L$} needs to switch to the highest
\alert{\bf remaining} priority of the threads that it blocks.
\end{itemize}\bigskip
\small this error is already known since around 1999
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\begin{textblock}{11}(2,1)
\alt<1>{\includegraphics[scale=0.25]{pics/p3.jpg}}
{\includegraphics[scale=0.125]{pics/p3.jpg}}
\alt<2>{\includegraphics[scale=0.25]{pics/p2.jpg}}
{\includegraphics[scale=0.125]{pics/p2.jpg}}
\alt<3>{\includegraphics[scale=0.153]{pics/p1.jpg}}
{\includegraphics[scale=0.076]{pics/p1.jpg}}
\alt<4>{\includegraphics[scale=0.25]{pics/p4.jpg}}
{\includegraphics[scale=0.125]{pics/p4.jpg}}
\alt<5>{\includegraphics[scale=0.088]{pics/p5.jpg}}
{\includegraphics[scale=0.044]{pics/p5.jpg}}
\end{textblock}
\begin{textblock}{13}(1,9)
\only<1>{
\begin{itemize}
\item by Rajkumar, 1991
\item \it ``it resumes the priority it had at the point of entry into the critical
section''
\end{itemize}}
\only<2>{
\begin{itemize}
\item by Jane Liu, 2000
\item {\it ``The job $J_l$ executes at its inherited
priority until it releases $R$; at that time, the
priority of $J_l$ returns to its priority
at the time when it acquires the resource $R$.''}\medskip
\item \small gives pseudo code and uses pretty bogus data structures
\item \small the interesting part is ``{\it left as an exercise}''
\end{itemize}}
\only<3>{
\begin{itemize}
\item by Laplante and Ovaska, 2011 (\$113.76)
\item \it ``when $[$the task$]$ exits the critical section that
caused the block, it reverts to the priority it had
when it entered that section''
\end{itemize}}
\only<4>{
\begin{itemize}
\item by Silberschatz, Galvin and Gagne (9th edition, 2013)
\item \it ``Upon releasing the
lock, the $[$low-priority$]$ thread will revert to its original
priority.''
\end{itemize}}
\only<5>{
\begin{itemize}
\item by Stallings (8th edition, 2014)
\item \it ``This priority change takes place as soon as the
higher-priority task blocks on the resource; it should end when
the resource is released by the lower-priority task.''
\end{itemize}}
\end{textblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Priority Scheduling}
\begin{itemize}
\item a scheduling algorithm that is widely used in real-time operating systems
\item has been ``proved'' correct by hand in a paper in 1990
\item but this algorithm turned out to be incorrect, despite its ``proof''\bigskip\pause
\item we (generalised) the algorithm and then {\bf really} proved that it is correct
\item we implemented this algorithm in a small OS called PINTOS (used for teaching at Stanford)
\item our implementation was faster than their reference implementation
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Lessons Learned}
\begin{itemize}
\item our proof-technique is adapted from security
protocols\bigskip
%\item do not venture outside your field of expertise
%\includegraphics[scale=0.03]{smiley.jpg}
%\bigskip
\item we solved the single-processor case; the multi-processor
case: no idea!
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Regular Expressions}
\begin{textblock}{6}(2,5)
\begin{tabular}{rrl@ {\hspace{13mm}}l}
\bl{$r$} & \bl{$::=$} & \bl{$\varnothing$} & null\\
& \bl{$\mid$} & \bl{$\epsilon$} & empty string\\
& \bl{$\mid$} & \bl{$c$} & character\\
& \bl{$\mid$} & \bl{$r_1 \cdot r_2$} & sequence\\
& \bl{$\mid$} & \bl{$r_1 + r_2$} & alternative / choice\\
& \bl{$\mid$} & \bl{$r^*$} & star (zero or more)\\
\end{tabular}
\end{textblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{The Derivative of a Rexp}
\large
If \bl{$r$} matches the string \bl{$c\!::\!s$}, what is a regular
expression that matches just \bl{$s$}?\bigskip\bigskip\bigskip\bigskip
\small
\bl{$der\,c\,r$} gives the answer, Brzozowski (1964), Owens (2005)
``\ldots have been lost in the sands of time\ldots''
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{}
\ldots{}whether a regular expression can match the empty string:
\begin{center}
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
\bl{$nullable(\varnothing)$} & \bl{$\dn$} & \bl{$false$}\\
\bl{$nullable(\epsilon)$} & \bl{$\dn$} & \bl{$true$}\\
\bl{$nullable (c)$} & \bl{$\dn$} & \bl{$false$}\\
\bl{$nullable (r_1 + r_2)$} & \bl{$\dn$} & \bl{$nullable(r_1) \vee nullable(r_2)$} \\
\bl{$nullable (r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{$nullable(r_1) \wedge nullable(r_2)$} \\
\bl{$nullable (r^*)$} & \bl{$\dn$} & \bl{$true$} \\
\end{tabular}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{The Derivative of a Rexp}
\begin{center}
\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}}
\bl{$der\, c\, (\varnothing)$} & \bl{$\dn$} & \bl{$\varnothing$} & \\
\bl{$der\, c\, (\epsilon)$} & \bl{$\dn$} & \bl{$\varnothing$} & \\
\bl{$der\, c\, (d)$} & \bl{$\dn$} & \bl{if $c = d$ then $\epsilon$ else $\varnothing$} & \\
\bl{$der\, c\, (r_1 + r_2)$} & \bl{$\dn$} & \bl{$der\, c\, r_1 + der\, c\, r_2$} & \\
\bl{$der\, c\, (r_1 \cdot r_2)$} & \bl{$\dn$} & \bl{if $nullable (r_1)$}\\
& & \bl{then $(der\,c\,r_1) \cdot r_2 + der\, c\, r_2$}\\
& & \bl{else $(der\, c\, r_1) \cdot r_2$}\\
\bl{$der\, c\, (r^*)$} & \bl{$\dn$} & \bl{$(der\,c\,r) \cdot (r^*)$} &\medskip\\\pause
\bl{$\textit{ders}\, []\, r$} & \bl{$\dn$} & \bl{$r$} & \\
\bl{$\textit{ders}\, (c\!::\!s)\, r$} & \bl{$\dn$} & \bl{$\textit{ders}\,s\,(der\,c\,r)$} & \\
\end{tabular}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Correctness}
It is a relative easy exercise in a theorem prover:
\begin{center}
\bl{$matches(r, s)$} if and only if \bl{$s \in L(r)$}
\end{center}\bigskip
\small
where \bl{$matches(r, s) \dn nullable(ders(r, s))$}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{\bl{$({\tt a}^?)^n \cdot {\tt a}^n$}}
\begin{center}
\begin{tikzpicture}
\begin{axis}[
xlabel={strings of \pcode{a}s},
ylabel={time in secs},
enlargelimits=false,
xtick={0,200,...,1000},
xmax=1000,
ytick={0,5,...,30},
scaled ticks=false,
axis lines=left,
width=9.5cm,
height=7cm,
legend entries={Python,Ruby,Scala V1,Scala V2},
legend pos=north west,
legend cell align=left
]
\addplot[blue,mark=*, mark options={fill=white}]
table {data/re-python.data};
\addplot[brown,mark=pentagon*, mark options={fill=white}]
table {data/re-ruby.data};
\addplot[red,mark=triangle*,mark options={fill=white}]
table {data/re1.data};
\addplot[green,mark=square*,mark options={fill=white}]
table {data/re2b.data};
\end{axis}
\end{tikzpicture}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{\bl{$({\tt a}^?)^n \cdot {\tt a}^n$}}
\begin{center}
\begin{tikzpicture}
\begin{axis}[
xlabel={strings of \pcode{a}s},
ylabel={time in secs},
enlargelimits=false,
xtick={0,3000,...,12000},
xmax=12000,
ymax=35,
ytick={0,5,...,30},
scaled ticks=false,
axis lines=left,
width=9cm,
height=7cm
]
\addplot[green,mark=square*,mark options={fill=white}] table {data/re2b.data};
\addplot[black,mark=square*,mark options={fill=white}] table {data/re3.data};
\end{axis}
\end{tikzpicture}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{POSIX Regex Matching}
Two rules:
\begin{itemize}
\item Longest match rule (``maximal munch rule''): The
longest initial substring matched by any regular expression
is taken as the next token.
\begin{center}
\bl{$\texttt{\Grid{iffoo\VS bla}}$}
\end{center}\medskip
\item Rule priority:
For a particular longest initial substring, the first regular
expression that can match determines the token.
\begin{center}
\bl{$\texttt{\Grid{if\VS bla}}$}
\end{center}
\end{itemize}\bigskip\pause
\small
\hfill Kuklewicz: most POSIX matchers are buggy\\
\footnotesize
\hfill \url{http://www.haskell.org/haskellwiki/Regex_Posix}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{POSIX Regex Matching}
\begin{itemize}
\item Sulzmann \& Lu came up with a beautiful
idea for how to extend the simple regular expression
matcher to POSIX matching/lexing (FLOPS 2014)\bigskip\bigskip
\begin{tabular}{@{\hspace{4cm}}c@{}}
\includegraphics[scale=0.20]{pics/sulzmann.jpg}\\[-2mm]
\hspace{0cm}\footnotesize Martin Sulzmann
\end{tabular}\bigskip\bigskip
\item the idea: define an inverse operation to the derivatives
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Regexes and Values}
Regular expressions and their corresponding values
(for \emph{how} a regular expression matched a string):
\begin{center}
\begin{columns}
\begin{column}{3cm}
\begin{tabular}{@{}rrl@{}}
\bl{$r$} & \bl{$::=$} & \bl{$\varnothing$}\\
& \bl{$\mid$} & \bl{$\epsilon$} \\
& \bl{$\mid$} & \bl{$c$} \\
& \bl{$\mid$} & \bl{$r_1 \cdot r_2$}\\
& \bl{$\mid$} & \bl{$r_1 + r_2$} \\
\\
& \bl{$\mid$} & \bl{$r^*$} \\
\\
\end{tabular}
\end{column}
\begin{column}{3cm}
\begin{tabular}{@{\hspace{-7mm}}rrl@{}}
\bl{$v$} & \bl{$::=$} & \\
& & \bl{$Empty$} \\
& \bl{$\mid$} & \bl{$Char(c)$} \\
& \bl{$\mid$} & \bl{$Seq(v_1,v_2)$}\\
& \bl{$\mid$} & \bl{$Left(v)$} \\
& \bl{$\mid$} & \bl{$Right(v)$} \\
& \bl{$\mid$} & \bl{$[]$} \\
& \bl{$\mid$} & \bl{$[v_1,\ldots\,v_n]$} \\
\end{tabular}
\end{column}
\end{columns}
\end{center}\pause
There is also a notion of a string behind a value: \bl{$|v|$}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Sulzmann \& Lu Matcher}
We want to match the string \bl{$abc$} using \bl{$r_1$}:
\begin{center}
\begin{tikzpicture}[scale=2,node distance=1.3cm,every node/.style={minimum size=8mm}]
\node (r1) {\bl{$r_1$}};
\node (r2) [right=of r1] {\bl{$r_2$}};
\draw[->,line width=1mm] (r1) -- (r2) node[above,midway] {\bl{$der\,a$}};\pause
\node (r3) [right=of r2] {\bl{$r_3$}};
\draw[->,line width=1mm] (r2) -- (r3) node[above,midway] {\bl{$der\,b$}};\pause
\node (r4) [right=of r3] {\bl{$r_4$}};
\draw[->,line width=1mm] (r3) -- (r4) node[above,midway] {\bl{$der\,c$}};\pause
\draw (r4) node[anchor=west] {\;\raisebox{3mm}{\bl{$\;\;nullable?$}}};\pause
\node (v4) [below=of r4] {\bl{$v_4$}};
\draw[->,line width=1mm] (r4) -- (v4);\pause
\node (v3) [left=of v4] {\bl{$v_3$}};
\draw[->,line width=1mm] (v4) -- (v3) node[below,midway] {\bl{$inj\,c$}};\pause
\node (v2) [left=of v3] {\bl{$v_2$}};
\draw[->,line width=1mm] (v3) -- (v2) node[below,midway] {\bl{$inj\,b$}};\pause
\node (v1) [left=of v2] {\bl{$v_1$}};
\draw[->,line width=1mm] (v2) -- (v1) node[below,midway] {\bl{$inj\,a$}};\pause
\draw[->,line width=0.5mm] (r3) -- (v3);
\draw[->,line width=0.5mm] (r2) -- (v2);
\draw[->,line width=0.5mm] (r1) -- (v1);
\draw (r4) node[anchor=north west] {\;\raisebox{-8mm}{\bl{$mkeps$}}};
\end{tikzpicture}
\end{center}
\only<10>{
The original ideas of Sulzmann and Lu are the \bl{\textit{mkeps}}
and \bl{\textit{inj}} functions (ommitted here).}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t,squeeze]
\frametitle{Sulzmann \& Lu Paper}
\begin{itemize}
\item I have no doubt the algorithm is correct ---
the problem is I do not believe their proof.
\begin{center}
\begin{bubble}[10cm]\small
``How could I miss this? Well, I was rather careless when
stating this Lemma :)\smallskip
Great example how formal machine checked proofs (and
proof assistants) can help to spot flawed reasoning steps.''
\end{bubble}
\end{center}\pause
\begin{center}
\begin{bubble}[10cm]\small
``Well, I don't think there's any flaw. The issue is how to
come up with a mechanical proof. In my world mathematical
proof $=$ mechanical proof doesn't necessarily hold.''
\end{bubble}
\end{center}\pause
\end{itemize}
\only<3>{%
\begin{textblock}{11}(1,4.4)
\begin{center}
\begin{bubble}[10.9cm]\small\centering
\includegraphics[scale=0.37]{pics/msbug.png}
\end{bubble}
\end{center}
\end{textblock}}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{\begin{tabular}{c}The Proof Idea\\[-1mm] by Sulzmann \& Lu
\end{tabular}}
\begin{itemize}
\item introduce an inductively defined ordering relation
\bl{$v \succ_r v'$} which captures the idea of POSIX matching
\item the algorithm returns the maximum of all possible
values that are possible for a regular expression.\pause
\bigskip\small
\item the idea is from a paper by Cardelli \& Frisch about
GREEDY matching (GREEDY $=$ preferring instant gratification to delayed
repletion):
\item e.g.~given \bl{$(a + (b + ab))^*$} and string \bl{$ab$}
\begin{center}
\begin{tabular}{ll}
GREEDY: & \bl{$[Left(a), Right(Left(b)]$}\\
POSIX: & \bl{$[Right(Right(Seq(a, b))))]$}
\end{tabular}
\end{center}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{}
\centering
\bl{\infer{\vdash Empty : \epsilon}{}}\hspace{15mm}
\bl{\infer{\vdash Char(c): c}{}}\bigskip
\bl{\infer{\vdash Seq(v_1, v_2) : r_1\cdot r_2}{\vdash v_1 : r_1 \quad \vdash v_2 : r_2}}
\bigskip
\bl{\infer{\vdash Left(v) : r_1 + r_2}{\vdash v : r_1}}\hspace{15mm}
\bl{\infer{\vdash Right(v): r_1 + r_2}{\vdash v : r_2}}\bigskip
\bl{\infer{\vdash [] : r^*}{}}\hspace{15mm}
\bl{\infer{\vdash [v_1,\ldots, v_n] : r^*}
{\vdash v_1 : r \quad\ldots\quad \vdash v_n : r}}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}<1>[c]
\frametitle{}
\small
%\begin{tabular}{@{}lll@{}}
%\bl{$POSIX(v, r)$} & \bl{$\dn$} & \bl{$\vdash v : r$}\\
% & & \bl{$\wedge \;\;(\forall v'.\;\; \vdash v' : r \,\wedge\, |v'| = |v|
% \Rightarrow v \succ_{\alert<2>{r}} v')$}
%\end{tabular}\bigskip\bigskip\bigskip
\centering
%\bl{\infer{Seq(v_1, v_2) \succ_{\alert<2>{r_1\cdot r_2}} Seq(v'_1, v'_2)}
% {v_1 = v'_1 \quad v_2 \succ_{\alert<2>{r_2}} v'_2}}\hspace{3mm}
%\bl{\infer{Seq(v_1, v_2) \succ_{\alert<2>{r_1\cdot r_2}} Seq(v'_1, v'_2)}
% {v_1 \not= v'_1 \quad v_1 \succ_{\alert<2>{r_1}} v'_1}}
%\bigskip
%\bl{\infer{Left(v) \succ_{\alert<2>{r_1 + r_2}} Left(v')}
% {v \succ_{\alert<2>{r_1}} v'}}\hspace{15mm}
%\bl{\infer{Right(v) \succ_{\alert<2>{r_1 + r_2}} Right(v')}
% {v \succ_{\alert<2>{r_2}} v'}}\bigskip\medskip
%\bl{\infer{Left(v) \succ_{\alert<2>{r_1 + r_2}} Right(v')}
% {length |v| \ge length |v'|}}\hspace{15mm}
%\bl{\infer{Right(v) \succ_{\alert<2>{r_1 + r_2}} Left(v')}
% {length |v| > length |v'|}}\bigskip
%\bl{$\big\ldots$}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Problems}
\begin{itemize}
\item Sulzmann: \ldots Let's assume \bl{$v$} is not
a $POSIX$ value, then there must be another one
\ldots contradiction.\bigskip\pause
\item Exists?
\begin{center}
\bl{$L(r) \not= \varnothing \;\Rightarrow\; \exists v.\;POSIX(v, r)$}
\end{center}\bigskip\bigskip\pause
\item in the sequence case
\bl{$Seq(v_1, v_2)\succ_{r_1\cdot r_2} Seq(v_1', v_2')$},
the induction hypotheses require
\bl{$|v_1| = |v'_1|$} and \bl{$|v_2| = |v'_2|$},
but you only know
\begin{center}
\bl{$|v_1| \;@\; |v_2| = |v'_1| \;@\; |v'_2|$}
\end{center}\pause\small
\item although one begins with the assumption that the two
values have the same flattening, this cannot be maintained
as one descends into the induction (alternative, sequence)
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Our Solution}
\begin{itemize}
\item a direct definition of what a POSIX value is, using
the relation \bl{$s \in r \to v$} (specification):\medskip
\begin{center}
\bl{\infer{[] \in \epsilon \to Empty}{}}\hspace{15mm}
\bl{\infer{c \in c \to Char(c)}{}}\bigskip\medskip
\bl{\infer{s \in r_1 + r_2 \to Left(v)}
{s \in r_1 \to v}}\hspace{10mm}
\bl{\infer{s \in r_1 + r_2 \to Right(v)}
{s \in r_2 \to v & s \not\in L(r_1)}}\bigskip\medskip
\bl{\infer{s_1 @ s_2 \in r_1 \cdot r_2 \to Seq(v_1, v_2)}
{\small\begin{array}{l}
s_1 \in r_1 \to v_1 \\
s_2 \in r_2 \to v_2 \\
\neg(\exists s_3\,s_4.\; s_3 \not= []
\wedge s_3 @ s_4 = s_2 \wedge
s_1 @ s_3 \in L(r_1) \wedge
s_4 \in L(r_2))
\end{array}}}
\bl{\ldots}
\end{center}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Properties}
It is almost trival to prove:
\begin{itemize}
\item Uniqueness
\begin{center}
If \bl{$s \in r \to v_1$} and \bl{$s \in r \to v_2$} then
\bl{$v_1 = v_2$}.
\end{center}\bigskip
\item Correctness
\begin{center}
\bl{$lexer(r, s) = v$} if and only if \bl{$s \in r \to v$}
\end{center}
\end{itemize}\bigskip\bigskip\pause
You can now start to implement optimisations and derive
correctness proofs for them. But we still do not know whether
\begin{center}
\bl{$s \in r \to v$}
\end{center}
is a POSIX value according to Sulzmann \& Lu's definition
(biggest value for \bl{$s$} and \bl{$r$})
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[t]
\frametitle{\Large\begin{tabular}{@{}c@{}}Pencil-and-Paper Proofs\\[-1mm]
in CS are normally incorrect\end{tabular}}
\begin{itemize}
\item case in point: in one of Roy's proofs he made the
incorrect inference
\begin{center}
if \bl{$\forall s.\; |v_2| \alert{\not}\in L(der\,c\,r_1) \cdot s$}
then \bl{$\forall s.\; c\,|v_2| \alert{\not}\in L(r_1) \cdot s$}
\end{center}\bigskip
while
\begin{center}
if \bl{$\forall s.\; |v_2| \in L(der\,c\,r_1) \cdot s$}
then \bl{$\forall s.\; c\,|v_2| \in L(r_1) \cdot s$}
\end{center}
is correct
\end{itemize}
\begin{textblock}{11}(12,11)
\includegraphics[scale=0.15]{pics/roy.jpg}
\end{textblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Proofs in Math~vs.~in CS}
\alert{\bf My theory on why CS-proofs are often buggy}
\\[-10mm]\mbox{}
\begin{center}
\begin{tabular}{@{}cc@{}}
\begin{tabular}{@{}p{5.6cm}}
\includegraphics[scale=0.43]{pics/icosahedron.png}\\[2mm]
{\bf Math}: \\
\raggedright\small
in math, ``objects'' can be ``looked'' at from all
``angles'';\\\smallskip
non-trivial proofs, but it seems
difficult to make mistakes
\end{tabular} &
\begin{tabular}{p{5cm}}
\includegraphics[scale=0.2]{pics/sel4callgraph.jpg}\\[3mm]
\raggedright
{\bf Code in CS}: \\
\raggedright\small
the call-graph of the seL4 microkernel OS;\\\smallskip
easy to make mistakes\\ \mbox{}\\
\end{tabular}
\end{tabular}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[c]
\frametitle{Conclusion}
\begin{itemize}
\item we replaced the POSIX definition of Sulzmann \& Lu by a
new definition (ours is inspired by work of Vansummeren,
2006)\medskip
\item their proof contained small gaps (acknowledged) but had
also fundamental flaws\medskip
\item now, its a nice exercise for theorem proving\medskip
\item some optimisations need to be applied to the algorithm
in order to become fast enough\medskip
\item can be used for lexing, is a small beautiful functional
program
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[b]
\frametitle{
\begin{tabular}{c}
\mbox{}\\[13mm]
% \alert{\Large Thank you very much again}\\
% \alert{\Large for the invitation!}\\
\alert{\LARGE Questions?}
\end{tabular}}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
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