\documentclass[dvipsnames,14pt,t]{beamer}
\usepackage{slides}
\usepackage{langs}
\usepackage{graph}
%\usepackage{grammar}
\usepackage{soul}
\usepackage{data}
\usepackage{proof}

% beamer stuff 
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\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 {re-python.data};
\addplot[brown,mark=pentagon*, mark options={fill=white}] 
  table {re-ruby.data};  
\end{axis}
\end{tikzpicture}
\end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}

\only<2->{
\begin{textblock}{3}(13,5)
\includegraphics[scale=0.33]{pics/skeleton.jpg}
\end{textblock}}

\begin{textblock}{3}(5.3,7)
\begin{tikzpicture}
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\begin{tikzpicture}
\node at (0,0) [single arrow, shape border rotate=90, fill=red,text=red]{\mbox{a}};
\end{tikzpicture}
\end{textblock}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{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);
  
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\node[anchor=base] at (6.5,2.8) 
     {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\centering  user: untrusted code\end{tabular}};

\draw[line width=1mm, red] (0.1,0.6) rectangle (2.1,4);
  \node[anchor=base] at (1.1,2.3) 
     {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\centering  developer ---\\ web server\end{tabular}};
  
\onslide<2->{
  \draw[line width=1mm, red, fill=red] (5.5,0.6) rectangle (7.5,1.8);
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  \node at (3.6,3.0) [single arrow, fill=red,text=white, 
                      minimum height=3.4cm]{\bf code};
  
\node at (3.6,1.3) [single arrow, fill=red,text=white, 
                      minimum height=3.4cm]{\bf certificate};
\node at (3.6,1.9) {\small\color{gray}{\mbox{}\hspace{-1mm}a proof in LF}};

\end{tikzpicture}
\end{center}
\end{block}
\end{textblock}

\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{frame}<2->[squeeze]
  \frametitle{} 
  
  \begin{columns}
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  \end{textblock}
  \end{column}
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  \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};
  \end{tikzpicture}}
  \end{textblock}
  
  \begin{textblock}{0}(0.4,12.8)
  \onslide<6->{
  \begin{bubble}[11cm]
  \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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{frame}[c]
  \frametitle{Real-Time Scheduling} 

  \begin{textblock}{11}(1,3)
  \begin{tabular}{@{\hspace{-10mm}}l}
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  \node at (-2.5,-1.5) {\textcolor{white}{a}};
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   \draw[fill, blue!50] (1,0) rectangle (3,0.6);
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   \draw[blue!100, very thick] (2,3) rectangle (3, 3.6);
   %\draw[red, <-, line width = 2mm] (2,-0.1) -- (2, -1);
  }
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   \draw[fill, blue!50] (1,0) rectangle (4,0.6);
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  \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<8->{\node [anchor=base] at (-1.5, 1.7) {\small medium pr.};}

  \end{tikzpicture}
  \end{tabular}
  \end{textblock}

  \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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{frame}[c]

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  }
  \only<2>{
    \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 $B_R$};
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    \draw[very thick,blue!100] (3,3) rectangle (4,3.6);
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  }
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    \draw[fill, blue!50] (1,0) rectangle (3,0.6);
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    \draw[very thick,blue!100] (7,3) rectangle (8,3.6);
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    \draw[blue!50, ->, line width = 2mm] (3,1) -- (3, 2.5);
  }
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    \draw[fill, blue!50] (1,0) rectangle (3,0.6);
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  }
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   \draw[fill, blue!50] (1,0) rectangle (3,0.6);
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    \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);
  }
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    \draw[fill, blue!50] (1,0) rectangle (3,0.6);
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    \node at (6.7,0.9) {\small $B_R$};
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  }
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  }
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    \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);
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    \draw[very thick, blue!50] (5,0) rectangle (7,0.6);
    \node at (1.5,0.9) {\small $A_L$};
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    \draw[very thick,blue!100] (1.5,0) -- (1.5,0.6);
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    \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 {re-python.data};
\addplot[brown,mark=pentagon*, mark options={fill=white}] 
  table {re-ruby.data};  
\addplot[red,mark=triangle*,mark options={fill=white}] 
  table {re1.data};  
\addplot[green,mark=square*,mark options={fill=white}] 
  table {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 {re2b.data};
\addplot[black,mark=square*,mark options={fill=white}] table {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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