
diff -r 5fa8344a5176 -r a35041d5707c Slides/slides02.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Slides/slides02.tex	Tue Jun 14 11:58:20 2016 +0100
@@ -0,0 +1,1543 @@
+\documentclass[dvipsnames,14pt,t]{beamer}
+\usepackage{slides}
+\usepackage{langs}
+\usepackage{graph}
+%\usepackage{grammar}
+\usepackage{soul}
+\usepackage{data}
+\usepackage{proof}
+
+% beamer stuff 
+\renewcommand{\slidecaption}{SMAL, 23.3.2016}
+\newcommand{\bl}[1]{\textcolor{blue}{#1}}
+
+\newcommand\grid[1]{%
+\begin{tikzpicture}[baseline=(char.base)]
+  \path[use as bounding box]
+    (0,0) rectangle (1em,1em);
+  \draw[red!50, fill=red!20]
+    (0,0) rectangle (1em,1em);
+  \node[inner sep=1pt,anchor=base west]
+    (char) at (0em,\gridraiseamount) {#1};
+\end{tikzpicture}}
+\newcommand\gridraiseamount{0.12em}
+
+\makeatletter
+\newcommand\Grid[1]{%
+  \@tfor\z:=#1\do{\grid{\z}}}
+\makeatother	
+
+\newcommand\Vspace[1][.3em]{%
+  \mbox{\kern.06em\vrule height.3ex}%
+  \vbox{\hrule width#1}%
+  \hbox{\vrule height.3ex}}
+
+\def\VS{\Vspace[0.6em]}
+
+
+
+\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}
+\node at (0,0) [single arrow, shape border rotate=90, fill=red,text=red]{\mbox{a}};
+\end{tikzpicture}
+\end{textblock}
+
+\begin{textblock}{3}(8.7,7)
+\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);
+  
+\draw[line width=1mm, red] (5.5,0.6) rectangle (7.5,4);
+\node[anchor=base] at (6.5,2.8) 
+     {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\centering  user: 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);
+  \node[anchor=base,white] at (6.5,1.1) 
+     {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\bf\centering proof- checker\end{tabular}};}
+
+  \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$2700 lines of code (1865 loc of\\ LF definitions; 
+803 loc in C including 2 library functions)\\[-3mm]
+\item<2-> 167 loc in C implement a type-checker\\ (proved correct by Harper and Pfenning)
+\end{itemize}
+\end{textblock}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \begin{frame}<2->[squeeze]
+  \frametitle{} 
+  
+  \begin{columns}
+  \tikzstyle{every node}=[node distance=25mm,text height=1.5ex, text depth=.25ex]
+  \tikzstyle{node1}=[rectangle, minimum size=10mm, rounded corners=3mm, very thick, 
+                     draw=black!50, top color=white, bottom color=black!20]
+  \tikzstyle{node2}=[rectangle, minimum size=12mm, rounded corners=3mm, very thick, 
+                     draw=red!70, top color=white, bottom color=red!50!black!20]
+  
+  \begin{column}{0.8\textwidth}
+  \begin{textblock}{0}(1,2)
+
+  \begin{tikzpicture}
+  \matrix[ampersand replacement=\&,column sep=7mm, row sep=5mm]
+  { \&[-10mm] 
+    \node (def1)   [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}}; \&
+    \node (proof1) [node1] {\large Proof}; \&
+    \node (alg1)   [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}}; \\
+    
+    \onslide<4->{\node {\begin{tabular}{c}\small 1st\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
+    \onslide<4->{\node (def2)   [node2] {\large Spec$^\text{+ex}$};} \&
+    \onslide<4->{\node (proof2) [node1] {\large Proof};} \&
+    \onslide<4->{\node (alg2)   [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\
+     
+    \onslide<5->{\node {\begin{tabular}{c}\small 2nd\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
+    \onslide<5->{\node (def3)   [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \&
+    \onslide<5->{\node (proof3) [node1] {\large Proof};} \&
+    \onslide<5->{\node (alg3)   [node2] {\large Alg$^\text{-ex}$};} \\
+
+    \onslide<6->{\node {\begin{tabular}{c}\small 3rd\\[-2.5mm] \footnotesize solution\end{tabular}};} \&
+    \onslide<6->{\node (def4)   [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}};} \&
+    \onslide<6->{\node (proof4) [node2] {\large\hspace{1mm}Proof\hspace{1mm}};} \&
+    \onslide<6->{\node (alg4)   [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}};} \\
+  };
+
+  \draw[->,black!50,line width=2mm] (proof1) -- (def1);
+  \draw[->,black!50,line width=2mm] (proof1) -- (alg1);
+  
+  \onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (def2);}
+  \onslide<4->{\draw[->,black!50,line width=2mm] (proof2) -- (alg2);}
+
+  \onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (def3);}
+  \onslide<5->{\draw[->,black!50,line width=2mm] (proof3) -- (alg3);}
+  
+  \onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (def4);}
+  \onslide<6->{\draw[->,black!50,line width=2mm] (proof4) -- (alg4);}
+
+  \onslide<3->{\draw[white,line width=1mm] (1.1,3.2) -- (0.9,2.85) -- (1.1,2.35) -- (0.9,2.0);} 
+  \end{tikzpicture}
+
+  \end{textblock}
+  \end{column}
+  \end{columns}
+
+
+  \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}
+  \begin{tikzpicture}[scale=1.1]
+  \node at (-2.5,-1.5) {\textcolor{white}{a}};
+  \node at (8,4) {\textcolor{white}{a}};
+  
+    
+  \only<1>{
+   \draw[fill, blue!50] (1,0) rectangle (3,0.6);
+  }
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+   \draw[fill, blue!50] (1,0) rectangle (2,0.6);
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+   \node at (2.5,0.9) {\small locks a resource};
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+   \draw[blue!50, very thick] (2,0) rectangle (4,0.6);
+   \draw[blue!100, very thick] (2,3) rectangle (3, 3.6);
+   %\draw[red, <-, line width = 2mm] (2,-0.1) -- (2, -1);
+  }
+  \only<5>{
+   \node at (2.5,0.9) {\small locks a resource};
+   \draw[fill, blue!50] (1,0) rectangle (4,0.6);
+   \draw[blue!100, fill] (4,3) rectangle (5, 3.6);
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+   \draw[red, <-, line width = 2mm] (2,-0.1) -- (2, -1);
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+   \draw[red, <-, line width = 2mm] (2.5,-0.1) -- (2.5, -1);
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+   \node at (2.5,0.9) {\small locks a resource}; 
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+   \draw[red, <-, line width = 2mm] (3.5,-0.1) -- (3.5, -1);
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+   \draw[red, <-, line width = 2mm] (4.5,-0.1) -- (4.5, -1);
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+   \draw[fill, blue!50] (1,0) rectangle (2.5,0.6);
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+   \draw[red, <-, line width = 2mm] (5.5,-0.1) -- (5.5, -1);
+   \node [anchor=base] at (6.3, 1.8) {\Large\ldots};
+  }
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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]
+
+  \begin{textblock}{11}(1,3)
+  \begin{tabular}{@{\hspace{-10mm}}l}
+  \begin{tikzpicture}[scale=1.1]
+  \node at (-2.5,-1.5) {\textcolor{white}{a}};
+  \node at (8,4) {\textcolor{white}{a}};
+  
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+    \draw[red, fill] (5,1.5) rectangle (6,2.1);
+    \draw[red, fill] (6.05,1.5) rectangle (7,2.1);
+  }
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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);
+    \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}
+
+
+%%% Local Variables:  
+%%% mode: latex
+%%% TeX-master: t
+%%% End: 
+




