Binary file slides/slides09.pdf has changed

--- a/slides/slides09.tex	Tue Nov 25 11:48:16 2014 +0000
+++ b/slides/slides09.tex	Tue Nov 25 15:54:28 2014 +0000
@@ -1165,7 +1165,7 @@
 
   
   \begin{columns}
-  \begin{column}{4cm}
+  \begin{column}{3cm}
   \begin{tabular}{cc|l}
   $e_1$ & $e_2$ & $e_1 + e_2$\\\hline{}
   - & - & -\\
@@ -1178,7 +1178,7 @@
   \end{tabular}
   \end{column}
 
-  \begin{column}{4cm}
+  \begin{column}{3cm}
   \begin{tabular}{cc|l}
   $e_1$ & $e_2$ & $e_1 * e_2$\\\hline{}
   - & - & +\\
@@ -1195,6 +1195,69 @@
   \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}[fragile]
+\mbox{}\\[-13mm]\mbox{}
+
+\small
+\begin{center}
+\bl{\begin{tabular}{lcl}
+$[n]_{env}$ & $\dn$ & 
+$\begin{cases}
+\{+\} & \text{if}\; n > 0\\
+\{-\} & \text{if}\; n < 0\\
+\{0\} & \text{if}\; n = 0
+\end{cases}$\\
+$[x]_{env}$ & $\dn$ & $env(x)$\\
+$[e_1 + e_2]_{env}$ & $\dn$ & $[e_1]_{env} \oplus [e_2]_{env}$\\
+$[e_1 * e_2]_{env}$ & $\dn$ & $[e_1]_{env} \otimes [e_2]_{env}$\\
+$[e_1 = e_2]_{env}$ & $\dn$ & $\{0, +\}$\\
+\end{tabular}}
+\end{center}
+
+\scriptsize
+\begin{lstlisting}[language=Scala,numbers=none,xleftmargin=-5mm]
+def aeval_exp(e: Exp, aenv: AEnv) : Set[Abst] = e match {
+  case Num(0) => Set(Zero)
+  case Num(n) if (n < 0) => Set(Neg)
+  case Num(n) if (n > 0) => Set(Pos)
+  case Var(x) => aenv(x)
+  case Plus(e1, e2) => 
+    aplus(aeval_exp(e1, aenv), aeval_exp(e2, aenv))
+  case Times(e1, e2) => 
+    atimes(aeval_exp(e1, aenv), aeval_exp(e2, aenv))
+  case Equ(e1, e2) => Set(Zero, Pos)
+}
+\end{lstlisting}
+
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+  \begin{frame}[c]
+  \frametitle{Sign Analysis}
+
+  \begin{itemize}
+  \item We want to find out whether \texttt{a} and \texttt{n}
+  are always positive?\medskip
+  \item After a few optimisations, we can indeed find this out.
+  \begin{itemize}
+  \item \texttt{if} returns \texttt{+} or \texttt{0}
+  \item branch into only one dircection if you know
+  \item if $x$ is \texttt{+}, then $x + \texttt{-1}$ 
+  cannot be negative 
+  \end{itemize}\bigskip
+  
+  \item What is this good for? Well, you do not need 
+  underflow checks (in order to prevent buffer-overflow
+  attacks).
+  \end{itemize}
+  
+  \end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
 
 \end{document}