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+\documentclass{beamer}
+
+\title{Implementing and Evaluating a Strategy-Iteration Based Static Analyser within the LLVM framework}
+\author{Carlo Zancanaro}
+\def\sid{309222257}
+\def\supervisor{Bernhard Scholz}
+
+\usepackage[numbers]{natbib}
+\usepackage{mdframed}
+\usepackage{algorithmicx}
+\usepackage{tikz}
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.4} 
+\algblockx[While]{While}{EndWhile}{\textbf{while}~}{\textbf{endwhile}}
+
+\newcommand\Z{\mathbb{Z}}
+\newcommand\CZ{\overline{\Z}}
+\def\newblock{\hskip .11em plus .33em minus .07em} % important line
+
+\begin{document}
+
+\begin{frame}
+  \maketitle
+\end{frame}
+
+\section{Introduction}
+\begin{frame}{Bugs are bad}
+  \begin{itemize}
+  \item<1->
+    Money - recently Knight Capital, US\$440 million lost in a day
+  \item<2->
+    Time - 50\% of development time is spent debugging\cite{debugging-book}
+  \item<3->
+    Security - buffer overflows and other security flaws
+  \item<4->
+    Lives - X-ray machines, helicopters, cars
+  \end{itemize}
+\end{frame}
+\begin{frame}{Static analysis is good}
+  \begin{itemize}
+  \item<1->
+    The more bugs we can catch at compile time, the better
+  \item<2->
+    We can't catch all bugs - Rice's theorem\cite{Rice}
+  \end{itemize}
+\end{frame}
+
+\section{Prior work}
+\begin{frame}[fragile]{Modelling programs}
+  \begin{center}
+    \begin{mdframed}
+      \begin{algorithmic}
+        \State $x = 0$
+        \State $y = 1$
+        \While {$x < 8$}
+          \Comment{value at start of this line}
+          \State $x = x + 2$
+          \State $y = y + 2$
+        \EndWhile
+      \end{algorithmic}
+    \end{mdframed}
+  \end{center}
+  \begin{align*}
+    \only<1>{~ \\ ~}
+    \only<2>{x &= \{0\} \\ y &= \{1\}}
+    \only<3>{x &= \{0, 2\} \\ y &= \{1, 3\}}
+    \only<4>{x &= \{0, 2, 4\} \\ y &= \{1, 3, 5\}}
+    \only<5>{x &= \{0, 2, 4, 6\} \\ y &= \{1, 3, 5, 7\}}
+    \only<6>{x &= \{0, 2, 4, 6, 8\} \\ y &= \{1, 3, 5, 7, 9\}}
+  \end{align*}
+\end{frame}
+
+\begin{frame}{Abstract interpretation}
+  Basic idea: simplify your domain
+
+  Instead of arbitrary subsets of $\Z$, something less precise:
+
+  \begin{itemize}
+  \item
+    signs\cite{CousotCousot77-1}: $x \in \{ \Z, \Z^+, \Z^-, 0\}$
+  \item
+    ranges\cite{CousotCousot77-1}: $x \le a; -x \le b, ~a,b \in \Z$
+  \item
+    zones\cite{mine:padoII}: $x - y \le c; \pm x \le c ~c \in Z$
+  \end{itemize}
+\end{frame}
+\begin{frame}[fragile]{Abstract interpretation}
+  \begin{figure}
+    \begin{tikzpicture}[scale=0.6,dot/.style={circle,fill,inner sep=1pt}]
+      \draw (0,0) -- coordinate (x axis mid) (10,0);
+      \draw (0,0) -- coordinate (y axis mid) (0,10);
+      
+      \foreach \x in {0,...,10}
+        \draw (\x,1pt) -- (\x,-3pt)
+          node[anchor=north] {\x};
+      \node[anchor=west] at (10,0) {$x$};
+      \foreach \y in {0,...,10}
+        \draw (1pt,\y) -- (-3pt,\y)
+          node[anchor=east] {\y};
+      \node[anchor=south] at (0,10) {$y$};
+
+      \foreach \v in {0,...,4}
+        \node [dot,radius=0.2pt] at (0 + 2*\v,1 + 2*\v) { };
+
+      \draw [draw=black, fill=gray, fill opacity=0.2]
+        (0,1) -- (0,9) -- (8,9) -- (8,1) -- (0,1);
+    \end{tikzpicture}
+    \caption{Comparison between concrete and abstract domains}
+  \end{figure}
+\end{frame}
+
+\begin{frame}{$\max$-strategy improvement}
+  \begin{itemize}
+  \item
+    Transform a program into equations
+  \item
+    Solve equations
+  \end{itemize}
+\end{frame}
+
+\begin{frame}[fragile]{$\max$-strategy example}
+  \begin{center}
+    \begin{mdframed}
+      \begin{algorithmic}
+        \State $x = 0$
+        \Comment $A$
+        \While {$x \le 8$} \Comment $B$
+          \State $x = x + 2$
+        \EndWhile
+        \State print$(x)$ \Comment $C$
+      \end{algorithmic}
+    \end{mdframed}
+  \end{center}
+  \alt<1>{\begin{align*}
+      ub(x)_A &\ge \infty \\
+      ub(x)_B &\ge 0 \\
+      ub(x)_B &\ge \min(ub(x)_B, 8) + 2 \\
+      ub(x)_C &\ge ub(x)_B
+    \end{align*}
+  }{\begin{align*}
+      ub(x)_A &= \infty \\
+      ub(x)_B &= \max(0, \min(ub(x)_B, 8) + 2) \\
+      ub(x)_C &= ub(x)_B \\
+      ~
+    \end{align*}}
+  \only<2>{}
+\end{frame}
+
+\begin{frame}[fragile]{$\max$-strategies}
+  A $\max$-strategy is a decision about which argument in a
+  $\max$-expression to choose.
+  \begin{align*}
+    ub(x)_A &= \infty \\
+    ub(x)_B &= \max(\alert<2>{0}, \alert<3>{\min(ub(x)_B, 8) + 2}) \\
+    ub(x)_C &= ub(x)_B
+  \end{align*}
+\end{frame}
+
+\begin{frame}[fragile]{Solver}
+  \begin{figure}
+    \begin{tikzpicture}
+      \node [rectangle, draw, align=center](max) at (0,0) {Maximiser \\ improve strategy};
+      \node [rectangle, draw, align=center](min) at (0,-3) {Minimiser \\ find assignment};
+
+      \path[]
+      (min) edge[bend left,arrows={-latex}]
+      node[anchor=east]{variable assignment}
+      (max)
+      
+      (max) edge[bend left,arrows={-latex}]
+      node[anchor=west]{$\max$-strategy}
+      (min);
+    \end{tikzpicture}
+    \caption{The high-level structure of the solver presented in
+      \cite{Gawlitza:2007:PFC:1762174.1762203}}
+  \end{figure}
+\end{frame}
+
+\section{Contribution}
+\begin{frame}[fragile]{Solver - enhancements}
+  The big idea: take into account data-dependencies
+  \begin{align*}
+    \alert<4>{x} &\alert<4>{= \max(0, \min(x - 1, \alert<3>{y}))} \\
+    \alert<2-4>{y} &= \max(0, x + 5, x) \\
+    z &= \max(0, z + 1, x)
+  \end{align*}
+\end{frame}
+
+\begin{frame}[fragile]{Solver - enhancements}
+  \begin{figure}
+    \begin{tikzpicture}
+      \node [rectangle, draw, align=center](max) at (0,0) {Maximiser \\ improve strategy};
+      \node [rectangle, draw, align=center](min) at (0,-3) {Minimiser \\ find assignment};
+
+      \path[]
+      (min) edge[bend left,arrows={-latex}]
+      node[anchor=east,align=right]{variable assignment \\ change set}
+      (max)
+      
+      (max) edge[bend left,arrows={-latex}]
+      node[anchor=west,align=left]{$\max$-strategy \\ change set}
+      (min);
+    \end{tikzpicture}
+    \caption{The high-level structure of our enhanced solver}
+  \end{figure}
+\end{frame}
+
+\section{Results}
+\begin{frame}{Implementation}
+  \begin{itemize}
+  \item
+    Implemented in C++
+  \item
+    Integrated into the LLVM/Clang static analysis framework
+  \end{itemize}
+\end{frame}
+
+\begin{frame}{Example system}
+  \begin{align*}
+    x_0 &= \max(-\infty, 0) \\
+    x_1 &= \max(-\infty, x_0) \\
+    x_2 &= \max(-\infty, x_1) \\
+    ... &~ ... \\
+    x_n &= \max(-\infty, x_{n-1})
+  \end{align*}
+\end{frame}
+
+\begin{frame}{Runtime improvements}
+  \begin{figure}
+    \begin{tikzpicture}
+      \begin{loglogaxis}[% 
+          scale only axis,
+          width=9cm,
+          height=5cm,
+          xmin=1, xmax=65536,
+          xlabel=size of equation system,
+          ymin=0, ymax=40,
+          ylabel=time (in seconds),
+          axis on top]
+        \addplot[
+          ybar,
+          bar width=0.5cm, 
+          bar shift=0in,
+          fill=red,
+          draw=black]
+        plot coordinates{ 
+          (1,0.000021982 )
+          (2,0.000037083 )
+          (4,0.000090918 )
+          (8,0.000331411 )
+          (16,0.001678790 )
+          (32,0.010238535 )
+          (64,0.069774912 )
+          (128,0.520062180 )
+          (256,3.983806321 )
+          (512,31.469653195 )
+        };
+
+      \end{loglogaxis}
+    \end{tikzpicture}
+    \caption{Performance of the naive algorithm}
+  \end{figure}
+\end{frame}
+
+\begin{frame}{Runtime improvements}
+  \begin{figure}
+    \begin{tikzpicture}
+      \begin{loglogaxis}[%
+          scale only axis,
+          width=9cm,
+          height=5cm,
+          xmin=1, xmax=65536,
+          xlabel=size of equation system,
+          ymin=0, ymax=40,
+          ylabel=time (in seconds),
+          axis on top]
+        \addplot[
+          ybar,
+          bar width=0.5cm, 
+          bar shift=0in,
+          fill=red,
+          draw=black] 
+        plot coordinates{ 
+          (1,0.000044508 )
+          (2,0.000063953 )
+          (4,0.000094303 )
+          (8,0.000176679 )
+          (16,0.000312958 )
+          (32,0.000642421 )
+          (64,0.001194996 )
+          (128,0.002451839 )
+          (256,0.004786286 )
+          (512,0.009280946 )
+          (1024,0.019159827 )
+          (2048,0.038067689 )
+          (4096,0.080535540 )
+          (8192,0.151181571 )
+          (16384,0.339113750 )
+          (32768,0.593622054 )
+          (65536,1.217525509 )
+        };
+
+      \end{loglogaxis}
+    \end{tikzpicture}
+    \caption{Performance of our improved algorithm}
+  \end{figure}
+\end{frame}
+
+\section{Future work}
+\begin{frame}{Future work}
+  \begin{itemize}
+    \item
+      Still slightly over-approximating dependencies
+    \item
+      LLVM/Clang integration is only a proof-of-concept
+  \end{itemize}
+\end{frame}
+
+\begin{frame}[fragile,allowframebreaks]{References}
+  \phantomsection
+  \bibliographystyle{abbrvnat}
+  \bibliography{references}
+\end{frame}
+
+\begin{frame}{Contributions}
+  \begin{itemize}
+    \item
+      Improvement of $\max$-strategy iteration algorithm, leveraging
+      sparsity of variable dependencies
+    \item
+      Implementation of a $\max$-strategy iteration based static
+      analyser in the LLVM/Clang framework
+  \end{itemize}
+\end{frame}
+
+\end{document}