\documentclass{beamer}

\usepackage{amsmath,graphicx,makeidx,listings,colortbl,helvet,ifthen,tikz,pgfpages,ifthen}

\begin{document}

\begin{frame}{Abstract Domain: {\bf Zones} (1/2)}
      \bf Goal: \emph{Tight bounds} on the possible values of \emph{integer variables}
      
      \qquad\qquad (like $x_1 \leq 42, \; x_2 \leq 13$)

      \bigskip
      \qquad$\leadsto$ buffer overflows, worst-case execution times, etc.
      
      \bigskip
      \bf Definition:
      Invariants of the form 
      \begin{align*}
        \bigwedge_x x \leq b_{x} \wedge \bigwedge_x -x \leq b_{-x} \wedge \bigwedge_{x,y} x - y \leq b_{x-y}
      \end{align*}
      That is:
               \emph{bounds} on \emph{variables} and
                \emph{differences of variables}.

      \bigskip
      \bf Applications:

      \smallskip
      Model-checking of timed automata \cite{DBLP:conf/eef/Yovine96,DBLP:conf/rtss/LarsenLPY97}
      
      \smallskip
      Static program analysis \cite{Sagiv01,DBLP:conf/pado/Mine01}

\end{frame}

%%%

\begin{frame}{Abstract Domain: {\bf Zones} (2/2)}
    \bf Our point of view:
    Zones are special \emph{template polyhedra}
    
    \smallskip
    \qquad$\leadsto$ template polyhedra: \cite{DBLP:conf/vmcai/SankaranarayananSM05}
 
    \bigskip
    \bf Given: 
    \emph{template constraint matrix} $T \in \{-1,0,1\}^{m\times n}$, where every row contains 
               \emph{at most one $1$}, and 
               \emph{at most one $-1$}. 
               Then:
\begin{align*}
  \gamma(b) 
    &:= 
    \{ x \in \Z^n \mid T x \leq b \}
    && 
    \text{for all } b \in \CZ^m
  \\
  \alpha(X) 
    &:= 
    \min \; \{ b \in \CZ^m \mid \gamma(b) \supseteq X \}
    && 
    \text{for all } X \subseteq \Z^n
\end{align*}	

  \bf Example:
  Let
  $
    T
    =
	      \begin{pmatrix}
	        1 & 0 \\
	        0 & 1 \\
	        -1 & 1
	      \end{pmatrix}
  $.
  Then:

  $
  \gamma\left(
    \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix}
  \right)
  =
  \left\{
  \bf Goal: 
  Find minimal \emph{zones} that are \emph{invariants}
 
  \bigskip
  \bf Example:
  \begin{center}
        
	$
	\begin{matrix}\mbox{\scalebox{0.8}{\begin{tikzpicture}
		 \node (start) [circle,draw] {$a$};
		 \node (null) [coordinate,right of = start] {};
		 \node (n1) [below of = start,circle,draw,yshift=-8mm]{$b$};
		 \path[ultra thick,->] (null) edge [] node [right] {} (start);
		 \path[ultra thick,->] (start) edge [] node [right,yshift=1mm] {$\stmtone$} (n1);
		 \path[ultra thick,->] (n1) edge [loop right,out=35,in=-35, distance=20mm] node [right] {
		   $\stmttwo$
		 } (n1);
	\end{tikzpicture}}}\end{matrix}
	\qquad\quad
	T 
	=
	      \begin{pmatrix}
	        1 & 0 \\
	        0 & 1 \\
	        -1 & 1
	      \end{pmatrix}
	$
  \end{center}

  \vspace*{-5mm}
  For program point $a$:

  \smallskip
  $
    \gamma \left( 
      \begin{pmatrix} \infty \\ \infty \\ \infty \end{pmatrix} 
    \right)
    =
    \left\{ 
      \begin{pmatrix} x_1\\x_2 \end{pmatrix} 
      \mid
	      \begin{pmatrix}
	        1 & 0 \\
	        0 & 1 \\
	        -1 & 1
	      \end{pmatrix}
      \begin{pmatrix} x_1\\x_2 \end{pmatrix} 
      \leq 
      \begin{pmatrix} \infty \\ \infty \\ \infty \end{pmatrix}            
    \right\}
  $

  \bigskip
  For program point $b$:

  \smallskip
  $
    \gamma \left( 
      \begin{pmatrix} 10 \\ 11 \\ 1 \end{pmatrix} 
    \right)
    =
    \left\{ 
      \begin{pmatrix} x_1\\x_2 \end{pmatrix} 
      \mid
	      \begin{pmatrix}
	        1 & 0 \\
	        0 & 1 \\
	        -1 & 1
	      \end{pmatrix}
      \begin{pmatrix} x_1\\x_2 \end{pmatrix} 
      \leq 
      \begin{pmatrix} 10 \\ 11 \\ 1 \end{pmatrix}            
    \right\}
  $
\end{frame}

%%%

\begin{frame}{How Can We Compute {\bf Small} Zones?}
  \begin{enumerate}
    \item<2->
    À la \cite{Sagiv01,DBLP:conf/pado/Mine01}
  
    \smallskip
    Abstr.\ Interpr.\ / widening / narrowing à la \cite{DBLP:conf/popl/CousotC77}
    
    \smallskip
    \emph{Properties}:
    \begin{itemize}
      \smallskip\item<4->
        Abstract transformers can be implemented in \emph{strongly polynomial time}
        (based on the Floyd-Warshall algorithm)
      \smallskip\item<5->
        \emph{Best} abstract transformers only for special classes of affine assignments 
        (like $x = x + d$ and $x = y + d$).
       \smallskip\item<6->
         Computed zones are \emph{not} necessarily \emph{minimal} 
    \end{itemize}
    \bigskip\item<7->
    À la \cite{DBLP:conf/csl/GawlitzaS07}
  
    \smallskip
    Abstr.\ Interpr.\ / strategy iteration for \emph{template polyhedra}
    
    \smallskip
    \emph{Properties}:
    \begin{itemize}
      \smallskip\item<9->
         Computed the \emph{minimal} zones that are invariant
      \smallskip\item<10->
         Arbitrary affine assignments (like $x = c_1 x_1 + \cdots + c_k x_k + d$)
      \smallskip\item<11->
         Proves that the corresponding decision problem is in $\mathbf{coNP}$.
           
         \qquad (it is also at least as hard as mean payoff games)
       \smallskip\item<12->
         At most exp. many iterations, each of which can be performed in \emph{weakly polynomial time} through \emph{linear programming}.
    \end{itemize}
  \end{enumerate}
\end{frame}

%%%

\begin{frame}{Obvious Question}
  \begin{center}
    \huge
    Can we do better?
  \end{center}

  \begin{center}  
  \bigskip
  \bf\huge Idea 
  
  \bigskip
  max-strategy improvement algorithm for \emph{template polyhedra}
  
  \vspace*{-2mm}
  \scalebox{2}{\rotatebox{-90}{$\leadsto$}}
  
  \smallskip
  max-strategy improvement algorithm for \emph{zones}

  \bigskip
  \bigskip
  \bf\huge Hope 
  
  \bigskip
  improvement step in \emph{weakly} polynomial time
  
  \vspace*{-2mm}
  \scalebox{2}{\rotatebox{-90}{$\leadsto$}}
  
  \smallskip
  improvement step in \emph{strongly} polynomial time
  \end{center}
\end{frame}

%%%

\outline{2}

%%%

\begin{frame}{Abstract Transformers}
      \emph{Statements} %
      %
      \begin{align*}
        T x \leq c \; \wedge \;  x' = A x + d
      \end{align*}

      \emph{Collecting Semantics} %
      %
      \begin{align*}
        \sem{T x \leq c \; \wedge \;  x' = A x + d} (X) 
        =
        \{ x' \mid x \in X \wedge Tx \leq c \wedge x' = Ax+d \} 
      \end{align*}

      \emph{Abstract Semantics} %
      %
      \begin{align*}
        \sem{T x \leq c \; \wedge \;  x' = A x + d}^\sharp (b)
          &= \alpha(\sem{T x \leq c \; \wedge \;  x' = A x + d} (\gamma( b )))
      \end{align*}
\end{frame}

%%%

\begin{frame}{First Step: {\bf Constraint System} over $\CZ$}
  
  \pos{1,1.0}{
	$
	\begin{matrix}\mbox{\scalebox{1}{\begin{tikzpicture}
		 \node (start) [circle,draw] {$a$};
		 \node (null) [coordinate,right of = start] {};
		 \node (n1) [below of = start,circle,draw,yshift=-8mm]{$b$};
		 \path[ultra thick,->] (null) edge [] node [right] {} (start);
		 \path[ultra thick,->] (start) edge [] node [right,yshift=1mm] {$\stmtone$} (n1);
		 \path[ultra thick,->] (n1) edge [loop right,out=35,in=-35, distance=20mm] node [right] 
		   {$\stmttwo$} (n1);
	\end{tikzpicture}}}\end{matrix}	
	\qquad
	T 
	=
	      \begin{pmatrix}
	        1 & 0 \\
	        0 & 1 \\
	        -1 & 1
	      \end{pmatrix}
	$
  }

  \pos{0.3,4}{\bf Least Solution of}
  
  \pos{0.3,4.5}{
  $
    \begin{array}{@{}r@{}l}
    {\vb^a_{1} &\geq \infty \\}
    {\vb^a_{2} &\geq \infty \\}
    {\vb^a_{3} &\geq \infty}
    \\[5pt]
    {\vb^b_{1} &\geq \pi_{1}(\sem{\stmtone}^\sharp (\vb^a_{1}, \vb^a_{2}, \vb^a_{3})) \\}
    {\vb^b_{2} &\geq \pi_{2}(\sem{\stmtone}^\sharp (\vb^a_{1}, \vb^a_{2}, \vb^a_{3})) \\}
    {\vb^b_{3} &\geq \pi_{3}(\sem{\stmtone}^\sharp (\vb^a_{1}, \vb^a_{2}, \vb^a_{3}))}
    \\[5pt]
    {\vb^b_{1} &\geq \pi_{1}(\sem{\stmttwo}^\sharp (\vb^b_{1}, \vb^b_{2}, \vb^b_{3})) \\}
    {\vb^b_{2} &\geq \pi_{2}(\sem{\stmttwo}^\sharp (\vb^b_{1}, \vb^b_{2}, \vb^b_{3})) \\}
    {\vb^b_{3} &\geq \pi_{3}(\sem{\stmttwo}^\sharp (\vb^b_{1}, \vb^b_{2}, \vb^b_{3})) }
    \end{array}
  $
  }
\end{frame}

%%%

\begin{frame}{Properties of the Abstract Semantics}
  \bf Observation:
  %
  \only<1-10>{%
      \begin{align*}
         &\pi_{k}(\sem{T x \leq c \; \wedge \;  x' = A x + d}^\sharp (b)) \\
         {=\;& \pi_{k}( \alpha(\sem{T x \leq c \; \wedge \;  x' = A x + d} (\gamma( b ))) ) \\}
         {=\;& \pi_{k}( \alpha(\sem{T x \leq c \; \wedge \;  x' = A x + d} ( \{ x \in \Z^n \mid Tx \leq b \} )) ) \\}
         {=\;& \pi_{k}( \alpha( \{ x' \mid x, x' \in \Z^n \;\wedge\; T x \leq b, c \; \wedge \;  x' = A x + d \} ) ) \\}
         {=\;& \pi_{k}( \alpha( \{ A x + d \mid x \in \Z^n \;\wedge\; T x \leq \min(b, c)  \} ) ) \\}
         {=\;& \sup \; \{ T_k(A x + d) \mid x \in \Z^n \;\wedge\; T x \leq \min(b, c)  \} \\}
         {=\;& T_k d + \sup \; \{ T_k A x \mid x \in \Z^n \;\wedge\;  T x \leq \min ( b, c )  \}  \\}
         {=\;& T_k d + \sup \; \{ T_k A x \mid x \in \R^n \;\wedge\;  T x \leq \min ( b, c )  \}  \\}
         {=\;& T_k d + \inf \; \{ (\min( b, c ))^\top y \mid y \in \R_{\geq 0}^n \wedge T^\top y = (T_1A)^\top y  \}  \\}
         {=\;& T_k d + \inf \; \{ (\min( b, c ))^\top y \mid y \in \Z_{\geq 0}^n \wedge T^\top y = (T_1A)^\top y  \}  \\}
      \end{align*}%
  }%
  \only<11->{%
      \begin{align*}
         &\pi_{k}(\sem{T x \leq c \; \wedge \;  x' = A x + d}^\sharp (b)) \\
         {=\;& T_k d + \inf \; \{ (\min( b, c ))^\top y \mid y \in \Z_{\geq 0}^n \wedge T^\top y = (T_1A)^\top y  \}  \\}
      \end{align*}
  }%
      
   {
      \bf Consequences:
      
      \bigskip
      $\pi_k \circ \sem{T x \leq c \; \wedge \;  x' = A x + d}^\sharp : \CZ^m \to \CZ$
      \begin{enumerate}
        \smallskip\item<13->
          is a point-wise \emph{minimum} of finitely many \emph{monotone} and \emph{affine} functions from the set $\CZ^M \to \CZ$.
        \smallskip\item<14->
          can be evaluated in \emph{strongly polynomial time}.
          
          \smallskip
            \qquad$\leadsto$ minimum cost flow network problem
            
            \qquad$\leadsto$ $\mathcal{O}( m \cdot \log m \cdot ( n + m \cdot \log m ))$
      \end{enumerate}
    }
    
%    \uncover<15>{
%      \bigskip
%      \bf So What?
%    }
\end{frame}

%%%

\begin{frame}{Minimum Cost Flow Network Problem}
  \begin{align*}
    \min  \;\; b^\top \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix} 
    \qquad\qquad
    \\[10pt]
    %
	      \begin{pmatrix}
	        1 & 0 & -1 \\
	        0 & 1 & 1\\
	        -1 & -1 & 0
	      \end{pmatrix}
	      \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}
	      &=
	      \begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix}
	      &
	      \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}
	      &\geq 
	      0
  \end{align*}
  
  \bigskip

  \begin{center}
	$
	\begin{matrix}\mbox{\scalebox{1}{\begin{tikzpicture}
		 \node (c1) [circle,draw] {$c_1$};
		 \node (c2) [circle,draw,right of = c1, xshift=10mm] {$c_2$};
		 \node (c3) [below of = c2,circle,draw,yshift=-8mm]{$c_3$};
		 \uncover{\path[ultra thick,->] (c1) edge [] node [left] {$y_1$} (c3);}
		 \uncover{\path[ultra thick,->] (c2) edge [] node [right] {$y_2$} (c3);}
		 \uncover{\path[ultra thick,->] (c1) edge [] node [above]  {$y_3$} (c2);}
	\end{tikzpicture}}}\end{matrix}	
	$
  \end{center}
  
\end{frame}

%%%

\begin{frame}{}
\vfill
  \begin{center}
    \huge\bf And now?  
  \end{center}
  \vfill
\end{frame}

%%%

\begin{frame}{A Simple Example}
\pos{0,1.3}{
   $
  \begin{array}{@{}p{8mm}@{\quad}r@{\,}l@{\qquad}r@{\,}l@{\quad}r@{\,}l@{\quad}}
    \centering $\Rot{\only<4-6>\neginfty\only<7-19>0\only<20->{42}}$
    & {x_1^+ &\geq \neginfty & x_1^+ &\geq 0 \qquad x_1^+ \geq x_3^+ + 1 } \\[0.2mm]
    \centering $\Rot{\only<4-6>\neginfty\only<7->0}$
    & {x_1^- &\geq \neginfty & x_1^- &\geq 0   \qquad x_1^- \geq x_3^- + (-1)} \\[0.2mm]
    \centering $\Rot{\only<4-10>\neginfty\only<11-19>0\only<20->{41}}$
    & {x_2^+ &\geq \neginfty & x_2^+ &\geq (x_1^- \geq -41) \;?\; \min \{ x_1^+, 41 \} } \\[0.2mm]
    \centering $\Rot{\only<4-10>\neginfty\only<11->0}$
    & {x_2^- &\geq \neginfty & x_2^- &\geq  (x_1^- \geq -41) \;?\; x_1^-} \\[0.2mm]
    \centering $\Rot{\only<4-15>\neginfty\only<16-19>0\only<20->{41}}$
    & {x_3^+ &\geq \neginfty & x_3^+ &\geq (x_2^- \geq -41 \;\&\; x_2^+ \geq 0 ) \;?\; \min \{ x_2^+, 41 \}} \\[0.2mm]
    \centering $\Rot{\only<4-15>\neginfty\only<16->0}$
    & {x_3^- &\geq \neginfty & x_3^- &\geq (x_2^- \geq -41 \;\&\; x_2^+ \geq 0 ) \;?\; \min \{ x_2^-, 0 \} }
  \end{array}
  $}
\pos{1,4.8}{\emph{Greatest Fixpoint Iteration:}}  
\pos{1,5.4}{
   $
  \begin{array}{|r||p{10mm}|p{10mm}|p{10mm}|p{10mm}|p{10mm}|p{10mm}|p{10mm}|@{}l@{}}
     & \centering 0 
     & \centering 1
     & \centering 2
     & \centering 3
     & \centering 4
     & \centering 5
     & \centering 6
     & \\
    \hline
    \hline
    x_1^+ 
      & \centering $\infty$ 
      & \centering $\only<3>\neginfty\only<6,9-10,13-15>0\only<18-19>\infty$
      & \centering $\only<10,14-15>0\only<19>{42}$
      & \centering $\only<15>0$
      & \centering 
      & \centering 
      & \centering
      &\\[0mm]\hline
    x_1^- 
      & \centering $\infty$ 
      & \centering $\only<3>\neginfty\only<6,9-10,13-15,18-19>0$
      & \centering $\only<10,14-15,19>0$
      & \centering $\only<15>0$
      & \centering 
      & \centering 
      & \centering
      &\\[0mm]\hline
    x_2^+ 
      & \centering $\infty$ 
      & \centering $\only<3,6>\neginfty\only<9-10,13-15,18-19>{41}$
      & \centering $\only<10,14-15>0\only<19>{41}$
      & \centering $\only<15>0$
      & \centering 
      & \centering 
      & \centering
      &\\[0mm]\hline
    x_2^- 
      & \centering $\infty$ 
      & \centering $\only<3,6>\neginfty\only<9-10,13-15,18-19>\infty$
      & \centering $\only<10,14-15,19>0$
      & \centering $\only<15>0$
      & \centering 
      & \centering 
      & \centering
      &\\[0mm]\hline
    x_3^+ 
      & \centering $\infty$ 
      & \centering $\only<3,6,9-10>\neginfty\only<13-15,18-19>{41}$
      & \centering $\only<10>\neginfty\only<14-15,19>{41}$
      & \centering $\only<15>0$
      & \centering 
      & \centering 
      & \centering
      &\\[0mm]\hline
    x_3^- 
      & \centering $\infty$ 
      & \centering $\only<3,6,9-10>\neginfty\only<13-15,18-19>0$
      & \centering $\only<10>\neginfty\only<14-15,19>0$
      & \centering $\only<15>0$
      & \centering 
      & \centering 
      & \centering
      &\\[0mm]\hline
  \end{array}
  $}
  
  \pos{1,1.3}{\uncover<2-4>{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (2,0.5);\end{tikzpicture}}}  
  \pos{3.4,1.3}{\uncover<5-16>{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (1.6,0.5);\end{tikzpicture}}}  
  \pos{5.3,1.3}{{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (2.5,0.5);\end{tikzpicture}}}  
  %
  \pos{1,1.8}{\uncover<2-4>{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (2,0.5);\end{tikzpicture}}}  
  \pos{3.4,1.8}{{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (1.6,0.5);\end{tikzpicture}}}  
  %
  \pos{1,2.3}{\uncover<2-7>{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (2,0.5);\end{tikzpicture}}}  
  \pos{3.4,2.3}{{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (6.0,0.5);\end{tikzpicture}}}  
  %
  \pos{1,2.8}{\uncover<2-7>{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (2,0.5);\end{tikzpicture}}}  
  \pos{3.4,2.8}{{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (6,0.5);\end{tikzpicture}}}  
  %
  \pos{1,3.3}{\uncover<2-11>{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (2,0.5);\end{tikzpicture}}}  
  \pos{3.4,3.3}{{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (7.5,0.5);\end{tikzpicture}}}  
  %
  \pos{1,3.8}{\uncover<2-11>{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (2,0.5);\end{tikzpicture}}}  
  \pos{3.4,3.8}{{\begin{tikzpicture}[fill opacity=0.25, opacity=0.25]\pgfsetstrokecolor{blue}\filldraw[fill=blue] (0,0) rectangle (7.5,0.5);\end{tikzpicture}}}  
  
  \only<21,22>{\brighten\brighten}

   \only<21>{\pos{1,1}{{\bf\huge 
     \hspace*{1.1cm}\scalebox{2}[3]{\Rot{No further}}
     
     \bigskip
     \scalebox{2}[3]{\Rot{Improvement}}
     
     \bigskip
     \hspace*{1.2cm}\scalebox{2}[3]{\Rot{possible!}}
   }}}
   \only<22>{\pos{0.3,1.5}{{\bf\huge 
     \bigskip
     \scalebox{2}[3.5]{\Rot{Least Solution}}
     
     \bigskip
     \bigskip
     \hspace*{3.5cm}\scalebox{2}[3.5]{\Rot{FOUND!}}
   }}}  
\end{frame}

\frame{\frametitle{Theorem}  
  \begin{itemize}
    \item<2->
      Our \emph{strategy improvement algorithm}
      computes \emph{least solutions}. %of certain fixpoint equation systems over $\mathbb{Z} \cup \{ \neginfty, \infty \}$.
    \bigskip
    \item<3->
      It can be used for performing \emph{precise zone analysis}.
      % i.e., no widening.
    \bigskip
    \item<4->
      The number of \emph{strategy improvement steps} is at most exponential.
    \bigskip
    \item<5->
      Each \emph{strategy improvement step} can be carried out by \emph{ordinary greatest fixpoint iteration}.
    \bigskip
    \item<6->
      The greatest fixpoint iterations can be performed in \emph{strongly polynomial time} 
      using a reduction
      to the \emph{minimum cost flow network problem}.
  \end{itemize}
}

%%%

\outline{3}

%%%

\begin{frame}{Conclusion}
  \begin{itemize}
    \pause\item
      \emph{Strategy iteration} is an interesting alternative to valued-based Approaches. 
    \pause\bigskip\item
      Computes \emph{minimal zones} instead of some small \emph{zones}.
    \pause\bigskip\item
      Extensions:
    \begin{itemize}
        \medskip
        \item[$\bullet$]
          \emph{Template-based} analysis:
          
          \emph{Linear}:
          \includegraphics[height=6mm]{figs/region_convex_poly}, i.e., $x + 2y \leq b_1 \wedge -y \leq b_2$
          (ESOP'07)
          
          \emph{Quadratic}:
          \includegraphics[height=6mm]{figs/region_convex}
          \;
          \includegraphics[height=6mm]{figs/region_general}, i.e., 
            $x^2 + 2xy \leq b_1 \wedge y^2 \leq b_2$
          (SAS'10)
        \bigskip  
        \item[$\bullet$]
          Strategy Iteration + \emph{SMT solving} (ESOP'11)
          
          \smallskip
          \quad From \;
%          
    $\begin{matrix}\text{         
	\scalebox{0.3}{
	\begin{tikzpicture}
		 \node (n1) [coordinate]{};
		 \node (n2) [coordinate,below of = n1,yshift=-10mm]{};
		 \node (n3) [coordinate,below of = n2,yshift=-10mm]{};
		 \path[-,ultra thick] (n1) edge [bend right] node [left,xshift=-2mm] {\huge $\mathbf{abstr.} (s_{1})$} (n2);
		 \path[-,ultra thick] (n1) edge [bend left] node [right,xshift=2mm] {\huge $\mathbf{abstr.} (s_{1}')$} (n2);
		 \path[->,ultra thick] (n2) edge [bend right] node [left,xshift=-2mm] {\huge $\mathbf{abstr.} (s_{2})$} (n3);
		 \path[->,ultra thick] (n2) edge [bend left] node [right,xshift=2mm] {\huge $\mathbf{abstr.} (s_{2}')$} (n3);
	\end{tikzpicture}
	}}\end{matrix}$
%
        \;  to \;
% 
    $\mathbf{abstr.} \begin{pmatrix}\text{         
	\scalebox{0.3}{
	\begin{tikzpicture}
		 \node (n1) [coordinate]{};
		 \node (n2) [coordinate,below of = n1,yshift=-10mm]{};
		 \node (n3) [coordinate,below of = n2,yshift=-10mm]{};
		 \path[-,ultra thick] (n1) edge [bend right] node [left,xshift=-2mm] {\huge $s_{1}$} (n2);
		 \path[-,ultra thick] (n1) edge [bend left] node [right,xshift=2mm] {\huge $s_{1}'$} (n2);
		 \path[->,ultra thick] (n2) edge [bend right] node [left,xshift=-2mm] {\huge $s_{2}$} (n3);
		 \path[->,ultra thick] (n2) edge [bend left] node [right,xshift=2mm] {\huge $s_{2}'$} (n3);
	\end{tikzpicture}
	}}\end{pmatrix}$
%
      
      \medskip\item[$\bullet$]
          \emph{Unbounded Time} Verification 
          for \emph{Cyber-Physical Systems}
          through Abstract Interpretation (ATVA'11, APLAS'11, Current Work)
    \end{itemize}
  \end{itemize}  

  \vspace*{-2mm}
  \begin{center}
    \Large\bf
    \pause Thanks for Your Attention! \pause Questions?
  \end{center}
\end{frame}

%%%

\begin{frame}[allowframebreaks]{References}
  \bibliographystyle{apalike2}
  \bibliography{bib}  
\end{frame}
\end{document}
