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+\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}
generated by cgit v1.2.3 (git 2.50.1) at 2025-10-31 10:56:50 +0000