From 2ee29b3426d2d79872fba35adbeb8d768983ffec Mon Sep 17 00:00:00 2001 From: Carlo Zancanaro Date: Thu, 19 Apr 2012 16:21:26 +1000 Subject: Add presentation; start a different implementation --- impl/Constant.h | 22 - impl/Equation.h | 47 - impl/EquationSystem.h | 40 - impl/Expression.cpp | 13 + impl/Expression.h | 20 +- impl/Max.h | 22 - impl/Min.h | 22 - impl/Operator.h | 14 + impl/Sum.h | 22 - impl/Term.h | 1 - impl/Variable.h | 24 - impl/main.cpp | 111 +- tex/presentation/main.aux | 79 + tex/presentation/main.log | 1349 + tex/presentation/main.nav | 57 + tex/presentation/main.out | 5 + tex/presentation/main.snm | 0 tex/presentation/main.tex | 290 + tex/presentation/main.toc | 6 + tex/presentation/missfont.log | 213 + tex/presentation/slides.aux | 104 + tex/presentation/slides.log | 81732 ++++++++++++++++ tex/presentation/slides.nav | 32 + tex/presentation/slides.out | 0 tex/presentation/slides.pdf | Bin 0 -> 240142 bytes tex/presentation/slides.snm | 0 tex/presentation/slides.tex | 655 + tex/presentation/slides.toc | 1 + tex/presentation/thomas/ASOPT.pdf | Bin 0 -> 7529 bytes tex/presentation/thomas/CNRS.pdf | Bin 0 -> 3086 bytes tex/presentation/thomas/Makefile | 7 + .../thomas/SAPLING november 2011/.DS_Store | Bin 0 -> 6148 bytes tex/presentation/thomas/TUM.pdf | 63 + tex/presentation/thomas/TUM2.pdf | Bin 0 -> 4514 bytes tex/presentation/thomas/VERIMAG.pdf | Bin 0 -> 4461 bytes tex/presentation/thomas/WWU.pdf | Bin 0 -> 5178 bytes tex/presentation/thomas/bib.bib | 3631 + tex/presentation/thomas/defs.tex | 250 + tex/presentation/thomas/defs.texshop | 1 + tex/presentation/thomas/figs/.svn/all-wcprops | 29 + tex/presentation/thomas/figs/.svn/entries | 164 + .../thomas/figs/.svn/prop-base/heater.pdf.svn-base | 5 + .../figs/.svn/prop-base/region_convex.pdf.svn-base | 5 + .../.svn/prop-base/region_convex_poly.pdf.svn-base | 5 + .../.svn/prop-base/region_general.pdf.svn-base | 5 + .../thomas/figs/.svn/text-base/heater.pdf.svn-base | Bin 0 -> 53919 bytes 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| 5 + tex/presentation/thomas/missfont.log | 46 + tex/presentation/thomas/pgfplots.sty | 50 + tex/presentation/thomas/proof.sty | 377 + tex/presentation/thomas/slides.log | 28 + tex/presentation/thomas/slides.tex | 780 + tex/presentation/thomas/slides.texshop | 1 + tex/presentation/thomas/strike.pdf | Bin 0 -> 17647 bytes tex/presentation/thomas/strike2.pdf | Bin 0 -> 6690 bytes tex/presentation/thomas/textpos.sty | 367 + tex/presentation/thomas/usyd.pdf | Bin 0 -> 26072 bytes 74 files changed, 182767 insertions(+), 229 deletions(-) delete mode 100644 impl/Constant.h delete mode 100644 impl/Equation.h delete mode 100644 impl/EquationSystem.h create mode 100644 impl/Expression.cpp delete mode 100644 impl/Max.h delete mode 100644 impl/Min.h create mode 100644 impl/Operator.h delete mode 100644 impl/Sum.h delete mode 100644 impl/Term.h delete mode 100644 impl/Variable.h create mode 100644 tex/presentation/main.aux create mode 100644 tex/presentation/main.log create mode 100644 tex/presentation/main.nav create mode 100644 tex/presentation/main.out create mode 100644 tex/presentation/main.snm create mode 100644 tex/presentation/main.tex create mode 100644 tex/presentation/main.toc create mode 100644 tex/presentation/missfont.log create mode 100644 tex/presentation/slides.aux create mode 100644 tex/presentation/slides.log create mode 100644 tex/presentation/slides.nav create mode 100644 tex/presentation/slides.out create mode 100644 tex/presentation/slides.pdf create mode 100644 tex/presentation/slides.snm create mode 100644 tex/presentation/slides.tex create mode 100644 tex/presentation/slides.toc create mode 100644 tex/presentation/thomas/ASOPT.pdf create mode 100644 tex/presentation/thomas/CNRS.pdf create mode 100644 tex/presentation/thomas/Makefile create mode 100644 tex/presentation/thomas/SAPLING november 2011/.DS_Store create mode 100644 tex/presentation/thomas/TUM.pdf create mode 100644 tex/presentation/thomas/TUM2.pdf create mode 100644 tex/presentation/thomas/VERIMAG.pdf create mode 100644 tex/presentation/thomas/WWU.pdf create mode 100644 tex/presentation/thomas/bib.bib create mode 100644 tex/presentation/thomas/defs.tex create mode 100644 tex/presentation/thomas/defs.texshop create mode 100644 tex/presentation/thomas/figs/.svn/all-wcprops create mode 100644 tex/presentation/thomas/figs/.svn/entries create mode 100644 tex/presentation/thomas/figs/.svn/prop-base/heater.pdf.svn-base create mode 100644 tex/presentation/thomas/figs/.svn/prop-base/region_convex.pdf.svn-base create mode 100644 tex/presentation/thomas/figs/.svn/prop-base/region_convex_poly.pdf.svn-base create mode 100644 tex/presentation/thomas/figs/.svn/prop-base/region_general.pdf.svn-base create mode 100644 tex/presentation/thomas/figs/.svn/text-base/heater.pdf.svn-base create mode 100644 tex/presentation/thomas/figs/.svn/text-base/region_convex.pdf.svn-base create mode 100644 tex/presentation/thomas/figs/.svn/text-base/region_convex_poly.pdf.svn-base create mode 100644 tex/presentation/thomas/figs/.svn/text-base/region_general.pdf.svn-base create mode 100644 tex/presentation/thomas/figs/heater.pdf create mode 100644 tex/presentation/thomas/figs/region_convex.pdf create mode 100644 tex/presentation/thomas/figs/region_convex_poly.pdf create mode 100644 tex/presentation/thomas/figs/region_general.pdf create mode 100644 tex/presentation/thomas/hand_rechts.pdf create mode 100644 tex/presentation/thomas/logos/.DS_Store create mode 100644 tex/presentation/thomas/main.aux create mode 100644 tex/presentation/thomas/main.idx create mode 100644 tex/presentation/thomas/main.log create mode 100644 tex/presentation/thomas/main.nav create mode 100644 tex/presentation/thomas/main.out create mode 100644 tex/presentation/thomas/main.pdf create mode 100644 tex/presentation/thomas/main.snm create mode 100644 tex/presentation/thomas/main.tex create mode 100644 tex/presentation/thomas/main.toc create mode 100644 tex/presentation/thomas/missfont.log create mode 100644 tex/presentation/thomas/pgfplots.sty create mode 100644 tex/presentation/thomas/proof.sty create mode 100644 tex/presentation/thomas/slides.log create mode 100644 tex/presentation/thomas/slides.tex create mode 100644 tex/presentation/thomas/slides.texshop create mode 100644 tex/presentation/thomas/strike.pdf create mode 100644 tex/presentation/thomas/strike2.pdf create mode 100644 tex/presentation/thomas/textpos.sty create mode 100644 tex/presentation/thomas/usyd.pdf diff --git a/impl/Constant.h b/impl/Constant.h deleted file mode 100644 index bab8cd9..0000000 --- a/impl/Constant.h +++ /dev/null @@ -1,22 +0,0 @@ -#ifndef CONSTANT_H -#define CONSTANT_H - -#include "Expression.h" - -template -struct Constant : public Expression { - Constant(const T& value) : _value(value) { } - - T value() const { - return this->_value; - } - - T eval(const std::map&) const { - return _value; - } - - private: - const T _value; -}; - -#endif diff --git a/impl/Equation.h b/impl/Equation.h deleted file mode 100644 index b861768..0000000 --- a/impl/Equation.h +++ /dev/null @@ -1,47 +0,0 @@ -#ifndef EQUATION_H -#define EQUATION_H - -#include "Variable.h" -#include "Expression.h" -#include -#include - -template -struct Equation { - Equation(const Variable& var, Expression* value) - : _var(var), _value(value) { } - - Variable var() const { - return _var; - } - Variable var(const Variable& var) { - Variable old_var = _var; - _var = var; - return _var; - } - - Expression* value() const { - return _value; - } - Expression* value(const Expression* value) { - Expression* old_value = _value; - _value = value; - return old_value; - } - - std::map solve(const std::map& curr) { - std::map result = curr; - T last; - do { - last = result[_var.name()]; - result[_var.name()] = _value->eval(result); - } while (last != result[_var.name()]); - return result; - } - - private: - Variable _var; - Expression* _value; -}; - -#endif diff --git a/impl/EquationSystem.h b/impl/EquationSystem.h deleted file mode 100644 index 9a393c8..0000000 --- a/impl/EquationSystem.h +++ /dev/null @@ -1,40 +0,0 @@ -#ifndef EQUATIONSYSTEM_H -#define EQUATIONSYSTEM_H - -#include "Equation.h" -#include - -template -struct EquationSystem { - bool add(Equation e) { - for (iterator it = equations.begin(); - it != equations.end(); - ++it) { - if ((*it).var().name() == e.var().name()) { - return false; - } - } - this->equations.push_back(e); - return true; - } - - std::map solve(std::map start) { - std::map result = start; - std::map comparison; - do { - comparison = result; - for (iterator it = equations.begin(); - it != equations.end(); - ++it) { - result = it->solve(result); - } - } while (comparison != result); - return result; - } - - private: - typedef typename std::list< Equation >::iterator iterator; - std::list< Equation > equations; -}; - -#endif diff --git a/impl/Expression.cpp b/impl/Expression.cpp new file mode 100644 index 0000000..61fe3ac --- /dev/null +++ b/impl/Expression.cpp @@ -0,0 +1,13 @@ +#include "Expression.h" + +template +const T +Expression :: minvalue(std::map< std::string, T> > m) { + return _operator.eval(_arguments, m); +} + +template +const T +Expression :: maxvalue(std::map< std::string, T> > m) { + return _operator.eval(_arguments, m); +} diff --git a/impl/Expression.h b/impl/Expression.h index 105ad28..38a5e65 100644 --- a/impl/Expression.h +++ b/impl/Expression.h @@ -1,12 +1,26 @@ #ifndef EXPRESSION_H #define EXPRESSION_H -#include #include +#include +#include +template +struct Expression; +#include "Operator.h" -template +template struct Expression { - virtual T eval(const std::map&) const = 0; + Expression(Operator* o, std::vector< Expression > a) + : _operator(o), _arguments(a) { } + virtual const T minvalue(std::map m) const { + return _operator->eval(_arguments, m); + } + virtual const T maxvalue(std::map m) const { + return _operator->eval(_arguments, m); + } + private: + Operator* _operator; + std::vector< Expression > _arguments; }; #endif diff --git a/impl/Max.h b/impl/Max.h deleted file mode 100644 index b3b5d65..0000000 --- a/impl/Max.h +++ /dev/null @@ -1,22 +0,0 @@ -#ifndef MAX_H -#define MAX_H - -#include "Expression.h" - -template -struct Max : public Expression { - Max(Expression* left, Expression* right) - : _left(left), _right(right) { } - - T eval(const std::map& mappings) const { - T left = _left->eval(mappings); - T right = _right->eval(mappings); - return left < right ? right : left; - } - - private: - Expression* _left; - Expression* _right; -}; - -#endif diff --git a/impl/Min.h b/impl/Min.h deleted file mode 100644 index 9f17545..0000000 --- a/impl/Min.h +++ /dev/null @@ -1,22 +0,0 @@ -#ifndef MIN_H -#define MIN_H - -#include "Expression.h" - -template -struct Min : public Expression { - Min(Expression* left, Expression* right) - : _left(left), _right(right) { } - - T eval(const std::map& mappings) const { - T left = _left->eval(mappings); - T right = _right->eval(mappings); - return left < right ? left : right; - } - - private: - Expression* _left; - Expression* _right; -}; - -#endif diff --git a/impl/Operator.h b/impl/Operator.h new file mode 100644 index 0000000..e83d495 --- /dev/null +++ b/impl/Operator.h @@ -0,0 +1,14 @@ +#ifndef OPERATOR_H +#define OPERATOR_H + +#include +#include +#include +#include "Expression.h" + +template +struct Operator { + virtual const T eval(const std::vector< Expression >&, const std::map&) const = 0; +}; + +#endif diff --git a/impl/Sum.h b/impl/Sum.h deleted file mode 100644 index bf04669..0000000 --- a/impl/Sum.h +++ /dev/null @@ -1,22 +0,0 @@ -#ifndef SUM_H -#define SUM_H - -#include "Expression.h" - -template -struct Sum : public Expression { - Sum(Expression* left, Expression* right) - : _left(left), _right(right) { } - - T eval(const std::map& mappings) const { - T left = _left->eval(mappings); - T right = _right->eval(mappings); - return left + right; - } - - private: - Expression* _left; - Expression* _right; -}; - -#endif diff --git a/impl/Term.h b/impl/Term.h deleted file mode 100644 index 8b13789..0000000 --- a/impl/Term.h +++ /dev/null @@ -1 +0,0 @@ - diff --git a/impl/Variable.h b/impl/Variable.h deleted file mode 100644 index 7433426..0000000 --- a/impl/Variable.h +++ /dev/null @@ -1,24 +0,0 @@ -#ifndef VARIABLE_H -#define VARIABLE_H - -#include -#include "Expression.h" - -template -struct Variable : public Expression { - Variable(const std::string& name) - : _name(name) { } - - std::string name() const { - return this->_name; - } - - T eval(const std::map& values) const { - return values.find(_name)->second; - } - - private: - const std::string _name; -}; - -#endif diff --git a/impl/main.cpp b/impl/main.cpp index 814a3ee..dc6cf45 100644 --- a/impl/main.cpp +++ b/impl/main.cpp @@ -1,36 +1,95 @@ -#include "Constant.h" -#include "Variable.h" -#include "Min.h" -#include "Max.h" -#include "Sum.h" -#include "Equation.h" -#include "EquationSystem.h" - #include +#include +#include +#include "Expression.h" +#include "Operator.h" using namespace std; -int main() { - std::map mappings; - mappings["x"] = 10; +struct Variable : public Operator { + Variable(const std::string& x) + : _value(x) { } + const float eval(const vector >& v, const map& m) const { + //assert(v.size() == 0); + map::const_iterator it = m.find(_value); + return (it == m.end() ? -INFINITY : it->second); + } + private: + std::string _value; +}; + +struct Const : public Operator { + Const(const float& x) + : _value(x) { } + const float eval(const vector >& v, const map& m) const { + //assert(v.size() == 0); + return _value; + } + private: + float _value; +}; + +struct Mult : public Operator { + Mult(const float& x) + : _value(x) { } + const float eval(const vector >& v, const map& m) const { + //assert(v.size() == 1); + return _value * v[0].minvalue(m); + } + private: + float _value; +}; - Equation e(Variable("x"), - new Max( - new Min( - new Constant(100), - new Sum( - new Variable("x"), - new Constant(100))), - new Constant(20))); +struct Maximum : public Operator { + const float eval(const vector >& v, const map& m) const { + //assert(v.size() == 1); + float value = -INFINITY; + for (vector >::const_iterator it = v.begin(); + it != v.end(); + ++it) { + const float result = it->maxvalue(m); + value = (value < result ? result : value); + } + return value; + } +}; - Equation e2(Variable("x"), - new Constant(200)); +struct Minimum : public Operator { + const float eval(const vector >& v, const map& m) const { + //assert(v.size() == 1); + float value = INFINITY; + for (vector >::const_iterator it = v.begin(); + it != v.end(); + ++it) { + const float result = it->maxvalue(m); + value = (value < result ? value : result); + } + return value; + } +}; - EquationSystem es; - es.add(e); - es.add(e2); - std::map result = es.solve(mappings); - cout << result["x"] << endl; +int main () { + typedef Expression E; + typedef vector Args; + map m; + //m["x"] = -10; + Args empty; + { + Args a; + { + Args b; + b.push_back(E(new Const(10), empty)); + b.push_back(E(new Const(20), empty)); + b.push_back(E(new Variable("x"), empty)); + a.push_back(E(new Minimum(), b)); + cout << E(new Minimum(), b).minvalue(m) << endl; + } + } + /*map m; + vector > empty; + vector > args; + args.push_back(Expression(new Const(10), empty)); + cout << Expression(new Mult(3), args).minvalue(m) << endl;*/ return 0; } diff --git a/tex/presentation/main.aux b/tex/presentation/main.aux new file mode 100644 index 0000000..7467836 --- /dev/null +++ b/tex/presentation/main.aux @@ -0,0 +1,79 @@ +\relax +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined 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b/tex/presentation/main.snm new file mode 100644 index 0000000..e69de29 diff --git a/tex/presentation/main.tex b/tex/presentation/main.tex new file mode 100644 index 0000000..286e4a7 --- /dev/null +++ b/tex/presentation/main.tex @@ -0,0 +1,290 @@ +\documentclass{beamer} + +\usetheme{Hannover} +\usepackage{amsmath,stmaryrd,listings} + +\title{{\bf Static Analysis \\ through \\Abstract Interpretation, \\Convex Optimization, and\\ Strategy Iteration}} + +\author{ + {\bf Thomas Martin Gawlitza} + \\[3pt] + joint work with + \\[3pt] + {\bf Helmut Seidl} +} + +\newcommand\N{\mathbb{N}} +\newcommand\Z{\mathbb{Z}} +\newcommand\CZ{\overline{\Z}} + +\let\max\undefined +\newcommand\max{\lor} +\let\min\undefined +\newcommand\min{\land} + +\begin{document} + +\begin{frame} + \maketitle +\end{frame} + +\begin{frame} + \tableofcontents +\end{frame} + +\section{Example} +\begin{frame}{Example} + \begin{eqnarray*} + x_1 &=& 0 \max (-1 + x_1 \min x_2) \\ + x_2 &=& 0 \max 5 + x_1 \max x_1 \\ + x_3 &=& 0 \max 1 + x_3 \max 0 + x_1 + \end{eqnarray*} + +\end{frame} + +\section{Definitions} +\begin{frame}{Preliminary Definitions} + \begin{center} + Strange notation: \\ + $x \max y = \text{max}(x, y)$ \\ + $x \min y = \text{min}(x, y)$ \\ + \uncover<2->{ + \bigskip + {\bf Monotone:} $x \leq y \implies f(x) \leq f(y) ~~ \forall x,y$ + } + + \uncover<3->{ + \bigskip + {\bf Assignment:} $\rho$ is always a variable assignment (from $X \rightarrow \CZ$) + } + \end{center} +\end{frame} + +\begin{frame}{Systems of Equations} + \begin{align*} + \varepsilon = \{ x_1 = e_1, x_2 = e_2, ... x_n = e_n \} + \end{align*} + \uncover<2->{ + \begin{align*} + \text{\bf Solutions: } & \rho = \llbracket \varepsilon \rrbracket \rho \\ + \text{\bf Presolutions: } & \rho \leq \llbracket \varepsilon \rrbracket \rho \\ + \text{\bf Fixpoint: } & f(x) = x \\ + \text{\bf Least fixpoint: } & \mu f \\ + \end{align*} + } + \uncover<3->{ + {\bf Evaluation:} + $\text{with } \rho ~ \epsilon ~ Vars(\varepsilon) \rightarrow \CZ$ \\ + \begin{align*} + (\llbracket \varepsilon \rrbracket \rho)(x) & := \llbracket e \rrbracket \rho \\ + \llbracket x \rrbracket \rho & := \rho (x) \\ + \llbracket f(e_1, ... e_k) \rrbracket & := f(\llbracket e_1 \rrbracket \rho, + ... \llbracket e_k \rrbracket \rho) + \end{align*} + } +\end{frame} + +\begin{frame}{Operators on $\CZ$} + Working in $\CZ = \Z \cup \{\infty, -\infty\}$ + + \bigskip + \begin{displaymath} + x +^{-\infty} y = \left\{ \begin{array}{ll} + -\infty & \text{if } -\infty \in \{x,y\} \\ + \infty & \text{if } -\infty \not\in \{x,y\} \text{ and } \infty \in \{x,y\} \\ + x + y & \text{if } x,y \in \Z \\ + \end{array} \right. + \end{displaymath} + \begin{displaymath} + x +^{\infty} y = \left\{ \begin{array}{ll} + \infty & \text{if } \infty \in \{x,y\} \\ + -\infty & \text{if } \infty \not\in \{x,y\} \text{ and } -\infty \in \{x,y\} \\ + x + y & \text{if } x,y \in \Z \\ + \end{array} \right. + \end{displaymath} + \begin{displaymath} + \begin{array}{lr} + x \cdot \infty = \infty \cdot x = \infty, + x \cdot -\infty = -\infty \cdot x = -\infty + & \forall x > 0 \\ + x \cdot \infty = \infty \cdot x = -\infty, + x \cdot -\infty = -\infty \cdot x = \infty + & \forall x < 0 \\ + 0 \cdot \infty = \infty \cdot 0 = + 0 \cdot -\infty = -\infty \cdot 0 = 0 & + \end{array} + \end{displaymath} +\end{frame} + +\begin{frame}{Expansivity} + $f$ is upward-expansive in $X'$ iff + + \bigskip + \begin{center} + $f(\rho \oplus \{x \mapsto \rho(x) + \delta\}) \geq f(\rho) + \delta)$ + ~~ $\forall x \in X', \rho \in X \rightarrow \CZ, \delta \in \N$ + \end{center} +\end{frame} + +\section{Max-strategies} +\begin{frame}{Max-strategies} + Assume for every $x = e \in \varepsilon$, $e$ is of the + form $e_1 \max e_2 \max ... \max e_k$. + + Then a $\max$-$strategy$ $\sigma$ maps each $e$ to one of its $e_k$. + + \bigskip + For all $\max$ strategies $\sigma$ the expression $e\sigma$ is defined as: + \begin{align*} + (e_1 \max ... \max e_k)\sigma & = (\sigma(e_1 \max ... \max e_k))\sigma \\ + (f(e_1,...e_k))\sigma & = f(e_1\sigma, ..., e_k\sigma) + \end{align*} + where $f \not = \max$ +\end{frame} + +\begin{frame}{Max-strategies} + We may need to change strategy throughout our evaluation. + + Consider the system: $\varepsilon = \{x_1 = x_1 + 1 \max 0\}$. + + Let: + \begin{align*} + \sigma_1 & = \{x_1 + 1 \max 0 \mapsto x_1 + 1\} \\ + \sigma_2 & = \{x_1 + 1 \max 0 \mapsto 0\} + \end{align*} + + \uncover<2->{ + This gives us: + } + \begin{align*} + \uncover<2->{ + \mu \llbracket \varepsilon (\sigma_1) \rrbracket & = \{x_1 \mapsto -\infty\} \\ + \mu \llbracket \varepsilon (\sigma_2) \rrbracket & = \{x_1 \mapsto 0\} \\ + } + \uncover<3->{ + \mu \llbracket \varepsilon \rrbracket & = \{x_1 \mapsto \infty\} + } + \end{align*} +\end{frame} + +\section{Bellman-Ford} +\begin{frame}{BF-functions} + $X$ a set. A monotone $f: (X \rightarrow \CZ) \rightarrow CZ$ is called + a \emph{Bellman-Ford function} iff, $\forall \rho, \rho': X \rightarrow \CZ$ + with $\rho' \geq \rho$ the following holds: + + \bigskip + If $f(\rho') > f(\rho)$ then $\exists x \in X$ and some $\delta \in \CZ\setminus\{-\infty\}$ such that: + \begin{align*} + \rho'(x) & > \rho(x) \\ + f(\rho') & = \rho'(x) + \delta \\ + f(\rho'') & \geq \rho''(x) + \delta ~~ \forall \rho'' \geq \rho' + \end{align*} +\end{frame} + +\begin{frame}{A quick lemma} + Let $\varepsilon$ be a system of \emph{BF-equations} with $n$ variables. \\ + Let $\rho^{(i)} = \llbracket\varepsilon\rrbracket^i(\_ \mapsto -\infty) ~~ \forall i \in \N$. \\ + The following holds for every $x \in Vars(\varepsilon)$: \\ + If there exists some $k > n$ with $\rho^{(k)}(x) > \rho^{(n)}(x)$, then + $\mu\llbracket\varepsilon\rrbracket(x) = \infty$. +\end{frame} + +\begin{frame}{The algorithm} + ~ \\ + {\bf Input:} A system of BF-equations with $n$ variables \\ + {\bf Output:} The least solution $\mu\llbracket\varepsilon\rrbracket$ of \varepsilon \\ + \begin{displaymath} + ~ \\ + \rho \leftarrow (\_ \mapsto -\infty) \\ + {\bf for} i = 1 {\bf to} n {\bf do} \rho \leftarrow \llbracket\varepsilon\rrbracket\rho \\ + \rho \leftarrow \rho' \text{ where } \rho'(x) = \left\{ \begin{array}{ll} + \rho(x) & \text{if } (\llbracket\varepsilon\rrbracket\rho)(x) \leq \rho(x) \\ + \infty & \text{if } (\llbracket\varepsilon\rrbracket\rho)(x) > \rho(x) + \end{array} \right. \forall x \in X \\ + {\bf for} i = 1 {\bf to} n-1 {\bf do} \rho \leftarrow \rho \max \llbracket\varepsilon\rrbracket\rho \\ + {\bf return} \rho \\ + ~ + \end{displaymath} +\end{frame} + +\begin{frame}{Example} + \begin{align*} + x & = 1 \\ + y & = y + x \max -10 \\ + z & = x \cdot^+ y + \end{align*} + Where $\cdot^+$ is defined by: + \begin{displaymath} + x \cdot^+ y = \left\{ \begin{array}{l l} + x \cdot y & \text{if } x, y > 0 \\ + -\infty & \text{if } x \le 0 \text{ or } y \le 0 + \end{array} \forall x,y \in \CZ + \end{displaymath} +\end{frame} + +\section{Max-strategy improvement} +\begin{frame}{Max-strategy improvement} + {\bf General idea:} + \begin{itemize} + \item Pick a max-strategy + \item Perform fixpoint iteration + \item Improve strategy + \item Repeat until we have a solution + \end{itemize} +\end{frame} + +\begin{frame}{What is an improvement?} + If we have $\varepsilon$ a system of equations, $\sigma$ a $\max$-strategy + for $\varepsilon$ and $\rho$ a presolution of $\varepsilon(\sigma)$, then we + call a $\max$-strategy $\sigma'$ is called an improvement of $\sigma$ with + respect to $\rho$ iff: + \begin{itemize} + \item if $\rho \not \in {\bf Sol}(\varepsilon)$, then $\llbracket\varepsilon(\sigma')\rrbracket\rho > \rho$ + \item for all $\max$-expression $e \in S_{\max}(\varepsilon)$ the following holds: \\ + If $\sigma'(e) \not = \sigma(e)$, then $\llbracket e\sigma'\rrbracket\rho > \llbracket e\sigma\rrbracket\rho$ + \end{itemize} +\end{frame} + +\begin{frame}{Algorithm} + ~ \\ + {\bf Input:} + \begin{itemize} + \item A system $\varepsilon$ of monotone equations over a complete linearly ordered set \\ + \item A $\max$-strategy $\sigma_{init}$ for $\varepsilon$ \\ + \item A pre-solution $\rho_{init}$ of $\varepsilon(\sigma_{init})$ with $\rho_{init} \le \mu\llbracket\varepsilon\rrbracket$ + \end{itemize} + {\bf Output:} The least solution $\mu\llbracket\varepsilon\rrbracket$ of $\varepsilon$ + \begin{displaymath} + ~ \\ + \sigma \leftarrow \sigma_{init} \\ + \rho \leftarrow \rho_{init} \\ + \text{while }(\rho \not \in {\bf Sol}(\varepsilon)) \{ \\ + ~~~~ \sigma \leftarrow P_{\max}(\sigma, \rho) \\ + ~~~~ \rho \leftarrow \mu_{\ge\rho} \llbracket\varepsilon(\sigma)\rrbracket \\ + \} \\ + {\bf return} \rho + ~ + \end{displaymath} +\end{frame} + +\begin{frame}{Another related lemma} + Whenever the $\max$-strategy improvement algorithm terminates, it + returns the least solution $\mu\llbracket\varepsilon\rrbracket$. +\end{frame} + +\begin{frame}{Example} + \begin{align*} + x_1 & = 0 \max x_1 + x_2 - 4 \\ + x_2 & = -1 \max ((x_1 + 1 \max 2 \cdot x_2) \min 5) + \end{align*} +\end{frame} + +\section{Feasibility} + +\section{Extended Integer Equations} + +\section{Abstract Interpretation over Zones} + + +\end{document} diff --git a/tex/presentation/main.toc b/tex/presentation/main.toc new file mode 100644 index 0000000..8e24cc6 --- /dev/null +++ b/tex/presentation/main.toc @@ -0,0 +1,6 @@ +\beamer@endinputifotherversion {3.10pt} +\beamer@sectionintoc {1}{Example}{3}{0}{1} +\beamer@sectionintoc {2}{Definitions}{4}{0}{2} +\beamer@sectionintoc {3}{Max-strategies}{12}{0}{3} +\beamer@sectionintoc {4}{Bellman-Ford}{16}{0}{4} +\beamer@sectionintoc {5}{Max-strategy improvement}{20}{0}{5} diff --git a/tex/presentation/missfont.log b/tex/presentation/missfont.log new file mode 100644 index 0000000..0a253cb --- /dev/null +++ b/tex/presentation/missfont.log @@ -0,0 +1,213 @@ +mktextfm phvr7t +mktextfm phvr7t +mktextfm phvr7t 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