Claude Michel - University of Nice Sophia Antipolis, Nice, France (original) (raw)

Papers by Claude Michel

Solving constraints over floating-point numbers is a critical issue in numerous applications nota... more Solving constraints over floating-point numbers is a critical issue in numerous applications notably in program verification. Capabilities of filtering algorithms for constraints over the floating-point numbers have been so far limited to 2b-consistency and its derivatives. Though safe, such filtering techniques suffer from the well known pathological problems of local consistencies, e.g., inability to efficiently handle multiple occurrences of the variables. These limitations also take roots in the strongly restricted floating-point arithmetic. To circumvent the poor properties of floating-point arithmetic, we propose in this paper to build various relaxations over the reals of the problem over the floats. We show that using linear programming (LP) to shrink the domains with safe linearisations of such relaxations can be very effective for boosting filtering techniques for constraints over the floats. Preliminary experiments on a limited but relevant set of benchmarks are very promising.

Global Filtering Algorithms Based on Linear Relaxations

Abstract: This paper introduces a ltering algorithm to prune the domainsof the variables in numer... more Abstract: This paper introduces a ltering algorithm to prune the domainsof the variables in numerical constraints. This algorithm is basedon the RLT (Reformulation-Linearization Technique) schema. In the reformulationphase, tight convex and concave approximations of nonlinearterms are generated, that's to say for bilinear terms, product of variables,power and univariate terms. New variables are introduced to linearizethe initial constraint system. A linear

Lecture Notes in Computer Science, 2012

interpretation based value analysis is a classical approach for verifying programs with floating-... more interpretation based value analysis is a classical approach for verifying programs with floating-point computations. However, state-of-the-art tools compute an over-approximation of the variable values that can be very coarse. In this paper, we show that constraint solvers can significantly refine the approximations computed with abstract interpretation tools. We introduce a hybrid approach that combines abstract interpretation and constraint programming techniques in a single static and automatic analysis. rAiCp, the system we developed is substantially more precise than Fluctuat, a state-of-the-art static analyser. Moreover, it could eliminate 13 false alarms generated by Fluctuat on a standard set of benchmarks.

Combining Local Consistencies with a New Global Filtering Algorithm on Linear Relaxations

Abstract: Numeric systems of constraints are widely used to model problems in numerousapplication... more Abstract: Numeric systems of constraints are widely used to model problems in numerousapplication areas ranging from robotics to chemistry. This paper introduces a new lteringalgorithm (GFLR) to prune the domains of the variables in such numeric applications. Roughlyspeaking, GFLR combines classical local consistencies and a new global ltering algorithmthat works on a linear relaxation of numeric constraints. We introduce a

Constraints in Software Testing, Verification and Analysis

Computing "sharp" upper bounds Statement of the problem Newton for under-constrained systems New ... more Computing "sharp" upper bounds Statement of the problem Newton for under-constrained systems New upper bounding strategie Experiments Conclusion Trends in global optimisation Performance Most successful systems (Baron, αBB,. . .) use local methods and linear relaxations → not rigorous (work with floats) Rigour Mainly rely on interval computation. .. available systems (e.g., Globsol) are quite slow Challenge: to combine the advantages of both approaches in an efficient and rigorous global optimisation framework 4 CSP & Global Optimization

Using constraint techniques for a safe and fast implementation of optimality-based reduction

Optimality-based reduction attempts to take advantage of the known bounds of the objective functi... more Optimality-based reduction attempts to take advantage of the known bounds of the objective function to reduce the domain of the variables, and thus to speed up the search of a global optimum. However, the basic algorithm is unsafe, and thus, the overall process may no longer be complete and may not reach the actual global optimum. Recently, Kearfott has proposed a safe implementation of optimality-based reduction. Unfortunately, his method suffers from some limitations and is rather slow. In this paper, we show how constraint programming filtering techniques can be used to implement optimality-based reduction in a safe and efficient way.

Solving constraints over the oating point numbers is a key issue in the process of software valid... more Solving constraints over the oating point numbers is a key issue in the process of software validation and verication. Techniques to solve such constraints on the basis of projection functions have been successfully developed. However, though correct, this approach can lead to slow convergence phenomena for very common constraints like addition and subtraction constraints. In this paper, we introduce new addition and subtraction constraints which, thanks to a new oating point subtraction property, directly compute optimal bounds for the domain of the variables at a low cost. Preliminary experiments have shown that these constraints can drastically speed up the ltering process.

2012 IEEE Fifth International Conference on Software Testing, Verification and Validation, 2012

interpretation-based value analysis is a classical approach for verifying programs with floating-... more interpretation-based value analysis is a classical approach for verifying programs with floating-point computations. However, state-of-the-art tools compute an overapproximation of the variable values that can be very coarse. Constraint solvers have recently been used to significantly refine the approximations computed by such tools. In this paper, we introduce a hybrid approach that combines abstract interpretation and constraint programming techniques in a single static and automatic analysis. First experiments showed that this approach can successfully analyze programs that could not be handled by abstract interpretation or constraint programming tools alone.

Verifying floating-point programs with constraint programming and abstract interpretation techniques

Automated Software Engineering, 2014

ABSTRACT Static value analysis is a classical approach for verifying programs with floating-point... more ABSTRACT Static value analysis is a classical approach for verifying programs with floating-point computations. Value analysis mainly relies on abstract interpretation and over-approximates the possible values of program variables. State-of-the-art tools may however compute over-approximations that can be rather coarse for some very usual program expressions. In this paper, we show that constraint solvers can significantly refine approximations computed with abstract interpretation tools. More precisely, we introduce a hybrid approach combining abstract interpretation and constraint programming techniques in a single static and automatic analysis. This hybrid approach benefits from the strong points of abstract interpretation and constraint programming techniques, and thus, it is more effective than static analysers and constraint solvers, when used separately. We compared the efficiency of the system we developed—named rAiCp—with state-of-the-art static analyzers: rAiCp produces substantially more precise approximations and is able to check program properties on both academic and industrial benchmarks.

Arxiv preprint arXiv:1109.0456, Sep 1, 2011

Abstract: Modern software systems, like GNU/Linux distributions or Eclipse-based development envi... more Abstract: Modern software systems, like GNU/Linux distributions or Eclipse-based development environment, are often deployed by selecting components out of large component repositories. Maintaining such software systems by performing component upgrades is a complex task, and the users need to have an expressive preferences language at their disposal to specify the kind of upgrades they are interested in. Recent research has shown that it is possible to develop solvers that handle preferences ...

Solving constraints over floating-point numbers is a critical issue in numerous applications nota... more Solving constraints over floating-point numbers is a critical issue in numerous applications notably in program verification. Capabilities of filtering algorithms for constraints over the floating-point numbers have been so far limited to 2b-consistency and its derivatives. Though safe, such filtering techniques suffer from the well known pathological problems of local consistencies, e.g., inability to efficiently handle multiple occurrences of the variables. These limitations also take roots in the strongly restricted floating-point arithmetic. To circumvent the poor properties of floating-point arithmetic, we propose in this paper to build various relaxations over the reals of the problem over the floats. We show that using linear programming (LP) to shrink the domains with safe linearisations of such relaxations can be very effective for boosting filtering techniques for constraints over the floats. Preliminary experiments on a limited but relevant set of benchmarks are very promising.

La mise en oeuvre effective de méthodes de vérification de programmes comportant des calculs sur ... more La mise en oeuvre effective de méthodes de vérification de programmes comportant des calculs sur les nombresà virgule flottante reste encore problématique. C'est pourquoi nous présentons dans cet article une nouvelle méthode de résolution de contraintes sur les nombresà virgule flottante qui consisteà les approximer sur les réels. Elle est basée sur la construction d'approximations sur les réels, précises et conservatives des solutions des contraintes sur les nombresà virgule flottante. Cette méthode permet de s'appuyer sur l'utilisation d'algorithmes de filtrage sur les réels pour résoudre des problèmes sur les nombresà virgule flottante. Il devient ainsi possible de repousser les limitations actuelles des solveurs de contraintes sur les nombresà virgules flottantes, telles que le problème du passageà l'échelle, pour générer des jeux de tests, ou vérifier des programmes plus conséquents que ceux traités jusqu'à maintenant.

Global Filtering Algorithms Based on Linear Relaxations

Abstract: This paper introduces a ltering algorithm to prune the domainsof the variables in numer... more Abstract: This paper introduces a ltering algorithm to prune the domainsof the variables in numerical constraints. This algorithm is basedon the RLT (Reformulation-Linearization Technique) schema. In the reformulationphase, tight convex and concave approximations of nonlinearterms are generated, that's to say for bilinear terms, product of variables,power and univariate terms. New variables are introduced to linearizethe initial constraint system. A linear

Siam Journal on Numerical Analysis, 2005

Numerical constraint systems are often handled by branch and prune algorithms that combine splitt... more Numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques, local consistencies, and interval methods. This paper first recalls the principles of Quad, a global constraint that works on a tight and safe linear relaxation of quadratic subsystems of constraints. Then, it introduces a generalization of Quad to polynomial constraint systems. It also introduces a method to get safe linear relaxations and shows how to compute safe bounds of the variables of the linear constraint system. Different linearization techniques are investigated to limit the number of generated constraints. QuadSolver, a new branch and prune algorithm that combines Quad, local consistencies, and interval methods, is introduced. QuadSolver has been evaluated on a variety of benchmarks from kinematics, mechanics, and robotics. On these benchmarks, it outperforms classical interval methods as well as constraint satisfaction problem solvers and it compares well with state-of-the-art optimization solvers.

This paper introduces a new filtering algorithm for handling systems of quadratic equations and i... more This paper introduces a new filtering algorithm for handling systems of quadratic equations and inequations. Such constraints are widely used to model distance relations in numerous application areas ranging from robotics to chemistry. Classical filtering algorithms are based upon local consistencies and thus, are unable to achieve a significant pruning of the domains of the variables occurring in quadratic constraints systems. The drawback of these approaches comes from the fact that the constraints are handled independently. We introduce here a global filtering algorithm that works on a tight linear relaxation of the quadratic constraints. First experimentations show that this new algorithm yields a much more effective pruning of the domains than local consistency filtering algorithms.

Computing Research Repository, 2008

Finding feasible points for which the proof succeeds is a critical issue in safe Branch and Bound... more Finding feasible points for which the proof succeeds is a critical issue in safe Branch and Bound algorithms which handle continuous problems. In this paper, we introduce a new strategy to compute very accurate approximations of feasible points. This strategy takes advantage of the Newton method for under-constrained systems of equations and inequalities. More precisely, it exploits the optimal solution of a linear relaxation of the problem to compute efficiently a promising upper bound. First experiments on the Coconuts benchmarks demonstrate that this approach is very effective.

This paper introduces a new framework for tackling constraints over the floating-point numbers. A... more This paper introduces a new framework for tackling constraints over the floating-point numbers. An important application area where such solvers are required is program analysis (e.g., structural test case generation, correctness proof of numeric operations). Albeit the floating-point numbers are a finite subset of the real numbers, classical CSP techniques are ineffective due to the huge size of the domains. Relations that hold over the real numbers may not hold over the floating-point numbers. Moreover, constraints that have no solutions over the reals may hold over the floats. Thus, interval-narrowing techniques, which are used in numeric CSP, cannot safely solve constraints systems over the floats. We analyse here the specific properties of the relations over the floats. A CSP over the floats is formally defined. We show how local-consistency filtering algorithms used in interval solvers can be adapted to achieve a safe pruning of such CSP. Finally, we illustrate the capabilities of a CSP over the floats for the generation of test data.

Journal of Computational and Applied Mathematics, 2007

Interval methods have shown their ability to locate and prove the existence of a global optima in... more Interval methods have shown their ability to locate and prove the existence of a global optima in a safe and rigorous way. Unfortunately, these methods are rather slow. Efficient solvers for optimization problems are based on linear relaxations. However, the latter are unsafe, and thus may overestimate, or worst, underestimate the very global minima. This paper introduces QuadOpt, an efficient and safe framework to rigorously bound the global optima as well as its location. QuadOpt uses consistency techniques to speed up the initial convergence of the interval narrowing algorithms. A lower bound is computed on a linear relaxation of the constraint system and the objective function. All these computations are based on a safe and rigorous implementation of linear programming techniques. First experimental results are very promising.

Solving constraints over floating-point numbers is a critical issue in numerous applications nota... more Solving constraints over floating-point numbers is a critical issue in numerous applications notably in program verification. Capabilities of filtering algorithms for constraints over the floating-point numbers have been so far limited to 2b-consistency and its derivatives. Though safe, such filtering techniques suffer from the well known pathological problems of local consistencies, e.g., inability to efficiently handle multiple occurrences of the variables. These limitations also take roots in the strongly restricted floating-point arithmetic. To circumvent the poor properties of floating-point arithmetic, we propose in this paper to build various relaxations over the reals of the problem over the floats. We show that using linear programming (LP) to shrink the domains with safe linearisations of such relaxations can be very effective for boosting filtering techniques for constraints over the floats. Preliminary experiments on a limited but relevant set of benchmarks are very promising.

Global Filtering Algorithms Based on Linear Relaxations

Abstract: This paper introduces a ltering algorithm to prune the domainsof the variables in numer... more Abstract: This paper introduces a ltering algorithm to prune the domainsof the variables in numerical constraints. This algorithm is basedon the RLT (Reformulation-Linearization Technique) schema. In the reformulationphase, tight convex and concave approximations of nonlinearterms are generated, that's to say for bilinear terms, product of variables,power and univariate terms. New variables are introduced to linearizethe initial constraint system. A linear

Lecture Notes in Computer Science, 2012

interpretation based value analysis is a classical approach for verifying programs with floating-... more interpretation based value analysis is a classical approach for verifying programs with floating-point computations. However, state-of-the-art tools compute an over-approximation of the variable values that can be very coarse. In this paper, we show that constraint solvers can significantly refine the approximations computed with abstract interpretation tools. We introduce a hybrid approach that combines abstract interpretation and constraint programming techniques in a single static and automatic analysis. rAiCp, the system we developed is substantially more precise than Fluctuat, a state-of-the-art static analyser. Moreover, it could eliminate 13 false alarms generated by Fluctuat on a standard set of benchmarks.

Combining Local Consistencies with a New Global Filtering Algorithm on Linear Relaxations

Abstract: Numeric systems of constraints are widely used to model problems in numerousapplication... more Abstract: Numeric systems of constraints are widely used to model problems in numerousapplication areas ranging from robotics to chemistry. This paper introduces a new lteringalgorithm (GFLR) to prune the domains of the variables in such numeric applications. Roughlyspeaking, GFLR combines classical local consistencies and a new global ltering algorithmthat works on a linear relaxation of numeric constraints. We introduce a

Constraints in Software Testing, Verification and Analysis

Computing "sharp" upper bounds Statement of the problem Newton for under-constrained systems New ... more Computing "sharp" upper bounds Statement of the problem Newton for under-constrained systems New upper bounding strategie Experiments Conclusion Trends in global optimisation Performance Most successful systems (Baron, αBB,. . .) use local methods and linear relaxations → not rigorous (work with floats) Rigour Mainly rely on interval computation. .. available systems (e.g., Globsol) are quite slow Challenge: to combine the advantages of both approaches in an efficient and rigorous global optimisation framework 4 CSP & Global Optimization

Using constraint techniques for a safe and fast implementation of optimality-based reduction

Optimality-based reduction attempts to take advantage of the known bounds of the objective functi... more Optimality-based reduction attempts to take advantage of the known bounds of the objective function to reduce the domain of the variables, and thus to speed up the search of a global optimum. However, the basic algorithm is unsafe, and thus, the overall process may no longer be complete and may not reach the actual global optimum. Recently, Kearfott has proposed a safe implementation of optimality-based reduction. Unfortunately, his method suffers from some limitations and is rather slow. In this paper, we show how constraint programming filtering techniques can be used to implement optimality-based reduction in a safe and efficient way.

Solving constraints over the oating point numbers is a key issue in the process of software valid... more Solving constraints over the oating point numbers is a key issue in the process of software validation and verication. Techniques to solve such constraints on the basis of projection functions have been successfully developed. However, though correct, this approach can lead to slow convergence phenomena for very common constraints like addition and subtraction constraints. In this paper, we introduce new addition and subtraction constraints which, thanks to a new oating point subtraction property, directly compute optimal bounds for the domain of the variables at a low cost. Preliminary experiments have shown that these constraints can drastically speed up the ltering process.

2012 IEEE Fifth International Conference on Software Testing, Verification and Validation, 2012

interpretation-based value analysis is a classical approach for verifying programs with floating-... more interpretation-based value analysis is a classical approach for verifying programs with floating-point computations. However, state-of-the-art tools compute an overapproximation of the variable values that can be very coarse. Constraint solvers have recently been used to significantly refine the approximations computed by such tools. In this paper, we introduce a hybrid approach that combines abstract interpretation and constraint programming techniques in a single static and automatic analysis. First experiments showed that this approach can successfully analyze programs that could not be handled by abstract interpretation or constraint programming tools alone.

Verifying floating-point programs with constraint programming and abstract interpretation techniques

Automated Software Engineering, 2014

ABSTRACT Static value analysis is a classical approach for verifying programs with floating-point... more ABSTRACT Static value analysis is a classical approach for verifying programs with floating-point computations. Value analysis mainly relies on abstract interpretation and over-approximates the possible values of program variables. State-of-the-art tools may however compute over-approximations that can be rather coarse for some very usual program expressions. In this paper, we show that constraint solvers can significantly refine approximations computed with abstract interpretation tools. More precisely, we introduce a hybrid approach combining abstract interpretation and constraint programming techniques in a single static and automatic analysis. This hybrid approach benefits from the strong points of abstract interpretation and constraint programming techniques, and thus, it is more effective than static analysers and constraint solvers, when used separately. We compared the efficiency of the system we developed—named rAiCp—with state-of-the-art static analyzers: rAiCp produces substantially more precise approximations and is able to check program properties on both academic and industrial benchmarks.

Arxiv preprint arXiv:1109.0456, Sep 1, 2011

Abstract: Modern software systems, like GNU/Linux distributions or Eclipse-based development envi... more Abstract: Modern software systems, like GNU/Linux distributions or Eclipse-based development environment, are often deployed by selecting components out of large component repositories. Maintaining such software systems by performing component upgrades is a complex task, and the users need to have an expressive preferences language at their disposal to specify the kind of upgrades they are interested in. Recent research has shown that it is possible to develop solvers that handle preferences ...

Solving constraints over floating-point numbers is a critical issue in numerous applications nota... more Solving constraints over floating-point numbers is a critical issue in numerous applications notably in program verification. Capabilities of filtering algorithms for constraints over the floating-point numbers have been so far limited to 2b-consistency and its derivatives. Though safe, such filtering techniques suffer from the well known pathological problems of local consistencies, e.g., inability to efficiently handle multiple occurrences of the variables. These limitations also take roots in the strongly restricted floating-point arithmetic. To circumvent the poor properties of floating-point arithmetic, we propose in this paper to build various relaxations over the reals of the problem over the floats. We show that using linear programming (LP) to shrink the domains with safe linearisations of such relaxations can be very effective for boosting filtering techniques for constraints over the floats. Preliminary experiments on a limited but relevant set of benchmarks are very promising.

La mise en oeuvre effective de méthodes de vérification de programmes comportant des calculs sur ... more La mise en oeuvre effective de méthodes de vérification de programmes comportant des calculs sur les nombresà virgule flottante reste encore problématique. C'est pourquoi nous présentons dans cet article une nouvelle méthode de résolution de contraintes sur les nombresà virgule flottante qui consisteà les approximer sur les réels. Elle est basée sur la construction d'approximations sur les réels, précises et conservatives des solutions des contraintes sur les nombresà virgule flottante. Cette méthode permet de s'appuyer sur l'utilisation d'algorithmes de filtrage sur les réels pour résoudre des problèmes sur les nombresà virgule flottante. Il devient ainsi possible de repousser les limitations actuelles des solveurs de contraintes sur les nombresà virgules flottantes, telles que le problème du passageà l'échelle, pour générer des jeux de tests, ou vérifier des programmes plus conséquents que ceux traités jusqu'à maintenant.

Global Filtering Algorithms Based on Linear Relaxations

Abstract: This paper introduces a ltering algorithm to prune the domainsof the variables in numer... more Abstract: This paper introduces a ltering algorithm to prune the domainsof the variables in numerical constraints. This algorithm is basedon the RLT (Reformulation-Linearization Technique) schema. In the reformulationphase, tight convex and concave approximations of nonlinearterms are generated, that's to say for bilinear terms, product of variables,power and univariate terms. New variables are introduced to linearizethe initial constraint system. A linear

Siam Journal on Numerical Analysis, 2005

Numerical constraint systems are often handled by branch and prune algorithms that combine splitt... more Numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques, local consistencies, and interval methods. This paper first recalls the principles of Quad, a global constraint that works on a tight and safe linear relaxation of quadratic subsystems of constraints. Then, it introduces a generalization of Quad to polynomial constraint systems. It also introduces a method to get safe linear relaxations and shows how to compute safe bounds of the variables of the linear constraint system. Different linearization techniques are investigated to limit the number of generated constraints. QuadSolver, a new branch and prune algorithm that combines Quad, local consistencies, and interval methods, is introduced. QuadSolver has been evaluated on a variety of benchmarks from kinematics, mechanics, and robotics. On these benchmarks, it outperforms classical interval methods as well as constraint satisfaction problem solvers and it compares well with state-of-the-art optimization solvers.

This paper introduces a new filtering algorithm for handling systems of quadratic equations and i... more This paper introduces a new filtering algorithm for handling systems of quadratic equations and inequations. Such constraints are widely used to model distance relations in numerous application areas ranging from robotics to chemistry. Classical filtering algorithms are based upon local consistencies and thus, are unable to achieve a significant pruning of the domains of the variables occurring in quadratic constraints systems. The drawback of these approaches comes from the fact that the constraints are handled independently. We introduce here a global filtering algorithm that works on a tight linear relaxation of the quadratic constraints. First experimentations show that this new algorithm yields a much more effective pruning of the domains than local consistency filtering algorithms.

Computing Research Repository, 2008

Finding feasible points for which the proof succeeds is a critical issue in safe Branch and Bound... more Finding feasible points for which the proof succeeds is a critical issue in safe Branch and Bound algorithms which handle continuous problems. In this paper, we introduce a new strategy to compute very accurate approximations of feasible points. This strategy takes advantage of the Newton method for under-constrained systems of equations and inequalities. More precisely, it exploits the optimal solution of a linear relaxation of the problem to compute efficiently a promising upper bound. First experiments on the Coconuts benchmarks demonstrate that this approach is very effective.

This paper introduces a new framework for tackling constraints over the floating-point numbers. A... more This paper introduces a new framework for tackling constraints over the floating-point numbers. An important application area where such solvers are required is program analysis (e.g., structural test case generation, correctness proof of numeric operations). Albeit the floating-point numbers are a finite subset of the real numbers, classical CSP techniques are ineffective due to the huge size of the domains. Relations that hold over the real numbers may not hold over the floating-point numbers. Moreover, constraints that have no solutions over the reals may hold over the floats. Thus, interval-narrowing techniques, which are used in numeric CSP, cannot safely solve constraints systems over the floats. We analyse here the specific properties of the relations over the floats. A CSP over the floats is formally defined. We show how local-consistency filtering algorithms used in interval solvers can be adapted to achieve a safe pruning of such CSP. Finally, we illustrate the capabilities of a CSP over the floats for the generation of test data.

Journal of Computational and Applied Mathematics, 2007

Interval methods have shown their ability to locate and prove the existence of a global optima in... more Interval methods have shown their ability to locate and prove the existence of a global optima in a safe and rigorous way. Unfortunately, these methods are rather slow. Efficient solvers for optimization problems are based on linear relaxations. However, the latter are unsafe, and thus may overestimate, or worst, underestimate the very global minima. This paper introduces QuadOpt, an efficient and safe framework to rigorously bound the global optima as well as its location. QuadOpt uses consistency techniques to speed up the initial convergence of the interval narrowing algorithms. A lower bound is computed on a linear relaxation of the constraint system and the objective function. All these computations are based on a safe and rigorous implementation of linear programming techniques. First experimental results are very promising.