Patrick Cousot_Proving Program Invariance and Termination by Parametric Abstraction, Lagrangian Rela

Patrick Cousot_Proving Program Invariance and Termination by Parametric Abstraction, Lagrangian Rela

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1 Proving Program Invariance and Termination by Parametric Abstraction, Lagrangian Relaxation and Semidefnite Programming Patrick Cousot ´ Ecole Normale Sup´ erieure 45 rue d’Ulm, 75230 Paris cedex 05 (France) Patrick.Cousot@ens.fr www.di.ens.fr/~cousot Abstract. In order to verify semialgebraic programs, we automatize the Floyd/Naur/Hoare proof method. The main task is to automatically infer valid invariants and rank functions. First we express the program semantics in polynomial form. Then the unknown rank function and invariants are abstracted in parametric form. The implication in the Floyd/Naur/Hoare veri±cation conditions is han- dled by abstraction into numerical constraints by Lagrangian relaxation. The remaining universal quanti±cation is handled by semide±nite pro- gramming relaxation. Finally the parameters are computed using semidef- inite programming solvers. This new approach exploits the recent progress in the numerical resolu- tion of linear or bilinear matrix inequalities by semide±nite programming using efficient polynomial primal/dual interior point methods generaliz- ing those well-known in linear programming to convex optimization. The framework is applied to invariance and termination proof of sequen- tial, nondeterministic, concurrent, and fair parallel imperative polyno- mial programs and can easily be extended to other safety and liveness properties. Keywords: Bilinear matrix inequality (BMI), Convex optimization, In- variance, Lagrangian relaxation, Linear matrix inequality (LMI), Live- ness, Parametric abstraction, Polynomial optimization, Proof, Rank func- tion, Safety, S-procedure, Semide±nite programming, Termination pre- condition, Termination, Program veri±cation. 1 Introduction Program verifcation is based on reasonings by induction (e.g. on program steps) which involves the discovery oF unknown inductive arguments (e.g. rank Func- tions, invariants) satisFying universally quantifed verifcation conditions. ±or static analysis the discovery oF the inductive arguments must be automated, which consists in solving the constraints provided by the verifcation conditions. Several methods have been considered: recurrence/di²erence equation resolu- tion; iteration, possibly with convergence acceleration; or direct methods (such R. Cousot (Ed.): VMCAI 2005, LNCS 3385, pp. 1–24, 2005. c ± Springer-Verlag Berlin Heidelberg 2005
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2 Patrick Cousot as elimination). All these methods involve some form of simpliFcation of the constraints by abstraction. In this paper, we explore parametric abstraction and direct resolution by Lagrangian relaxation into semidefnite programming . This is applied to termi- nation (a typical liveness property) of semialgebraic programs. The extension to invariance (a typical safety property) is sketched.
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Patrick Cousot_Proving Program Invariance and Termination by Parametric Abstraction, Lagrangian Rela

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