Canonized Rewriting and Ground AC Completion Modulo Shostak Theories : Design and Implementation

AC-completion efficiently handles equality modulo associative and commutative function symbols. When the input is ground, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground AC-completion for deciding formulas in the combination of the theory of equality with user-defined AC symbols, uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our algorithm, called AC(X), is obtained by augmenting in a modular way ground AC-completion with the canonizer and solver present for the theory X. This integration rests on canonized rewriting, a new relation reminiscent to normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is proved sound, complete and terminating, and is implemented to extend the core of the Alt-Ergo theorem prover.Comment: 30 pages, full version of the paper TACAS'11 paper "Canonized Rewriting and Ground AC-Completion Modulo Shostak Theories" accepted for publication by LMCS (Logical Methods in Computer Science

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International audienceThe Model Checking Modulo Theories (MCMT) framework is a powerful model checking technique for verifying safety properties of parameterized transition systems. In MCMT, logical formulas are used to represent both transitions and sets of states and safety properties are verified by an SMT-based backward reachability analysis. To be fully automated , the class of formulas handled in MCMT is restricted to cubes, i.e. existentially quantified conjunction of literals. While being very expressive , cubes cannot define properties with a global termination condition , usually described by a universally quantified formula. In this paper we describe BRWP, an extension of the backward reach-ability of MCMT for reasoning about validity properties expressed as universal cubes, that is formulas of the form ∃i∀j.C(i, j), where C(i, j) is a conjunction of literals. Our approach consists in a tight cooperation between the backward reachability loop and a deductive verification engine based on weakest-precondition calculus (WP). To provide evidence for the applicability of our new algorithm, we show how to make Cubicle, a model checker based on MCMT, cooperates with the Why3 platform for deductive program verification

Renforcement du noyau d un démonstrateur SMT (Conception et implantation de procédures de décisions efficaces)

Cette thèse s'intéresse à la démonstration automatique de la validité de formules mathématiques issues de la preuve de programmes. Elle se focalise tout particulièrement sur la Satisfiabilité Modulo Théories (SMT): un jeune domaine de recherche qui a connu de grands progrès durant la dernière décennie. Les démonstrateurs de cette famille ont des applications diverses dans la conception de microprocesseurs, la preuve de programmes, le model-checking, etc.Les démonstrateurs SMT offrent un bon compromis entre l'expressivité et l'efficacité. Ils reposent sur une coopération étroite d'un solveur SAT avec une combinaison de procédures de décision pour des théories spécifiques comme la théorie de l'égalité libre avec des symboles non interprétés, l'arithmétique linéaire sur les entiers et les rationnels, et la théorie des tableaux.L'objectif de cette thèse est d'améliorer l'efficacité et l'expressivité du démonstrateur SMT Alt-Ergo. Pour cela, nous proposons une nouvelle procédure de décision pour la théorie de l'arithmétique linéaire sur les entiers. Cette procédure est inspirée par la méthode de Fourier-Motzkin, mais elle utilise un simplexe sur les rationnels pour effectuer les calculs en pratique. Nous proposons également un nouveau mécanisme de combinaison, capable de raisonner dans l'union de la théorie de l'égalité libre, la théorie AC des symboles associatifs et commutatifs et une théorie arbitraire deShostak. Ce mécanisme est une extension modulaire et non intrusive de la procédure de completion close modulo AC avec la théorie de Shostak. Aussi, nous avons étendu Alt-Ergo avec des procédures de décision existantes pour y intégrer d'autres théories intéressantes comme la théorie de types de données énumérés et la théorie des tableaux. Enfin, nous avons exploré des techniques de simplification de formules en amont et l'amélioration de son solveur SAT.This thesis tackles the problem of automatically proving the validity of mathematical formulas generated by program verification tools. In particular, it focuses on Satisfiability Modulo Theories (SMT): a young research topic that has seen great advances during the last decade. The solvers of this family have various applications in hardware design, program verification, model checking, etc.SMT solvers offer a good compromise between expressiveness and efficiency. They rely on a tight cooperation between a SAT solver and a combination of decision procedures for specific theories, such as the free theory of equality with uninterpreted symbols, linear arithmetic over integers and rationals, or the theory of arrays.This thesis aims at improving the efficiency and the expressiveness of the Alt-Ergo SMT solver. For that, we designed a new decision procedure for the theory of linear integer arithmetic. This procedure is inspired by Fourier-Motzkin's method, but it uses a rational simplex to perform computations in practice. We have also designed a new combination framework, capable of reasoning in the union of the free theory of equality, the AC theory of associative and commutativesymbols, and an arbitrary signature-disjoint Shostak theory. This framework is a modular and non-intrusive extension of the ground AC completion procedure with the given Shostak theory. In addition, we have extended Alt-Ergo with existing decision procedures to integrate additional interesting theories, such as the theory of enumerated data types and the theory of arrays. Finally, we have explored preprocessing techniques for formulas simplification as well as the enhancement of Alt-Ergo's SAT solver.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF

A Non-linear Arithmetic Procedure for Control-Command Software Verification

International audienceState-of-the-art (semi-)decision procedures for non-linear real arithmetic address polynomial inequalities by mean of symbolic methods, such as quantifier elimination, or numerical approaches such as interval arithmetic. Although (some of) these methods offer nice completeness properties, their high complexity remains a limit, despite the impressive efficiency of modern implementations. This appears to be an obstacle to the use of SMT solvers when verifying, for instance, functional properties of control-command programs. Using off-the-shelf convex optimization solvers is known to constitute an appealing alternative. However, these solvers only deliver approximate solutions, which means they do not readily provide the soundness expected for applications such as software verification. We thus investigate a-posteriori validation methods and their integration in the SMT framework. Although our early prototype, implemented in the Alt-Ergo SMT solver, often does not prove competitive with state of the art solvers, it already gives some interesting results, particularly on control-command programs

A Collaborative Framework for Non-Linear Integer Arithmetic Reasoning in Alt-Ergo

In this paper, we describe a collaborative framework for reasoning modulo simple properties of non-linear integer arithmetic. This framework relies on the AC(X) combination method and on interval calculus. The first component is used to handle equalities of linear integer arithmetic and associativity and commutativity properties of non-linear multiplication. The interval calculus component is used - in addition to standard linear operations over inequalities - to refine bounds of non-linear terms and to inform the SAT solver about judicious case-splits on bounded intervals. The framework has been implemented in the Alt-Ergo theorem prover. We show its effectiveness on a set of formulas generated from deductive program verification

SAT-MICRO: petit mais costaud !

National audienceLe problème SAT, qui consiste à déterminer si une formule booléenne est satisfaisable, est un des problèmes NP-complets les plus célèbres et aussi un des plus étudiés. Basés initialement sur la procédure DPLL, les SAT-solvers modernes ont connu des progrès spectaculaires ces dix dernières années dans leurs performances, essentiellement grâce à deux optimisations: le retour en arrière non-chronologique et l'apprentissage par analyse des clauses conflits. Nous proposons dans cet article une étude formelle du fonctionnement de ces techniques ainsi qu'une réalisation en OCaml d'un SAT-solver, baptisé SAT-MICRO, intégrant ces optimisations ainsi qu'une mise en forme normale conjonctive paresseuse. Le fonctionnement de SAT-MICRO est décrit par un ensemble de règles d'inférence et la taille de son code, 70 lignes au total, permet d'envisager sa certification complète

Jouez à Faire Consensus Avec MITTEN (démonstration)

National audienceCet article présente Mitten, un outil pour décrire finement et jouer des scénarios sur une implémentation de Tenderbake-le prochain protocole de consensus à la pBFT de la blockchain Tezos. Mitten est paramétrable pour filtrer et examiner les messages selon des scénarios particuliers écrits dans un DSL construit sur OCaml. Grâce à Mitten, nous avons pu écrire et simuler des scénarios subtils pour reproduire des comportements difficilement atteignables en temps normal. Nous avons également pu simuler des situations permettant d'exhiber des bugs dans l'implémentation en cours et de tester des correctifs proposés

Applying SMT Solvers to the Test Template Framework

The Test Template Framework (TTF) is a model-based testing method for the Z notation. In the TTF, test cases are generated from test specifications, which are predicates written in Z. In turn, the Z notation is based on first-order logic with equality and Zermelo-Fraenkel set theory. In this way, a test case is a witness satisfying a formula in that theory. Satisfiability Modulo Theory (SMT) solvers are software tools that decide the satisfiability of arbitrary formulas in a large number of built-in logical theories and their combination. In this paper, we present the first results of applying two SMT solvers, Yices and CVC3, as the engines to find test cases from TTF's test specifications. In doing so, shallow embeddings of a significant portion of the Z notation into the input languages of Yices and CVC3 are provided, given that they do not directly support Zermelo-Fraenkel set theory as defined in Z. Finally, the results of applying these embeddings to a number of test specifications of eight cases studies are analysed.Comment: In Proceedings MBT 2012, arXiv:1202.582