Efficient Generation of Craig Interpolants in Satisfiability Modulo Theories

The problem of computing Craig Interpolants has recently received a lot of interest. In this paper, we address the problem of efficient generation of interpolants for some important fragments of first order logic, which are amenable for effective decision procedures, called Satisfiability Modulo Theory solvers. We make the following contributions. First, we provide interpolation procedures for several basic theories of interest: the theories of linear arithmetic over the rationals, difference logic over rationals and integers, and UTVPI over rationals and integers. Second, we define a novel approach to interpolate combinations of theories, that applies to the Delayed Theory Combination approach. Efficiency is ensured by the fact that the proposed interpolation algorithms extend state of the art algorithms for Satisfiability Modulo Theories. Our experimental evaluation shows that the MathSAT SMT solver can produce interpolants with minor overhead in search, and much more efficiently than other competitor solvers.Comment: submitted to ACM Transactions on Computational Logic (TOCL

Living Without Beth and Craig: Definitions and Interpolants in Description Logics with Nominals and Role Inclusions

The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (PBDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, the CIP and PBDP transform potentially hard existence problems into deduction problems in the underlying logic. Description Logics with nominals and/or role inclusions do not enjoy the CIP nor PBDP, but interpolants and explicit definitions have many potential applications in ontology engineering and ontology-based data management. In this article we show the following: even without Craig and Beth, the existence of interpolants and explicit definitions is decidable in description logics with nominals and/or role inclusions such as ALCO, ALCH and ALCHIO. However, living without Craig and Beth makes this problem harder than deduction: we prove that the existence problems become 2ExpTime-complete, thus one exponential harder than validity. The existence of explicit definitions is 2ExpTime-hard even if one asks for a definition of a nominal using any symbol distinct from that nominal, but it becomes ExpTime-complete if one asks for a definition of a concept name using any symbol distinct from that concept name.Comment: We have added results on description logics with role inclusions and an ExpTime-completeness result for the explicit definability of concept names. The title has been modified by adding role inclusions. This paper has been accepted for AAAA 202

Living Without Beth and Craig: Definitions and Interpolants in Description and Modal Logics with Nominals and Role Inclusions

The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (PBDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, the CIP and PBDP reduce potentially hard existence problems to entailment in the underlying logic. Description (and modal) logics with nominals and/or role inclusions do not enjoy the CIP nor the PBDP, but interpolants and explicit definitions have many applications, in particular in concept learning, ontology engineering, and ontology-based data management. In this article we show that, even without Beth and Craig, the existence of interpolants and explicit definitions is decidable in description logics with nominals and/or role inclusions such as $\mathcal {ALCO}$ , $\mathcal {ALCH}$ and $\mathcal {ALCHOI}$ and corresponding hybrid modal logics. However, living without Beth and Craig makes these problems harder than entailment: the existence problems become 2ExpTime-complete in the presence of an ontology or the universal modality, and coNExpTime-complete otherwise. We also analyze explicit definition existence if all symbols (except the one that is defined) are admitted in the definition. In this case the complexity depends on whether one considers individual or concept names. Finally, we consider the problem of computing interpolants and explicit definitions if they exist and turn the complexity upper bound proof into an algorithm computing them, at least for description logics with role inclusions. </jats:p

Vérification relationnelle pour des programmes avec des données entières

Les travaux présentés dans cette thèse sont lies aux problèmes de vérification de l'atteignabilité et de la terminaison de programmes qui manipulent des données entières non-bornées. On décrit une nouvelle méthode de vérification basée sur une technique d'accélération de boucle, qui calcule, de manière exacte, la clôture transitive d'une relation arithmétique. D'abord, on introduit un algorithme d'accélération de boucle qui peut calculer, en quelques secondes, des clôtures transitives pour des relations de l'ordre d'une centaine de variables. Ensuite, on présente une méthode d'analyse de l'atteignabilité, qui manipule des relations entre les variables entières d'un programme, et applique l'accélération pour le calcul des relations entrée-sortie des procédures, de façon modulaire. Une approche alternative pour l'analyse de l'atteignabilité, présentée également dans cette thèse, intègre l'accélération avec l'abstraction par prédicats, afin de traiter le problème de divergence de cette dernière. Ces deux méthodes ont été évaluées de manière pratique, sur un nombre important d'exemples, qui étaient, jusqu'a présent, hors de la portée des outils d'analyse existants. Dernièrement, on a étudié le problème de la terminaison pour certaines classes de boucles de programme, et on a montré la décidabilité pour les relations étudiées. Pour ces classes de relations arithmétiques, on présente un algorithme qui s'exécute en temps au plus polynomial, et qui calcule l'ensemble d'états qui peuvent générer une exécution infinie. Ensuite on a intégré cet algorithme dans une méthode d'analyse de la terminaison pour des programmes qui manipulent des données entières.This work presents novel methods for verification of reachability and termination properties of programs that manipulate unbounded integer data. Most of these methods are based on acceleration techniques which compute transitive closures of program loops. We first present an algorithm that accelerates several classes of integer relations and show that the new method performs up to four orders of magnitude better than the previous ones. On the theoretical side, our framework provides a common solution to the acceleration problem by proving that the considered classes of relations are periodic. Subsequently, we introduce a semi-algorithmic reachability analysis technique that tracks relations between variables of integer programs and applies the proposed acceleration algorithm to compute summaries of procedures in a modular way. Next, we present an alternative approach to reachability analysis that integrates predicate abstraction with our acceleration techniques to increase the likelihood of convergence of the algorithm. We evaluate these algorithms and show that they can handle a number of complex integer programs where previous approaches failed. Finally, we study the termination problem for several classes of program loops and show that it is decidable. Moreover, for some of these classes, we design a polynomial time algorithm that computes the exact set of program configurations from which non-terminating runs exist. We further integrate this algorithm into a semi-algorithmic method that analyzes termination of integer programs, and show that the resulting technique can verify termination properties of several non-trivial integer programs.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF

Interpolant Generation for UTVPI

The problem of computing Craig interpolants in SMT has recently received a lot of interest, mainly for its applications in formal verification. Efficient algorithms for interpolant generation have been presented for some theories of interest –including that of equality and uninterpreted functions (EUF), linear arithmetic over the rationals (LA(Q)), and some fragments of linear arithmetic over the integers (LA(Z))– and they are successfully used within model checking tools. In this paper we address the problem of computing interpolants in the theory of Unit-Two-Variable-Per-Inequality (UTVPI). This theory is a very useful fragment of LA(Z), since it is the most expressive fragment of LA(Z) with a polynomial decision procedure, and many HWand SWverification queries are naturally expressed in this fragment. We present an efficient graph-based algorithm for interpolant generation in UTVPI, which exploits the power of modern SMT techniques. We have implemented our new algorithm within the MATHSAT SMT solver. Our experimental evaluation demonstrates both the efficiency and the usefulness of the new algorithm