Combining Decision Algorithms for Matching in the Union of Disjoint Equational Theories

AbstractThis paper addresses the problem of systematically building a matching algorithm for the union of two disjoint theoriesE1∪E2provided that matching algorithms are known in both theoriesE1andE2. In general, the blind use of combination techniques introduces unification. Two different restrictions are considered in order to reduce this unification to matching. First, we show that combining matching algorithms (with linear constant restriction) is always sufficient for solving a pure fragment of combined matching problems. Second, the investigated method is complete for the largest class of theories where unification is not needed, including regular collapse-free theories and linear theories. Syntactic conditions are given to define this class of theories in which solving the combined matching problem is performed in a modular way

Combinable Extensions of Abelian Groups

The design of decision procedures for combinations of theories sharing some arithmetic fragment is a challenging problem in verification. One possible solution is to apply a combination method à la Nelson-Oppen, like the one developed by Ghilardi for unions of non-disjoint theories. We show how to apply this non-disjoint combination method with the theory of abelian groups as shared theory. We consider the completeness and the effectiveness of this non-disjoint combination method. For the completeness, we show that the theory of abelian groups can be embedded into a theory admitting quantifier elimination. For achieving effectiveness, we rely on a superposition calculus modulo abelian groups that is shown complete for theories of practical interest in verification

A Gentle Non-Disjoint Combination of Satisfiability Procedures (Extended Version)

A satisfiability problem is often expressed in a combination of theories, and a natural approach consists in solving the problem by combining the satisfiability procedures available for the component theories. This is the purpose of the combination method introduced by Nelson and Oppen. However, in its initial presentation, the Nelson-Oppen combination method requires the theories to be signature-disjoint and stably infinite (to guarantee the existence of an infinite model). The notion of gentle theory has been introduced in the last few years as one solution to go beyond the restriction of stable infiniteness, but in the case of disjoint theories. In this paper, we adapt the notion of gentle theory to the non-disjoint combination of theories sharing only unary predicates (plus constants and the equality). Like in the disjoint case, combining two theories, one of them being gentle, requires some minor assumptions on the other one. We show that major classes of theories, i.e.\ Löwenheim and Bernays-Schönfinkel-Ramsey, satisfy the appropriate notion of gentleness introduced for this particular non-disjoint combination framework

A Rewriting Approach to the Combination of Data Structures with Bridging Theories

International audienceWe introduce a combination method à la Nelson-Oppen to solve the satisfiability problem modulo a non-disjoint union of theories connected with bridging functions. The combination method is particularly useful to handle verification conditions involving functions defined over inductive data structures. We investigate the problem of determining the data structure theories for which this combination method is sound and complete. Our completeness proof is based on a rewriting approach where the bridging function is defined as a term rewrite system, and the data structure theory is given by a basic congruence relation. Our contribution is to introduce a class of data structure theories that are combinable with a disjoint target theory via an inductively defined bridging function. This class includes the theory of equality, the theory of absolutely free data structures, and all the theories in between. Hence, our non-disjoint combination method applies to many classical data structure theories admitting a rewrite-based satisfiability procedure

Timed Specification For Web Services Compatibility Analysis

AbstractWeb services are becoming one of the main technologies for designing and building complex inter-enterprise business applications. Usually, a business application cannot be fulfilled by one Web service but by coordinating a set of them. In particular, to perform a coordination, one of the important investigations is the compatibility analysis. Two Web services are said compatible if they can interact correctly. In the literature, the proposed frameworks for the services compatibility checking rely on the supported sequences of messages. The interaction of services depends also on other properties, such that the exchanged data flow. Thus, considering only supported sequences of messages seems to be insufficient. Other properties on which the services interaction can rely on, are the temporal constraints. In this paper, we focus our interest on the compatibility analysis of Web services regarding their (1) supported sequences of messages, (2) the exchanged data flow, (3) constraints related to the exchanged data flow and (4) the temporal requirements. Based on these properties, we study three compatibility classes: (i) absolute compatibility, (ii) likely compatibility and (iii) absolute incompatibility

A Constraint-based Approach to Web Services Provisioning

In this paper we consider the provisioning problem of Web services. Our framework is based on the existence of an abstract composition, i.e., the way some services of different types can be combined together in order to achieve a given task. Our approach consists in instantiating this abstract representation of a composite Web service by selecting the most appropriate concrete Web services. This instantiation is based on constraint programming techniques which allows us to match the Web services according to a given request. Our proposal performs this instantiation in a distributed manner, i.e., the solvers for each service type are solving some constraints at one level, and they are forwarding the rest of the request (modified by the local solution) to the next services. When a service cannot provision part of the composition, a distributed backtrack mechanism enables to change previous solutions (i.e., provisions). A major interest of our approach is to preserve privacy: solutions are not sent to the whole composition, services know only the services to which they are connected, and parts of the request that are already solved are removed from the next requests. We introduce a specific data structure, namely Message Treatment Structure, for modeling the problem. We show the interest of this data structure to express the general principles of our framework and the related algorithms

Automatic Decidability for Theories Modulo Integer Offsets

Many verification problems can be reduced to a satisfiability problem modulo theories. For building satisfiability procedures the rewriting-based approach uses a general calculus for equational reasoning named superposition. Schematic superposition, in turn, provides a mean to reason on the derivations computed by superposition. Until now, schematic superposition was only studied for standard superposition. We present a schematic superposition calculus modulo a fragment of arithmetics, namely the theory of Integer Offsets. This new schematic calculus is used to prove the decidability of the satisfiability problem for some theories extending Integer Offsets. We illustrate our theoretical contribution on theories representing extensions of classical data structures, e.g., lists and records. An implementation in the rewriting-based Maude system constitutes a practical contribution. It enables automatic decidability proofs for theories of practical use

Generalizing CASL Specification Components and Preserving Rewrite Proofs

We propose the theoretical basis of a tool for the generation of reusable CASL specification components by generalization of existing ones. The underlying idea is, given a component and a set of semantic properties that it satisfies and that we want to preserve, to find a parameterized, more general, component satisfying the following conditions: the original component is one of its possible instantiations, and any of its instantiations satisfy the stated properties. We present here both the definition of the generalization operation for CASL and the problem of preserving properties in the generalized component. To guarantee the preservation of properties, we propose to preserve their proofs, concentrating on the use of rewrite proofs. This technique provides a simple way to find sufficient conditions for the preservation of the corresponding properties. This work is being integrated in the specification component development tool FERUS, under development for the CASL language, using ELAN as the rewrite proof engine

Non-Disjoint Unions of Theories and Combinations of Satisfiability Procedures: First Results

In this paper we outline a theoretical framework for the combination of decision procedures for the satisfiability of constraints with respect to a constrainttheory. We describe a general combination method which, given a procedure that decides constraint satisfiability with respect to a constraint theory ${\cal T}_1$ and one that decides constraint satisfiability with respect to a constraint theory ${\cal T}_2$, is able to produce a procedure that (semi-)decides constraint satisfiability with respect to the union of ${\cal T}_1$ and ${\cal T}_2$. We also provide some model-theoretic conditions on the constraint language and the component constraint theories for the method to be sound and complete, with special emphasis on the case in which the signatures of ${\cal T}_1$ and ${\cal T}_2$ are non-disjoint

Automatic Decidability for Theories with Counting Operators

International audienceThe notion of schematic paramodulation has been introduced to reason on properties of (standard) paramodulation. We present a schematic paramodulation calculus modulo a fragment of arithmetics, namely the theory of Integer Offsets. This new schematic calculus is used to prove the decidability of the satisfiability problem for some theories equipped with counting operators. We illustrate our theoretical contribution on theories representing extensions of classical data structures, e.g., lists and records