Abstract. The problem of computing Craig interpolants in SAT and 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 their combination-and they are successfully used within model checking tools. For the theory of linear arithmetic over the integers (LA(Z)), however, the problem of finding an interpolant is more challenging, and the task of developing efficient interpolant generators for the full theory LA(Z) is still the objective of ongoing research.In this paper we try to close this gap. We build on previous work and present a novel interpolation algorithm for SMT(LA(Z)), which exploits the full power of current state-of-the-art SMT(LA(Z)) solvers. We demonstrate the potential of our approach with an extensive experimental evaluation of our implementation of the proposed algorithm in the MATHSAT SMT solver. Motivations, Related Work and GoalsGiven two formulas A and B such that A∧B is inconsistent, a Craig interpolant (simply "interpolant" hereafter) for (A, B) is a formula I s.t. A entails I, I ∧ B is inconsistent, and all uninterpreted symbols of I occur in both A and B.Interpolation in both SAT and SMT has been recognized to be a substantial tool for formal verification. For instance, in the context of software model checking based on counter-example-guided-abstraction-refinement (CEGAR) interpolants of quantifier-free formulas in suitable theories are computed for automatically refining abstractions in order to rule out spurious counterexamples. Consequently, the problem of computing interpolants in SMT has received a lot of interest in the last years (e.g., [14,17,19,11,4,10,13,7,8,3,12]). In the recent years, efficient algorithms and tools for interpolant generation for quantifier-free formulas in SMT have been presented for some theories of interest, including that of equality and uninterpreted functions (EUF) [14,7], linear arithmetic over the rationals (LA(Q)) [14,17,4], and for their combination [19,17,4,8], and they are successfully used within model-checking tools.For the theory of linear arithmetic over the integers (LA(Z)), however, the problem of finding an interpolant is more challenging. In fact, it is not always possible to

Abstract. The problem of computing Craig interpolants in SAT and 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 interestincluding that of equality and uninterpreted functions (E UF), linear arithmetic over the rationals (LA(Q)), and their combination-and they are successfully used within model checking tools. For the theory of linear arithmetic over the integers (LA(Z)), however, the problem of finding an interpolant is more challenging, and the task of developing efficient interpolant generators for the full theory LA(Z) is still the objective of ongoing research. In this article we try to close this gap. We build on previous work and present a novel interpolation algorithm for SMT(LA(Z)), which exploits the full power of current stateof-the-art SMT(LA(Z)) solvers. We demonstrate the potential of our approach with an extensive experimental evaluation of our implementation of the proposed algorithm in the MathSAT SMT solver.

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International audienceIn distributed development of modern systems, contracts play a vital role in ensuring interoperability of components and adherence to specifications. It is therefore often desirable to verify the satisfaction of an overall property represented as a contract, given the satisfaction of smaller properties also represented as contracts. When the verification result is negative, designers must face the issue of refining the subproperties and components. This is an instance of the classical synthesis problems: " can we construct a model that satisfies some given specification? " In this work, we propose two strategies enabling designers to synthesize or refine a set of contracts so that their composition satisfies a given contract. We develop a generic algebraic method and show how it can be applied in different contract models to support top-down component-based development of distributed systems

In the distributed development of modern IT systems, contracts play a vital role in ensuring interoperability of components and adherence to specifications. The design of embedded systems, however, is made more complex by the heterogeneous nature of components, which are often described using different models and interaction mechanisms. Composing such components is generally not well-defined, making design and verification difficult. Several denotational frameworks have been proposed to handle heterogeneity using a variety of approaches. However, the application of heterogeneous modeling frameworks to contract-based design has not yet been investigated.In this work, we develop an operational model with precise heterogeneous denotational semantics, based on tag machines, that can represent heterogeneous composition, and provide conditions under which composition can be captured soundly and completely. The operational framework is implemented in a prototype tool which we use for experimental evaluation. We then construct a full contract model and introduce heterogeneous composition, refinement, dominance, and compatibility between contracts, altogether enabling a formalized and rigorous design process for heterogeneous systems. Besides, we also develop a generic algebraic method to synthesize or refine a set of contracts so that their composition satisfies a given contract.