Combination of Uniform Interpolants for DAPs Verification

Abstract

In this chapter, we attack the problem of computing covers in theory combinations. Theory combination is an important topic in automated reasoning, and computing combined covers is particularly interesting in view of DAP verification. We would like to investigate a general method that works for combination of DB schemas for which cover algorithms exist: in this way, whenever we get such algorithms for the component theories, we can easily combine them so as to get a combined cover algorithm for the combined DB schema. In this chapter, we theoretically study the problem of combined covers from a general perspective, by identifying the equality interpolating property as the sufficient (and, in some sense, also necessary) condition. We do so by exhibiting a concrete combined algorithm (called $\mathsf{ConvexCombCover}$) that modularly exploits the cover algorithms for the component theories. The key feature of this algorithm relies on the extensive usage of the Beth Definability property. We conclude the chapter by giving a combined cover algorithm (called $\mathsf{TameCombCover}$) for possibly non-convex theories, in case the combination is ‘tame’: $\mathsf{TameCombCover}$ has important applications in DAP verification because it can be used in case of DB extended-schemas.