Model Completeness, Uniform Interpolants and Superposition Calculus

Abstract

Uniform interpolants have been largely studied in non-classical propositional logics since the nineties; a successive research line within the automated reasoning community investigated uniform quantifier-free interpolants (sometimes referred to as “covers”) in first-order theories. This further research line is motivated by the fact that uniform interpolants offer an effective solution to tackle quantifier elimination and symbol elimination problems, which are central in model checking infinite state systems. This was first pointed out in ESOP 2008 by Gulwani and Musuvathi, and then by the authors of the present contribution in the context of recent applications to the verification of data-aware processes. In this paper, we show how covers are strictly related to model completions, a well-known topic in model theory. We also investigate the computation of covers within the Superposition Calculus, by adopting a constrained version of the calculus and by defining appropriate settings and reduction strategies. In addition, we show that computing covers is computationally tractable for the fragment of the language used when tackling the verification of data-aware processes. This observation is confirmed by analyzing the preliminary results obtained using the mcmt tool to verify relevant examples of data-aware processes. These examples can be found in the last version of the tool distribution.

Interpolation and Uniform Interpolation in Quantifier-Free Fragments of Combined First-Order Theories

“…In this context, complex dynamic systems that can interact with a persistent data storage are verified against some property of interest via sophisticated techniques based on SMT-solving and on automated reasoning; specifically, the presence of the 'data' component, which is usually represented as relational databases that the 'process' component can query and update, requires the development of suitable techniques for eliminating (to some extent) quantifiers binding variables that range over the content of such databases. This task can be effectively and efficiently performed by computing (combined) uniform interpolants [31,32,39,40]. This motivated the implementation of algorithms for computing combined uniform interpolants in the state-of-the-art MCMT model checker [72].…”

“…Proofs are omitted (sometimes in favor of intuitive justifications); however, they can all be found in the original papers. More precisely, proofs of the results from Section 3 are in [36], proofs of the results from Section 4 are in [37], and proofs of the results from Section 5 are in [31,32,[38][39][40].…”

“…This section presents results contained in [31,32,[38][39][40]64,65]. First, we analyze a strong form of of quantifier-free interpolation and its relationship with model-completeness [31,32]; we then show that, for convex theories, the same hypotheses allowing the transfer of the existence of ordinary interpolants also allow the transfer of the existence of these stronger interpolants [39,40].…”

“…That is why investigating uniform interpolation in E U F can be interesting. In fact, E U F does enjoy uniform interpolation; the result was stated on various occasions in the literature (including, e.g., [30]), but the first proofs were only published in the conference paper [31] and its journal version [32]. Alternative proofs are reported in [38,65].…”

“…In these contributions, examples of computations of uniform interpolants were supplied, and some algorithms were also sketched. The first formal proofs about the existence of uniform interpolants in E U F (the theory of pure equality in an arbitrary signature) were however published only in [31,32]. The usefulness of uniform interpolants in model checking was already stressed in [30] and further motivated by our recent line of research on the verification of data-aware processes [31,[33][34][35].…”