Combination of convex theories: Modularity, deduction completeness, and explanation

AbstractDecision procedures are key components of theorem provers and constraint satisfaction systems. Their modular combination is of prime interest for building efficient systems, but their effective use is often limited by poor interface capabilities, when such procedures only provide a simple “sat/unsat” answer. In this paper, we develop a framework to design cooperation schemas between such procedures while maintaining modularity of their interfaces. First, we use the framework to specify and prove the correctness of classic combination schemas by Nelson–Oppen and Shostak. Second, we introduce the concept of deduction complete satisfiability procedures, we show how to build them for large classes of theories, then we provide a schema to modularly combine them. Third, we consider the problem of modularly constructing explanations for combinations by re-using available proof-producing procedures for the component theories

Combining Proof-Producing Decision Procedures

International audienceConstraint solvers are key modules in many systems with reasoning capabilities (e.g., automated theorem provers). To incorporate constraint solvers in such systems, the capability of producing conflict sets or explanations of their results is crucial. For expressiveness, constraints are usually built out in unions of theories and constraint solvers in such unions are obtained by modularly combining solvers for the component theories. In this paper, we consider the problem of modularly constructing conflict sets for a combined theory by re-using available proof-producing procedures for the component theories. The key idea of our solution to this problem is the concept of explanation graph, which is a labelled, acyclic and undirected graph capable of recording the entailment of some equalities. Explanation graphs allow us to record explanations computed by a proof-producing procedure and to refine the Nelson-Oppen combination method to modularly build conflict sets for disjoint unions of theories. We also study how the computed conflict sets relate to an appropriate notion of minimality