Conflict-driven satisfiability for theory combination: lemmas, modules, and proofs

Search-based satisfiability procedures try to build a model of the input formula by simultaneously proposing candidate models and deriving new formulae implied by the input. Conflict-driven procedures perform nontrivial inferences only when resolving conflicts between formulae and assignments representing the candidate model. CDSAT (Conflict-Driven SATisfiability) is a method for conflict-driven reasoning in unions of theories. It combines solvers for individual theories as theory modules within a solver for the union of the theories. In this article, we add lemma learning to CDSAT; we show that theory modules for several theories of practical interest fulfill the requirements for completeness and termination of CDSAT; and we present two ways to enrich CDSAT with proof generation. First, we present a proof-carrying CDSAT transition system that produces proof objects in memory accommodating multiple proof formats. Alternatively, we apply to CDSAT the LCF approach to proofs from interactive theorem proving, by defining a kernel of reasoning primitives that guarantees that CDSAT proofs are correct by construction.