Abstract
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 non-trivial inferences only when resolving conflicts between formulæ and assignments representing the candidate model. CDSAT (Conflict-Driven SATisfiability) is a method for conflict-driven reasoning in unions of theories. It combines inference systems for individual theories as theory modules within a solver for the union of the theories. This article augments CDSAT with a more general lemma learning capability and with proof generation. Furthermore, theory modules for several theories of practical interest are shown to fulfill the requirements for completeness and termination of CDSAT. Proof generation is accomplished by a proof-carrying version of the CDSAT transition system that produces proof objects in memory accommodating multiple proof formats. Alternatively, one can apply to CDSAT the LCF approach to proofs from interactive theorem proving, by defining a kernel of reasoning primitives that guarantees the correctness by construction of CDSAT proofs.
Highlights
The satisfiability problem is one of checking if a given formula has a model
Many SAT solvers employ a conflict-driven search strategy, known as ConflictDriven Clause Learning (CDCL), in which the solver extends a partial assignment until it satisfies all clauses, or a conflict arises as the assignment falsifies a clause
Conflict-driven satisfiability procedures work by building partial assignments, detecting conflicts when the assignment falsifies the input formula, and performing conflict-driven inferences to explain conflicts and reorient the search
Summary
The satisfiability problem is one of checking if a given formula has a model. In the propositional case (SAT) the input is usually a formula in conjunctive normal form (a set of clauses), and a model is an assignment of truth values to propositional variables that satisfies all the clauses. We prove that if all modules are black-boxes, CDSAT emulates the equality-sharing method (covering MBTC), and we demonstrate the role of the leading theory by considering the case where at-most cardinality constraints need to be enforced. The DPLL(T ) or CDCL(T ) paradigm naturally supports the generation of proofs by resolution, where the theory lemmas are plugged in as leaves with black-box subproofs [3,12,27,38] This style has been implemented in solvers such as Z3 [3], veriT [2,27], and CVC4 [38] and extended in several ways (e.g., [2,38]). In CDSAT, the CDCL-based SAT solver loses its centrality as the only conflict-driven component, and all theory modules contribute directly to the proof, including new terms. Lemma learning and proof generation for CDSAT appeared in a conference version [10] of the present article
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