On Deciding Satisfiability by DPLL( $\Gamma+{\mathcal T}$ ) and Unsound Theorem Proving

Abstract

Applications in software verification often require determining the satisfiability of first-order formulae with respect to some background theories. During development, conjectures are usually false. Therefore, it is desirable to have a theorem prover that terminates on satisfiable instances. Satisfiability Modulo Theories (SMT) solvers have proven highly scalable, efficient and suitable for integrated theory reasoning. Superposition-based inference systems are strong at reasoning with equalities, universally quantified variables, and Horn clauses. We describe a calculus that tightly integrates Superposition and SMT solvers. The combination is refutationally complete if background theory symbols only occur in ground formulae, and non-ground clauses are variable inactive. Termination is enforced by introducing additional axioms as hypotheses. The calculus detects any unsoundness introduced by these axioms and recovers from it.