Extracting (Easily) Checkable Proofs from a Satisfiability Solver that Employs both Preorder and Postorder Resolution

In many applications the desired outcome of satisfiability checking is that the formula is unsatisfiable: A satisfying assignment essentially exhibits a bug and unsatisfiability implies a lack of bugs, at least for the property being verified. Current high-performance satisfiability checkers are unable to provide proof of unsatisfiability. Since bugs have been discovered in many solvers long after being put into service, an uncheckable decision poses a significant problem if important economic or safety decisions are to be based upon it. Current tableau-based systems that are able to produce proofs are unable to process propositional formulas of practical size. This paper describes modifications of the classical backtracking-search satisfiability algorithm of Davis, Putnam, Loveland and Logemann (DPLL) that are designed to extract checkable proofs of practical length when the formula is believed to be unsatisfiable. It is known that the purely postorder resolution proofs extractable from standard DPLL (called “tree resolution” proofs) are exponentially longer than nontree proofs in the worst case. Experience also shows them to be of impractical length. This paper describes an efficient method to integrate postorder resolution with preorder reasoning methods, including binary-clause reasoning, equivalent-literal identification, and variableelimination resolution, to produce nontree proofs. Preliminary experiments show that the resulting proofs are much shorter, but that memory has to be managed extremely carefully and the time is usually longer compared to the best “complete” programs that do not produce proofs.