Improved Single Pass Algorithms for Resolution Proof Reduction - (Poster Presentation)

An unsatisfiability proof is a series of applications of proof rules on an input formula to deduce false. Unsatisfiability proofs for a Boolean formula can find many applications in verification. For instance, one application is automatic learning of abstractions for unbounded model checking by analyzing proofs of program safety for bounded steps [6, 5, 4]. We can also learn unsatisfiable cores from unsatisfiability proofs, which are useful in locating errors in inconsistent specifications [10]. These proofs can be used by higher order theorem provers as sub-proofs of another proof [2]. One of the most widely used proof rules for Boolean formulas is the resolution rule, i.e., if a∨b and ¬a∨c holds then we can deduce b∨c. In the application of the rule, a is known as pivot. A resolution proof is generated by applying resolution rule on the clauses of an unsatisfiable Boolean formula to deduce false. Modern SAT solvers (Boolean satisfiability checkers) implement some variation of DPLL that is enhanced with conflict driven clause learning [9, 8]. Without incurring large additional cost on the solvers, we can generate a resolution proof from a run of the solvers on an unsatisfiable formula [11]. Due to the nature of the algorithms employed by SAT solvers, a generated resolution proof may contain redundant parts and a strictly smaller resolution proof can be obtained. Applications of the resolution proofs are sensitive to the proof size. Since minimizing resolution proofs is a hard problem [7], there has been significant interest in finding algorithms that partially minimize the resolution proofs generated by SAT solvers. In [1], two low complexity algorithms for optimizing the proofs are presented. Our work is focused on one of the two, namely Recycle-Pivots. Lets consider a resolution step that produces a clause using some pivot p. The resolution step is called redundant if each deduction sequence from the clause to false contains a resolution step with the pivot p. A redundant resolution can easily be removed by local modifications in the proof structure. After removing a redundant resolution step, a strictly smaller proof is obtained. Recycle-Pivots traverses the proofs single time to remove the redundant resolutions partially. From each clause, the algorithm starts from the clause and follows the deduction sequences to find equal pivots. The algorithm stops looking for equal pivots if it reaches to a clause that is used to deduce more than one clause. In this work, we …