Boundary Points and Resolution

Chapter

Abstract

We use the notion of boundary points to study resolution proofs. Given a CNF formula F, an l(x)-boundary point is a complete assignment falsifying only clauses of F having the same literal l(x) of variable x. An l(x)-boundary point p mandates a resolution on variable x. Adding the resolvent of this resolution to F eliminates p as an l(x)-boundary point. Any resolution proof has to eventually eliminate all boundary points of F. Hence one can study resolution proofs from the viewpoint of boundary point elimination. We use equivalence checking formulas to compare proofs of their unsatisfiability built by a conflict-driven SAT-solver and very short proofs tailored to these formulas. We show experimentally that in contrast to proofs generated by this SAT-solver, almost every resolution of a specialized proof eliminates a boundary point. This implies that one may use the share of resolutions eliminating boundary points as a metric of proof quality. We argue that obtaining proofs with a high value of this metric requires taking into account the formula structure. We show that for any unsatisfiable CNF formula there always exists a proof consisting only of resolutions eliminating cut boundary points (which are a relaxation of the notion of boundary points). This result enables building resolution SAT-solvers that are driven by elimination of cut boundary points.

This chapter is an extended version of the conference paper [9].

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Northeastern UniversityBastonUSA

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