# Boundary Points and Resolution

Eugene Goldberg and Panagiotis Manolios.

*Advanced Techniques in Logic Synthesis, Optimizations and Applications*, © Springer

# Overview

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.

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