Counting the Number of Equivalent Binary Resolution Proofs

Abstract

A binary resolution proof is represented by a binary resolution tree (brt) with clauses at the nodes and resolutions being performed at the internal nodes. A rotation in a brt can be performed on two adjacent internal nodes if the result of reversing the order of the resolutions does not affect the clause recorded at the node closer to the root. Two brts are saidto be rotationally equivalent if one can be obtained from the other by a sequence of rotations. Let c(T) be the number of brts rotationally equivalent to T. It is shown that if T has n resolutions, all on distinct atoms, and m merges or factors between literals, then c(T) ≥ 2(su)2n-gQ(mlog(n/m))

Moreover c(T) can be as large as n!/(m+1). A-ordering, lock resolution andthe rank/activity restriction avoidcalculating equivalent brts. A dynamic programming polynomial-time algorithm is also given to calculate c(T) if T has no merges or factors.

Research supported by a grant from NSERC of Canada