An optimality result for clause form translation

Abstract

The exponential complexity in size of the standard clause form translation is often considered as a serious drawback of the resolution method. Fortunately, a polynomial translation is possible by first introducing definitions, one for each subformula of the conjecture. This exhaustiveness can however be proved inefficient when the length of proofs is considered. In order to improve this interesting technique, we first generalize it to renamings, which consist in introducing definitions only for a subset of subformulas, resulting in a wide set of possible clause forms from a single conjecture. We show how a simple and efficient algorithm yields a renaming which, on equivalence-free conjectures, minimizes the number of clauses among these clause forms. This translation has been tested on the famous challenge problem by P. Andrews, yielding a spectacular reduction in search space and time, and therefore is one of the more simple and general technique to efficiently produce a resolution proof for this problem.