On the Polynomial Transparency of Resolution

In this paper a framework is developed for measuring the complexities of deductions in an ab­ stract and computationally perspicuous man­ ner. As a notion of central importance appears the so-called polynomial transparency of a cal­ culus. If a logic calculus possesses this prop­ erty, then the complexity of any deduction can be correctly measured in terms of its inference steps. The resolution calculus lacks this prop­ erty. It is proven that the number of inference steps of a resolution proof does not give a rep­ resentative measure of the actual complexity of the proof, even if only shortest proofs are considered. We use a class of formulae which have proofs with a polynomial number of inference steps, but for which the size of any proof is exponential. The polynomial intransparency of resolution is due to the renaming of derived clauses, which is a fundamental de­ duction mechanism. This result motivates the development of new data structures for the rep­ resentation of logical formulae.