Proof Complexity - Polynomiality of Proofs

Polynomiality of Proofs

Different propositional proof system for theorem proving in propositional logic, such as the sequent calculus, the cutting-plane method, resolution, the DPLL algorithm, etc. produce different proofs when applied to the same formula. Proof complexity measures the efficiency of a method in terms of the size of the proofs it produces.

Two points make the study of proof complexity non-trivial:

1. the size of a proof depends on the formula that is to be proved inconsistent;
2. proof methods are generally families of algorithms, as some of their steps are not univocally specified; for example, resolution is based on iteratively choosing a pair of clauses containing opposite literals and producing a new clause that is a consequence of them; since several such pairs may be available at each step, the algorithm has to choose one; these choices affect the proof length.

The first point is taken into account by comparing the size of a proof of a formula with the size of the formula. This comparison is made using the usual assumptions of computational complexity: first, a polynomial proof size/formula size ratio means that the proof is of size similar to that of the formula; second, this ratio is studied in the asymptotic case as the size of the formula increases.

The second point is taken into account by considering, for each formula, the shortest possible proof the considered method can produce.

The question of polynomiality of proofs is whether a method can always produce a proof of size polynomial in the size of the formula. If such a method exists, then NP would be equal to coNP: this is why the question of polynomiality of proofs is considered important in computational complexity. For some methods, the existence of formulae whose shortest proofs are always superpolynomial has been proved. For other methods, it is an open question.