Proof complexity

In computer science, proof complexity is a measure of efficiency of automated theorem proving methods that is based on the size of the proofs they produce. The methods for proving contradiction in propositional logic are the most analyzed. The two main issues considered in proof complexity are whether a proof method can produce a polynomial proof of every inconsistent formula, and whether the proofs produced by one method are always of size similar to those produced by another method.

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:

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.

A second question about proof complexity is whether a method is more efficient than another. Since the proof size depends on the formula, it is possible that one method can produce a short proof of a formula and only long proofs of another formula, while a second method can have exactly the opposite behavior. The assumptions of measuring the size of the proofs relative to the size of the formula and considering only the shortest proofs are also used in this context.

...
Wikipedia