Proof complexity is concerned with the mathematical analysis of the informal concept of a feasible proof when the qualification ‘feasible’ is interpreted in a complexity-theoretic sense. The most important measure of complexity of a proof is its length when it is thought of as a string over a finite alphabet. The basic question that proof complexity studies is to estimate (from below as well as from above) the minimal possible length of a proof of a formula. Measuring the complexity of a proof by its length may seem crude at first but it is analogous to measuring the complexity of an algorithm by the length of time it takes.
In the context of propositional logic the main question is whether there exists a proof system in which every propositional tautology has a short proof, a proof bounded in length by a polynomial in the length of the formula. With a suitably general definition of what a ‘proof system’ is, the question is equivalent to the problem whether the computational complexity class NP is closed under complementation.
In the setting of first-order logic one considers theories whose principal axiom scheme is the scheme of induction but accepted only for predicates on binary strings that have limited computational complexity. These are the so-called bounded arithmetic theories. A typical question is this: Can we prove more universally valid properties of strings if we assume induction for NP predicates than if we have only induction for polynomial-time predicates?
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