Proof complexity (11w5103)

Proof complexity is a research area that studies the concept of complexity from the point of view of logic. In proof complexity, an important question is: “how difficult is it to prove a theorem?” There are various ways that one can measure the complexity of a theorem. We can ask what is the length of the shortest proof of a theorem in a given formal system (size of the proofs) or how strong a theory is needed to prove the theorem (that is, how complex are the concepts involved in the proof). The former is studied in the context of proof systems (in particular, propositional proof systems), the latter in bounded arithmetic. Naturally, the length of a shortest proof of a theorem very much depends on the type of proof system in which it is being proved. For a proof system, we also would like to know if there is an efficient algorithm that would produce a proof of any tautology, and whether it would produce a shortest such proof. These questions, besides their mathematical and philosophical significance, have practical applications in automated theorem proving. From the computational point of view, the question of proving tautologies is a co-NP question: that is, a counterexample to a formula which is not a tautology would be short and easily verifiable. Moreover, it is known that the existence of a propositional proof system in which all tautologies have short proofs is equivalent to proving that NP is closed under complementation. This establishes an important link between proof complexity and a major open problem in computational complexity theory. There are other connections between computational and proof complexity (for example, circuit lower bounds and proof system lower bounds), although in some cases the proof complexity counterparts of computational complexity results are still unresolved. A related, uniform side of proof complexity is the study of weak systems of arithmetic (in particular, bounded arithmetic). Here the complexity of a proof of a theorem is defined in terms of the complexity of concepts involved in that proof. For example, the weaker systems of arithmetic cannot operate with concepts such as the Pigeonhole Principle. A recent subarea of proof complexity, called bounded reverse mathematics, studies the complexity of reasoning needed to prove a given theorem: that is, what is the weakest theory in which a given mathematical theorem can be proven. Proof complexity historically was developed during the 1960’s and 1970’s, as an outgrowth of research on computer-based theorem provers. At first, researchers in proof complexity concentrated primarily on lower bounds on proof size, and targeted lower bounds on computational complexity (for instance, Cook’s characterization of the NP=?coNP problem in terms of proof complexity) and independence results in formal