Twelve Problems in Proof Complexity

Abstract

Proof complexity is a research area that studies the concept of complexity from the point of view of logic. Although it is very much connected with computational complexity, the goals are different. In proof complexity we are studying the question how difficult is to prove a theorem? There are various ways how one can measure the “complexity” of a theorem. We may ask what is the length of the shortest proof of the theorem in a given formal system. Thus the complexity is the size of proofs. This corresponds to questions in computational complexity about the size of circuits, the number of steps of Turing machines etc. needed to compute a given function. But we may also ask how strong theory is needed to prove the theorem. This also has a counterpart in computational complexity—the questions about the smallest complexity class to which a given set or function belongs.