Finding Bugs in Short Proofs: The Metamathematics of Resolution Lower Bounds

We study the *refuter* problems for proof complexity lower bounds. Suppose ϕ is a hard tautology that does not admit any length-s proof in some proof system P. In the corresponding refuter problem, we are given (query access to) a purported length-s proof π in P that claims to have proved ϕ, and our goal is to find an invalid derivation step within π. As suggested by witnessing theorems in bounded arithmetic, the *computational complexity* of these refuter problems is closely tied to the *metamathematics* of the underlying lower bounds.