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

Abstract

We study the *refuter* problems for proof complexity lower bounds. Suppose $\varphi$ 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 $\pi$ in $P$ that claims to have proved $\varphi$, and our goal is to find an invalid derivation step within $\pi$. 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. We focus on refuter problems corresponding to lower bounds for *resolution*, which is arguably the single most studied system in proof complexity. To capture the complexity of refuter problems for resolution *size* lower bounds, we introduce a new class $\mathrm{rwPHP}(\mathsf{PLS})$ in decision-tree $\mathsf{TFNP}$, which can be seen as a randomized version of $\mathsf{PLS}$. Interpreted in bounded arithmetic, our results show that the theory $\mathsf{T}^1_2(\alpha) + \mathrm{dwPHP}(\mathsf{PV}(\alpha))$ characterizes the "reasoning power" required to prove (the "easiest") resolution size lower bounds. As a corollary, we obtain surprisingly efficient proofs of resolution lower bounds. In particular, we show that many resolution size lower bounds can be proved in low-width *random resolution* [Pudl\'ak--Thapen, CCC'17].