Hard QBFs for Merge Resolution

TL;DR

This paper establishes the first exponential lower bounds for the proof size in Merge Resolution (MRes), revealing its limitations and distinguishing it from other QBF proof systems.

Contribution

It provides the first genuine QBF proof size lower bounds for MRes, demonstrating its limitations and contrasting it with other resolution systems.

Findings

MRes has exponential lower bounds for certain QBFs.

Results are derived from circuit complexity and combinatorial arguments.

MRes's capabilities are largely orthogonal to other QBF resolution models.

Abstract

We prove the first genuine QBF proof size lower bounds for the proof system Merge Resolution (MRes [Olaf Beyersdorff et al., 2020]), a refutational proof system for prenex quantified Boolean formulas (QBF) with a CNF matrix. Unlike most QBF resolution systems in the literature, proofs in MRes consist of resolution steps together with information on countermodels, which are syntactically stored in the proofs as merge maps. As demonstrated in [Olaf Beyersdorff et al., 2020], this makes MRes quite powerful: it has strategy extraction by design and allows short proofs for formulas which are hard for classical QBF resolution systems. Here we show the first genuine QBF exponential lower bounds for MRes, thereby uncovering limitations of MRes. Technically, the results are either transferred from bounds from circuit complexity (for restricted versions of MRes) or directly obtained by combinatorial arguments (for full MRes). Our results imply that the MRes approach is largely orthogonal to other QBF resolution models such as the QCDCL resolution systems QRes and QURes and the expansion systems $\forall$Exp+Res and IR.