Hard QBFs for Merge Resolution

Abstract

We prove the first genuine quantified Boolean formula (QBF) proof size lower bounds for the proof system Merge Resolution (MRes), a refutational proof system for prenex QBFs 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. This makes MRes quite powerful: it has strategy extraction by design and allows short proofs for formulas that 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 ∀ Exp + Res and IR.