Extending Merge Resolution to a Family of Proof Systems

TL;DR

This paper generalizes the Merge Resolution proof system for false QBFs by replacing merge maps with arbitrary complete representations, analyzing their proof-theoretic properties, and establishing lower bounds for these generalized systems.

Contribution

It introduces MRes-R, a family of proof systems extending Merge Resolution with arbitrary representations, and explores their proof-theoretic properties and lower bounds.

Findings

MRes-R systems are sound and complete for false QBFs.

eFrege+$orall$red can simulate all MRes-R proof systems.

Certain formulas remain hard for all MRes-R systems, extending known lower bounds.

Abstract

Merge Resolution (MRes [Beyersdorff et al. J. Autom. Reason.'2021]) is a recently introduced proof system for false QBFs. It stores the countermodels as merge maps. Merge maps are deterministic branching programs in which isomorphism checking is efficient making MRes a polynomial time verifiable proof system. In this paper, we introduce a family of proof systems MRes-R in which, the countermodels are stored in any pre-fixed complete representation R, instead of merge maps. Hence corresponding to each such R, we have a sound and refutationally complete QBF-proof system in MRes-R. To handle arbitrary representations for the strategies, we introduce consistency checking rules in MRes-R instead of isomorphism checking. As a result these proof systems are not polynomial time verifiable. Consequently, the paper shows that using merge maps is too restrictive and can be replaced with arbitrary representations leading to several interesting proof systems. Exploring proof theoretic properties of MRes-R, we show that eFrege+$\forall$red simulates all valid refutations from proof systems in MRes-R. In order to simulate arbitrary representations in MRes-R, we first represent the steps used by the proof systems as a new complete structure. Consequently, the corresponding proof system belonging to MRes-R is able to simulate all proof systems in MRes-R. Finally, we simulate this proof system via eFrege+$\forall$red using the ideas from [Chew et al. ECCC.'2021]. On the lower bound side, we show that the completion principle formulas from [Jonata et al. Theor. Comput. Sci.'2015] which are shown to be hard for regular MRes in [Beyersdorff et al. FSTTCS.'2020], are also hard for any regular proof system in MRes-R. Thereby, the paper lifts the lower bound of regular MRes to an entire class of proof systems, which use some complete representation, including those undiscovered, instead of merge maps.