Understanding Gentzen and Frege Systems for QBF

Abstract

Recently Beyersdorff, Bonacina, and Chew [10] introduced a natural class of Frege systems for quantified Boolean formulas (QBF) and showed strong lower bounds for restricted versions of these systems. Here we provide a comprehensive analysis of the new extended Frege system from [10], denoted EF + ∀red, which is a natural extension of classical extended Frege EF.Our main results are the following: Firstly, we prove that the standard Gentzen-style system G * 1 p-simulates EF + ∀red and that G * 1 is strictly stronger under standard complexity-theoretic hardness assumptions.Secondly, we show a correspondence of EF + ∀red to bounded arithmetic: EF + ∀red can be seen as the non-uniform propositional version of intuitionistic S 1 2 . Specifically, intuitionistic S 1 2 proofs of arbitrary statements in prenex form translate to polynomial-size EF + ∀red proofs, and EF + ∀red is in a sense the weakest system with this property.Finally, we show that unconditional lower bounds for EF + ∀red would imply either a major breakthrough in circuit complexity or in classical proof complexity, and in fact the converse implications hold as well. Therefore, the system EF + ∀red naturally unites the central problems from circuit and proof complexity.Technically, our results rest on a formalised strategy extraction theorem for EF + ∀red akin to witnessing in intuitionistic S 1 2 and a normal form for EF + ∀red proofs.

Understanding cutting planes for QBFs

“…It is interesting to note that while feasible interpolation is the only technique known for propositional Cutting Planes, we have two conceptually different lower bound methods -and hence more hard formulas in QBF. This is in line with recent findings in [17] showing that lower bounds for QBF Frege either stem from circuit lower bounds (for NC 1 ) or from propositional Frege lower bounds. Our results here illustrate the same paradigm for CP+∀red: lower bounds arise either from lower bounds for a fragment of TC 0 (via strategy extraction) or via propositional lower bound methods for Cutting Planes (feasible interpolation).…”

“…Strategy extraction as a lower bound technique was first devised for Q-Res [11] and subsequently extended to QBF Frege systems [9,17]. The technique applies to calculi that allow to efficiently extract winning strategies for the universal player from a refutation (or alternatively Skolem functions for the existential variables from a proof of a true QBF).…”

“…Proof complexity research of these systems resulted in a complete understanding of the relative complexity of QBF resolution systems [11,4], and the transfer of propositional techniques to QBF systems was thoroughly assessed [14,13,15]. In addition, stronger QBF Frege and Gentzen systems were defined and investigated [27,9,17].…”

Are Short Proofs Narrow? QBF Resolution Is Not So Simple

“…Especially for lower bounds it requires new ideas and techniques. We remark that in this direction, a new and 'genuine QBF technique' based on strategy extraction was recently developed, showing lower bounds for Q-Res [Beyersdorff et al 2015] and indeed much stronger systems Beyersdorff and Pich 2016].…”

A Game Characterisation of Tree-like Q-resolution Size

“…Indeed, this interpolation technique [31] also applies to QBF resolution-type systems [11]. Recently, the papers [8,10,18] introduce a new lower bound technique for QBF systems based on strategy extraction. Conceptually, strategy extraction and feasible interpolation both import lower bounds from circuit complexity and translate them into size of proofs lower bounds.…”