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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.

We aim to understand inherent reasons for lower bounds for QBF proof systems and revisit and compare two previous approaches in this direction. The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff and Pich, LICS’16). Here, we show a refined version of strategy extraction and thereby for any QBF proof system obtain a trichotomy for hardness: (1) via circuit lower bounds, (2) via propositional Resolution lower bounds, or (3) “genuine” QBF lower bounds. The second approach tries to explain QBF lower bounds through quantifier alternations in a system called relaxing QU-Res (Chen, ACM TOCT 2017). We prove a strong lower bound for relaxing QU-Res, which at the same time exhibits significant shortcomings of that model. Prompted by this, we introduce a hierarchy of new systems that improve Chen’s model and prove a strict separation for the complexity of proofs in this hierarchy. We show that lower bounds in our new model correspond to the trichotomy obtained via strategy extraction.

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