Understanding Gentzen and Frege Systems for QBF

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 ${\text{G}}_1^{\ast}$ p-simulates EF + ∀red and that ${\text{G}}_1^{\ast}$ 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 $S_2^1$. Specifically, intuitionistic $S_2^1$ 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 $S_2^1$ and a normal form for EF + ∀red proofs.