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 EE + Vred, which is a natural extension of classical extended Frege EE. Our main results are the following: Firstly, we prove that the standard Gentzen-style system G*
₁
p-simulates EE + Vred and that G*
₁
is strictly stronger under standard complexity-theoretic hardness assumptions. Secondly, we show a correspondence of EE + Vred to bounded arithmetic: EE + Vred can be seen as the non-uniform propositional version of intuitionistic S12 . Specifically, intuitionistic S12 proofs of arbitrary statements in prenex form translate to polynomial-size EE + Vred proofs, and EE + Vred is in a sense the weakest system with this property. Finally, we show that unconditional lower bounds for EE + Vred 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 EE + Vred naturally unites the central problems from circuit and proof complexity. Technically, our results rest on a formalised strategy extraction theorem for EE + Vred akin to witnessing in intuitionistic S12 and a normal form for EE + Vred proofs.