Recently Beyersdorff, Bonacina, and Chew (ITCS'16) 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 Beyersdorff et al., 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 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.