Some subsystems of constant-depth Frege with parity

We consider three relatively strong families of subsystems of AC0[2]-Frege proof systems, i.e., propositional proof systems using constant-depth formulas with an additional parity connective, for which exponential lower bounds on proof size are known. In order of increasing strength, the subsystems are (i) constant-depth proof systems with parity axioms and the (ii) treelike and (iii) daglike versions of systems introduced by Krajíček which we call PKcd(⊕). In a PKcd(⊕)-proof, lines are disjunctions (cedents) in which all disjuncts have depth at most d, parities can only appear as the outermost connectives of disjuncts, and all but c disjuncts contain no parity connective at all. We prove that treelike PKO(1)O(1)(⊕) is quasipolynomially but not polynomially equivalent to constant-depth systems with parity axioms. We also verify that the technique for separating parity axioms from parity connectives due to Impagliazzo and Segerlind can be adapted to give a superpolynomial separation between daglike PKO(1)O(1)(⊕) and AC0[2]-Frege; the technique is inherently unable to prove superquasipolynomial separations. We also study proof systems related to the system Res-Lin introduced by Itsykson and Sokolov. We prove that an extension of treelike Res-Lin is polynomially simulated by a system related to daglike PKO(1)O(1)(⊕), and obtain an exponential lower bound for this system.Peer ReviewedPostprint (author's final draft

Relativized Propositional Calculus

Proof systems for the Relativized Propositional Calculus are defined and compared.Comment: 8 page

Frege systems for quantified Boolean logic

We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C. Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC0[p] circuits. In contrast, any non-trivial lower bound for propositional AC0[p]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.This research was supported by grant nos. 48138 and 60842 from the John Templeton Foundation, EPSRC grant EP/L024233/1, and a Doctoral Prize Fellowship from EPSRC (third author). The second author was funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 279611 and under the European Union’s Horizon 2020 Research and Innovation Programme/ERC grant agreement no. 648276 AUTAR. The fourth author was supported by the Austrian Science Fund (FWF) under project number P28699 and by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant Agreement no. 61507. Part of this work was done when Beyersdorff and Pich were at the University of Leeds and Bonacina at Sapienza University Rome.Peer ReviewedPostprint (published version

Mathematical Logic: Proof Theory, Constructive Mathematics

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