Some Subsystems of Constant-Depth Frege with Parity

Abstract

We consider three relatively strong families of subsystems of AC 0 [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 PK c d (⊕). In a PK c d (⊕)-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 PK O (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 PK O (1) O (1) (⊕) and AC 0 [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 PK O(1) O(1) (⊕), and obtain an exponential lower bound for this system.