Propositional Logic for Circuit Classes

Abstract. By introducing a parallel extension rule that is aware of inde-pendence of the introduced extension variables, a calculus for quantified propositional logic is obtained where heights of derivations correspond to heights of appropriate circuits. Adding an uninterpreted predicate on bit-strings (analog to an oracle in relativised complexity classes) this statement can be made precise in the sense that the height of the most shallow proof that a circuit can be evaluated is, up to an additive con-stant, the height of that circuit. The main tool for showing lower bounds on proof heights is a variant of an iteration principle studied by Takeuti. This reformulation might be of independent interest, as it allows for polynomial size formulae in the relativised language that require proofs of exponential height. 1 Introduction and Related Work In systems like “extended Frege ” there is a rule that allows one to introduce a new variable by a defining clause p ↔ A. If several variables are to be introduced