Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic

Abstract

AbstractA proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (1)Feasible interpolation theorems for the following proof systems:(a)resolution(b)a subsystem of LK corresponding to the bounded arithmetic theory (α)(c)linear equational calculus(d)cutting planes.(2)New proofs of the exponential lower bounds (for new formulas)(a)for resolution ([15])(b)for the cutting planes proof system with coefficients written in unary ([4]).(3)An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds in the bounded arithmetic theory (α).In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle.