Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic

TLDR: A new proof of the interpolation theorem based on a communication complexity approach is given which allows a similar estimate for a larger class of proofs and several corollaries are derived.

Abstract: A 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.