T i 2 = S i +1 2 implies ∑ p i +1 ⊆ Δ p i +1 ⧸poly. S 2 and IΔ 0 ƒ are not finitely axiomatizable. The main tool is a Herbrand-type witnessing theorem for ∃∀∃ П b i -formulas provable in T i 2 where the witnessing functions are □ p i +1.

LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives ¬ and $\bigwedge, \bigvee$. Then for every d ≥ 0 and n ≥ 2, there is a set Td n of depth d sequents of total size O which are refutable in LK by depth d + 1 proof of size exp) but such that every depth d refutation must have the size at least exp). The sets Td n express a weaker form of the pigeonhole (...) principle. (shrink)

We consider the problem about the length of proofs of the sentences $\operatorname{Con}_S(\underline{n})$ saying that there is no proof of contradiction in S whose length is ≤ n. We show the relation of this problem to some problems about propositional proof systems.

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 S 2 2 (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 S 2 2 (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. (shrink)