No 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.

We consider the problem about the length of proofs of the sentences 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.

We investigate the proof complexity, in (extensions of) resolution and in bounded arithmetic, of the weak pigeonhole principle and of the Ramsey theorem. In particular, we link the proof complexities of these two principles. Further we give lower bounds to the width of resolution proofs and to the size of (extensions of) tree-like resolution proofs of the Ramsey theorem. We establish a connection between provability of WPHP in fragments of bounded arithmetic and cryptographic assumptions (the existence of one-way functions). In particular, we show that functions violating WPHP 2n n are one-way and, on the other hand, one-way permutations give rise to functions violating PHP n+1 n , and strongly collision-free families of hash functions give rise to functions violating WPHP 2n n (all in suitable models of bounded arithmetic). Further we formulate a few problems and conjectures; in particular, on the structured PHP (introduced here) and on the unrelativised WPHP.

The so-called weak form of Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Qj(x) = 0, does not have a solution in the algebraic closure if and only if 1 is in the ideal generated by the polynomials (?,(*)• We shall prove a lower bound on the degrees of polynomials P,(x) such that £, P,(x)Q t (x) = 1.This result has the following application. The modular counting principle states that no finite set whose cardinality is not divisible by q can be partitioned into ^-element classes. For each fixed cardinality N, this principle can be expressed as a propositional formula Count^fo,...) with underlying variables x e , where e ranges over <7-element subsets of N. Ajtai [4] proved recently that, whenever p,q are two different primes, the propositional formulas Count$ n+I do not have polynomial size, constant-depth Frege proofs from substitution instances of Count/?, where m^O (modp). We give a new proof of this theorem based on the lower bound for Hilbert's Nullstellensatz. Furthermore our technique enables us to extend the independence results for counting principles to composite numbers p and q. This improved lower bound together with new upper bounds yield an exact characterization of when Count, can be proved efficiently from Count p , for all values of p and q.