Proof complexity of natural formulas via communication arguments

Abstract

A canonical communication problem Search (φ) is defined for every unsatisfiable CNF φ: an assignment to the variables of φ is partitioned among the communicating parties, they are to find a clause of φ falsified by this assignment. Lower bounds on the randomized k-party communication complexity of Search (φ) in the number-on-forehead (NOF) model imply tree-size lower bounds, rank lower bounds, and size-space tradeoffs for the formula φ in the semantic proof system Tcc (k, c) that operates with proof lines that can be computed by k-party randomized communication protocol using at most c bits of communication [9]. All known lower bounds on Search (φ) (e.g. [1, 9, 13]) are realized on ad-hoc formulas φ (i.e. they were introduced specifically for these lower bounds). We introduce a new communication complexity approach that allows establishing proof complexity lower bounds for natural formulas. First, we demonstrate our approach for two-party communication and apply it to the proof system Res(⊕) that operates with disjunctions of linear equalities over F2 [14]. Let a formula PMG encode that a graph G has a perfect matching. If G has an odd number of vertices, then PMG has a tree-like Res(⊕)-refutation of a polynomial-size [14]. It was unknown whether this is the case for graphs with an even number of vertices. Using our approach we resolve this question and show a lower bound 2Ω(n) on size of tree-like Res(⊕)-refutations of PMKn+2,n. Then we apply our approach for k-party communication complexity in the NOF model and obtain a [EQUATION] lower bound on the randomized k-party communication complexity of Search (BPHPM2n) w.r.t. to some natural partition of the variables, where BPHPM2n is the bit pigeonhole principle and M = 2n + 2n(1--1/k). In particular, our result implies that the bit pigeonhole requires exponential tree-like Th(k) proofs, where Th(k) is the semantic proof system operating with polynomial inequalities of degree at most k and k = O(log1--ϵ n) for some ϵ > 0. We also show that BPHP2n+12n superpolynomially separates tree-like Th(log1--ϵ m) from tree-like Th(log m), where m is the number of variables in the refuted formula.