Bounded-depth Frege complexity of Tseitin formulas for all graphs

Abstract

We prove that there is a constant K such that Tseitin formulas for a connected graph G requires proofs of size 2tw(G)Ω(1/d) in depth-d Frege systems for d<Klog⁡nlog⁡log⁡n, where tw(G) is the treewidth of G. This extends Håstad's recent lower bound from grid graphs to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2tw(G)O(1/d)poly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution.