Hard examples for the bounded depth Frege proof system

Abstract

We prove exponential lower bounds on the size of a bounded depth Frege proof of a Tseitin graph-based contradiction, whenever the underlying graph is an expander. This is the first example of a contradiction, naturally formalized as a 3-CNF, that has no short bounded depth Frege proofs. Previously, lower bounds of this type were known only for the pigeonhole principle and for Tseitin contradictions based on complete graphs.

Our proof is a novel reduction of a Tseitin formula of an expander graph to the pigeonhole principle, in a manner resembling that done by Fu and Urquhart for complete graphs.

In the proof we introduce a general method for removing extension variables without significantly increasing the proof size, which may be interesting in its own right.