Poly-Logarithmic Frege Depth Lower Bounds Via An Expander Switching Lemma

We show that any polynomial-size Frege refutation of a certain linear-size unsatisfiable 3-CNF formula over n variables must have depth Omega(root log n). This is an exponential improvement over the previous best results (Pitassi et al. 1993, Krajicek et al. 1995, Ben-Sasson 2002) which give Omega(log log n) lower bounds.The 3-CNF formulas which we use to establish this result are Tseitin contradictions on 3-regular expander graphs. In more detail, our main result is a proof that for every d, any depth-d Frege refutation of the Tseitin contradiction over these n-node graphs must have size n(Omega((log n)/d2)). A key ingredient of our approach is a new switching lemma for a carefully designed random restriction process over these expanders. These random restrictions reduce a Tseitin instance on a 3-regular n-node expander to a Tseitin instance on a random subgraph which is a topological embedding of a 3-regular n'-node expander, for some n' which is not too much less than n. Our result involves Omega (root log n) iterative applications of this type of random restriction.