Exponential Separations between Restricted Resolution and Cutting Planes Proof Systems

We prove an exponential lower bound for tree-like Cutting Planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between tree-like and dag-like proofs for both Cutting Planes and resolution; in both cases only superpolynomial separations were known before. In order to prove this, we extend the lower bounds on the depth of monotone circuits of Raz and McKenzie (FOCS 1997) to monotone real circuits.In the case of resolution, we further improve this result by giving an exponential separation of tree-like resolution from (dag-like) regular resolution proofs. In fact, the refutation provided to give the upper bound respects the stronger restriction of being a Davis-Putnam resolution proof. This extends the corresponding superpolynomial separation of Urquhart (Bull. Symb. Logic 1, 1995).Finally, we prove an exponential separation between Davis-Putnam resolution and unrestricted resolution proofs; only a superpolynomial separation was previously known from Goerdt (Ann. Math. Artificial Intelligence 6, 1992).