Near Optimal Separation Of Tree-Like And General Resolution

We present the best known separation between tree-like and general resolution, improving on the recent exp(n ^∈) separation of [2]. This is done by constructing a natural family of contradictions, of size n, that have O(n)-size resolution refutations, but only exp(Ω(n/log n))- size tree-like refutations. This result implies that the most commonly used automated theorem procedures, which produce tree-like resolution refutations, will perform badly on some inputs, while other simple procedures, that produce general resolution refutations, will have polynomial run-time on these very same inputs. We show, furthermore that the gap we present is nearly optimal. Specifically, if S (S _T ) is the minimal size of a (tree-like) refutation, we prove that S _T = exp(O(S log log S/log S)).