Near-Optimal Separation of Treelike and General Resolution

We present the best known separation between tree-like and general resolution, improving on the recent $exp(n^{\varepsilon}$ separation of [BEGJ98]. This is done by constructing a natural family of contradictions, of size $n$, that have $O(n)$-size resolution refutations, but only $exp(\Omega(n / log n))$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 refuations, 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 $S_T = exp(O(S log log S/ log S))$.