Exponential Lower Bounds on the Size of Clause Based Semantic Derivations

We consider the clause{based version of the general model of semantic derivations proposed by Kraj cek. Resolution refutation proof is a special deterministic version of fanin-2 clause{based derivation. We prove the following combinatorial lower bound on the length of such derivations. Let F be a k-partite hypergraph, with at most b points in each part such that no point belongs to more than d edges and any two edges share at most points. If jFj k(d + 1)=2 then no CNF containing such a hypergraph among its clauses, can have a fanin-l semantic derivation of length smaller than exp k 2 b(l+). When applied to the generalized pigeonhole principle PHP m n and to blocking principles for nite geometries, this directly yields exponential lower bounds on the length of their semantic derivations, including the exp ? (n 2 =(lm)) lower bound for the length of fanin-l clause{based semantic derivation of PHP m n .