We give the first time-space tradeoff lower bounds for Resolution proofs that apply to superlinear space. In particular, we show that there are formulas of size N that have Resolution refutations of space and size each roughly Nlog2N (and like all formulas have Resolution refutations of space N) for which any Resolution refutation using space S and length T requires T(N058log2NS)(loglogNlogloglogN) . By downward translation, a similar tradeoff applies to all smaller space bounds.

We also show somewhat stronger time-space tradeoff lower bounds for Regular Resolution, which are also the first to apply to superlinear space. Namely, for any space bound S at most 2o(N14) there are formulas of size N, having clauses of width 4, that have Regular Resolution proofs of space S and slightly larger size T=O(NS), but for which any Regular Resolution proof of space S1− requires length T^{\Omega(\log\log N/\log\log\log N)}$.