On Space and Depth in Resolution

Abstract

We show that the total space in resolution, as well as in any other reasonable proof system, is equal (up to a polynomial and ${(\log n)^{O(1)}}$ factors) to the minimum refutation depth. In particular, all these variants of total space are equivalent in this sense. The same conclusion holds for variable space as long as we penalize for excessively (that is, super-exponential) long proofs, which makes the question about equivalence of variable space and depth about the same as the question of (non)-existence of “supercritical” tradeoffs between the variable space and the proof length. We provide a partial negative answer to this question: for all ${s(n) \leq n^{1/2}}$ there exist CNF contradictions ${\tau_n}$ that possess refutations with variable space s(n) but such that every refutation of ${\tau_n}$ with variable space ${o(s^2)}$ must have double exponential length ${2^{2^{\Omega(s)}}}$. We also include a much weaker tradeoff result between variable space and depth in the opposite range ${s(n) \ll \log n}$ and show that no supercritical tradeoff is possible in this range.