Total Space in Resolution

ABSTRACT. We show Ω(n^2) lower bounds on the total space used in resolution refutations of random k-CNFs over n variables, and of the graph pigeonhole principle and the bit pigeonhole principle for n holes. This answers the long-standing open problem of whether there are families of k-CNF formulas of size O(n) requiring total space Ω(n^2) in resolution, and gives the first truly quadratic lower bounds on total space. The results follow from a more general theorem showing that, for formulas satisfying certain conditions, in every resolution refutation there is a memory configuration containing many clauses of large width.