Game Characterizations and the PSPACE-Completeness of Tree Resolution Space

The Prover/Delayer game is a combinatorial game that can be used to prove upper and lower bounds on the size of Tree Resolution proofs, and also perfectly characterizes the space needed to compute them. As a proof system, Tree Resolution forms the underpinnings of all DPLL-based SAT solvers, so it is of interest not only to proof complexity researchers, but also to those in the area of propositional reasoning. In this paper, we prove the PSPACE-Completeness of the Prover/Delayer game as well as the problem of predicting Tree Resolution space requirements, where space is the number of clauses that must be kept in memory simultaneously during the computation of a refutation. Since in practice memory is often a limiting resource, researchers developing SAT solvers may wish to know ahead of time how much memory will be required for solving a certain formula, but the present result shows that predicting this is at least as hard as it would be to simply find a refutation.