Towards an Optimal Separation of Space and Length in Resolution

摘要

Most state-of-the-art satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of a largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space has been quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether these measures are unrelated in the sense that short proofs can be arbitrarily complex with respect to space. In this paper, we present some evidence indicating that the latter case should hold and provide a roadmap for how such an optimal separation result could be obtained. We do so by proving a tight bound of $Theta(sqrt{n})$ on the space needed for so-called pebbling contradictions over pyramid graphs of size $n$. This yields the first polynomial lower bound on space that is not a consequence of a corresponding lower bound on width, as well as an improvement of the weak separation of space and width of Nordstr?m (STOC 2006)from logarithmic to polynomial. A preliminary version of this paper appeared in the Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC'08).