Towards an Optimal Separation of Space and Length in Resolution

Abstract

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, but while strong results have been proven on length our understanding of space is still quite poor. For instance, it remains open whether the fact that a formula is provable in short length implies that it is also provable in small space or whether on the contrary 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 that the true answer should be that the latter case holds. We do this 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, another well-studied measure in resolution, as well as an improvement of the weak separation in (Nordstrom 2006) of space and width from logarithmic to polynomial. Also, continuing the line of research initiated by (Ben-Sasson 2002) into trade-offs between different proof complexity measures, we present a simplified proof of the recent length-space trade-off result in (Hertel and Pitassi 2007), and show how our ideas can be used to prove a couple of other exponential trade-offs in resolution.