TY - GEN

T1 - Towards an optimal separation of space and length in resolution

AU - Nordström, Jakob

AU - Håstad, Johan

PY - 2008

Y1 - 2008

N2 - 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, fn 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 the 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 is still 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 on the contrary these measures are completely 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 and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Θ(√n) on the space needed for so-called pebbling contradictions over pyramid graphs of size n. 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.

AB - 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, fn 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 the 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 is still 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 on the contrary these measures are completely 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 and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Θ(√n) on the space needed for so-called pebbling contradictions over pyramid graphs of size n. 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.

KW - Length

KW - Lower bound

KW - Pebbling

KW - Proof complexity

KW - Resolution

KW - Separation

KW - Space

UR - http://www.scopus.com/inward/record.url?scp=57049099276&partnerID=8YFLogxK

U2 - 10.1145/1374376.1374478

DO - 10.1145/1374376.1374478

M3 - Article in proceedings

AN - SCOPUS:57049099276

SN - 9781605580470

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 701

EP - 710

BT - STOC'08

PB - Association for Computing Machinery (ACM)

T2 - 40th Annual ACM Symposium on Theory of Computing, STOC 2008

Y2 - 17 May 2008 through 20 May 2008

ER -