TY - GEN

T1 - Short proofs may be spacious

T2 - 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

AU - Ben-Sasson, Eli

AU - Nordström, Jakob

PY - 2008

Y1 - 2008

N2 - A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negatively in the strongest possible way. We show that there are families of 6-CNF formulas of size n, for arbitrarily large n, that have resolution proofs of length O(n) but for which any proof requires space Ω(n/log n). This is the strongest asymptotic separation possible since any proof of length O(n) can always be transformed into a proof in space O(n/log n). Our result follows by reducing the space complexity of so called pebbling formulas over a directed acyclic graph to the black-white pebbling price of the graph. The proof is somewhat simpler than previous results (in particular, those reported in [Nordström 2006, Nordström and Håstad 2008]) as it uses a slightly different flavor of pebbling formulas which allows for a rather straightforward reduction of proof space to standard black-white pebbling price.

AB - A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space. In this paper we resolve the question by answering it negatively in the strongest possible way. We show that there are families of 6-CNF formulas of size n, for arbitrarily large n, that have resolution proofs of length O(n) but for which any proof requires space Ω(n/log n). This is the strongest asymptotic separation possible since any proof of length O(n) can always be transformed into a proof in space O(n/log n). Our result follows by reducing the space complexity of so called pebbling formulas over a directed acyclic graph to the black-white pebbling price of the graph. The proof is somewhat simpler than previous results (in particular, those reported in [Nordström 2006, Nordström and Håstad 2008]) as it uses a slightly different flavor of pebbling formulas which allows for a rather straightforward reduction of proof space to standard black-white pebbling price.

UR - http://www.scopus.com/inward/record.url?scp=57949109817&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2008.42

DO - 10.1109/FOCS.2008.42

M3 - Article in proceedings

AN - SCOPUS:57949109817

SN - 9780769534367

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 709

EP - 718

BT - Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

Y2 - 25 October 2008 through 28 October 2008

ER -