Short Proofs May Be Spacious: An Optimal Separation of Space and Length in Resolution Eli Ben-Sasson ∗ Computer Science Department Technion — Israel Institute of Technology Haifa, 32000, Israel eli@cs.technion.ac.il Jakob Nordstr¨ om † Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology ‡ Cambridge, MA 02139, USA jakobn@mit.edu November 23, 2008 Abstract 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 asymp- totic 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 pre- vious results (in particular, those reported in [Nordstr¨ om 2006, Nordstr¨ om and H˚ astad 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. 1 Introduction Resolution length and space Perhaps the single most studied proof system in propositional proof complexity is resolution. This system made its first appearance in 1937 in [Bla37] and began to be investi- gated in connection with automated theorem proving in the 1960s [DLL62, DP60, Rob65]. Because of the simplicity of resolution—there is only one derivation rule—and because all lines in a proof are clauses, this proof system readily lends itself to proof search algorithms. ∗ Research supported in part by an Alon Fellowship and grants by the Israeli Science Foundation and by the US-Israel Binational Science Foundation. † Research supported in part by the Ericsson Research Foundation, the Foundation Olle Engkvist Byggma”stare, and the Foun- dation Blanceflor Boncompagni-Ludovisi, ne’e Bildt. ‡ This work performed while at the Royal Institute of Technology (KTH) and while visiting the Technion. Electronic Colloquium on Computational Complexity, Report No. 2 (2009) ISSN 1433-8092