Understanding Space in Proof Complexity: Separations and Trade-offs via Substitutions

For current state-of-the-art satisfiability algorithms based on the DPLL procedure and clause learning, the two main bottlenecks are the amounts of time and memory used. In the field of proof complexity, these resources correspond to the length and space of resolution proofs for formulas in conjunctive normal form (CNF). There has been a long line of research investigating these proof complexity measures, but while strong results have been established for length, our understanding of space and how it relates to length has remained quite poor. In particular, the question whether resolution proofs can be optimized for length and space simultaneously, or whether there are trade-offs between these two measures, has remained essentially open apart from a few results in restricted settings. In this paper, we remedy this situation by proving a host of length-space trade-off results for resolution in a completely general setting. Our collection of trade-offs cover almost the whole range of values for the space complexity of formulas, and most of the trade-offs are superpolynomial or even exponential and essentially tight. Using similar techniques, we show that these trade-offs in fact extend (albeit with worse parameters) to the exponentially stronger k-DNF resolution proof systems, which operate with formulas in disjunctive normal form with terms of bounded arity k. We also answer the open question whether the k-DNF resolution systems form a strict hierarchy with respect to space in the affirmative. Our key technical contribution is the following, somewhat surprising, theorem: Any CNF formula F can be transformed by simple variable substitution into a new formula F ′ such that if F has the right properties, F ′ can be proven in essentially the same length as F , whereas on the other hand the minimal number of lines one needs to keep in memory simultaneously in any proof of F ′ is lower-bounded by the minimal number of variables needed simultaneously in any proof of F . Applying this theorem to so-called pebbling formulas defined in terms of pebble games on directed acyclic graphs, we obtain our results. The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 240258 and was supported by the Israeli Science Foundation and by the US-Israel Binational Science Foundation under grant number 2006104. Research supported by the Royal Swedish Academy of Sciences, the Ericsson Research Foundation, the Sweden-America Foundation, the Foundation Olle Engkvist Byggmästare, the Sven and Dagmar Salén Foundation, and the Foundation Blanceflor Boncompagni-Ludovisi, née Bildt.