Time-space tradeoffs in resolution: superpolynomial lower bounds for superlinear space

  • Authors:

  • Paul Beame;Christopher Beck;Russell Impagliazzo

  • Affiliations:

  • University of Washington, Seattle, WA, USA;Princeton University, Princeton, NJ, USA;University of California San Diego & Institute for Advanced Study, San Diego, CA, USA

  • Venue:
  • STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
  • Year:
  • 2012

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Abstract

We give the first time-space tradeoff lower bounds for Resolution proofs that apply to superlinear space. In particular, we show that there are formulas of size N that have Resolution refutations of space and size each roughly Nlog2 N (and like all formulas have Resolution refutations of space N) for which any Resolution refutation using space S and length T requires T ≥ (N0.58 log2 N/S)Ω(log log N/log log log N). By downward translation, a similar tradeoff applies to all smaller space bounds. We also show somewhat stronger time-space tradeoff lower bounds for Regular Resolution, which are also the first to apply to superlinear space. Namely, for any space bound S at most 2o(N1/4) there are formulas of size $N$, having clauses of width 4, that have Regular Resolution proofs of space S and slightly larger size T=O(NS), but for which any Regular Resolution proof of space S1-ε requires length TΩ(log log N/ log log log N).