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Random resolution refutations

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Abstract

We study the random resolution refutation system defined in Buss et al. (J Symb Logic 79(2):496–525, 2014). This attempts to capture the notion of a resolution refutation that may make mistakes but is correct most of the time. By proving the equivalence of several different definitions, we show that this concept is robust. On the other hand, if \({{\bf P} \neq {\bf NP}}\), then random resolution cannot be polynomially simulated by any proof system in which correctness of proofs is checkable in polynomial time. We prove several upper and lower bounds on the width and size of random resolution refutations of explicit and random unsatisfiable CNF formulas. Our main result is a separation between polylogarithmic width random resolution and quasipolynomial size resolution, which solves the problem stated in Buss et al. (2014). We also prove exponential size lower bounds on random resolution refutations of the pigeonhole principle CNFs, and of a family of CNFs which have polynomial size refutations in constant-depth Frege.

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Acknowledgements

We would like to thank Pavel Hrubeš and Jan Krajíček for discussions and valuable suggestions. In particular, the proof of Proposition 3.6 generalizes an idea of Hrubeš. We are grateful to Michal Garlík for pointing out some errors in an earlier version of the paper.

Part of this research was supported by the European Research Council under the Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement 339691. The Institute of Mathematics of the Czech Academy of Sciences is supported by RVO:67985840.

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  1. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Pavel Pudlák & Neil Thapen

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  1. Pavel Pudlák
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  2. Neil Thapen
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Correspondence to Pavel Pudlák.

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Pudlák, P., Thapen, N. Random resolution refutations. comput. complex. 28, 185–239 (2019). https://doi.org/10.1007/s00037-019-00182-7

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