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 $${{\bf P} \neq {\bf NP}}$$P≠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.

中文翻译：

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随机解析反驳

我们研究了 Buss 等人定义的随机分辨率反驳系统。(J Symb Logic 79(2):496–525, 2014)。这试图捕捉可能会犯错误但在大多数情况下是正确的决议反驳的概念。通过证明几个不同定义的等价性，我们表明这个概念是稳健的。另一方面，如果 $${{\bf P} \neq {\bf NP}}$$P≠NP，则任何证明正确性可在多项式时间内检查的证明系统都不能以多项式模拟随机分辨率。我们证明了显式和随机不可满足 CNF 公式的随机分辨率反驳的宽度和大小的几个上限和下限。我们的主要结果是多对数宽度随机分辨率和拟多项式尺寸分辨率之间的分离，这解决了 Buss 等人所述的问题。(2014)。