In this paper we provide a probabilistic characterization of the random Max $r$-Sat problem. We study the variance of the number of clauses satisfied by a random assignment, and the covariance of the numbers of clauses satisfied by a random pair of assignments of an arbitrary distance. Closed-form formulas for the expected value and the variance of these quantities are provided. We asymptotically and probabilistically analyze these formulas and use them to gain insights on the similarity of instances.

Based on the above probabilistic characterization, we show that the correlation between the numbers of clauses satisfied by a random pair of assignments of distance $d=cn$, $0\le c\le 1$, converges in probability to $({(1-c)}^r-1∕2^r)∕(1-1∕2^r)$. Our main result is that the so-called normalized autocorrelation length of Max $r$-Sat converges in probability to $(1-1∕2^r)∕r$. The latter quantity is of interest in the area of landscape analysis as a way to better understand problems and assess their hardness for local search heuristics. A former result regarding the same quantity only expressed it in terms of Walsh coefficients. All our results apply to random $r$-Sat as well.

在本文中，我们提供了随机最大值的概率表征 $\mathrm{[R}$-星期六的问题。我们研究了随机分配所满足的子句数的方差，以及任意距离的随机分配对所满足的子句数的协方差。提供了期望值和这些数量的方差的封闭式公式。我们渐进地和概率地分析这些公式，并使用它们来获取实例相似性的见解。

基于以上概率表征，我们表明由距离的随机分配对满足的子句数之间的相关性 $d=Cñ$， $0\le C\le \mathrm{1个}$，收敛到 $（{（\mathrm{1个}-C）}^{\mathrm{[R}}-\mathrm{1个}∕{\mathrm{2个}}^{\mathrm{[R}}）∕（\mathrm{1个}-\mathrm{1个}∕{\mathrm{2个}}^{\mathrm{[R}}）$。我们的主要结果是Max的所谓归一化自相关长度 $\mathrm{[R}$-Sat收敛到的概率 $（\mathrm{1个}-\mathrm{1个}∕{\mathrm{2个}}^{\mathrm{[R}}）∕\mathrm{[R}$。后一种量在景观分析领域中很有意义，可以更好地理解问题并评估其局部搜索启发式方法的难度。以前关于相同数量的结果仅用沃尔什系数表示。我们所有的结果都适用于随机$\mathrm{[R}$-也坐。