The basic random k-SAT problem is: given a set of n Boolean variables, and m clauses of size k picked uniformly at random from the set of all such clauses on our variables, is the conjunction of these clauses satisfiable? Here we consider a variation of this problem where there is a bias towards variables occurring positive—that is, variables occur negated w.p. and positive otherwise—and study how the satisfiability threshold depends on p. For this model breaks many of the symmetries of the original random k-SAT problem, for example, the distribution of satisfying assignments in the Boolean cube is no longer uniform. For any fixed k, we find the asymptotics of the threshold as p approaches 0 or . The former confirms earlier predictions based on numerical studies and heuristic methods from statistical physics.

中文翻译：

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有偏随机 k-SAT

基本的随机k- SAT 问题是：给定一组n 个布尔变量，从我们变量的所有此类子句集合中随机均匀挑选m个大小为k 的子句，这些子句的合取是否可满足？在这里，我们考虑这个问题的一个变体，其中对出现正值的变量有偏见——也就是说，变量出现负 wp和正反例——并研究可满足性阈值如何取决于p。因为这个模型打破了原始随机k- SAT 问题的许多对称性，例如，布尔立方体中满足分配的分布不再是均匀的。对于任何固定k，当p接近 0 或 时，我们找到阈值的渐近线。前者证实了早期基于数值研究和统计物理学启发式方法的预测。