A k-CNF (conjunctive normal form) formula is a regular (k, s)-CNF one if every variable occurs s times in the formula, where k ⩾ 2 and s > 0 are integers. Regular (3, s)-CNF formulas have some good structural properties, so carrying out a probability analysis of the structure for random formulas of this type is easier than conducting such an analysis for random 3-CNF formulas. Some subclasses of the regular (3, s)-CNF formula have also characteristics of intractability that differ from random 3-CNF formulas. For this purpose, we propose strictly d-regular (k, 2s)-CNF formula, which is a regular (k, 2s)-CNF formula for which d ⩾ 0 is an even number and each literal occurs \(s - {d \over 2}\) or \(s + {d \over 2}\) times (the literals from a variable x are x and ¬x, where x is positive and ¬x is negative). In this paper, we present a new model to generate strictly d-regular random (k, 2s)-CNF formulas, and focus on the strictly d-regular random (3, 2s)-CNF formulas. Let F be a strictly d-regular random (3, 2s)-CNF formula such that 2s > d. We show that there exists a real number s₀ such that the formula F is unsatisfiable with high probability when s > s₀, and present a numerical solution for the real number s₀. The result is supported by simulated experiments, and is consistent with the existing conclusion for the case of d = 0. Furthermore, we have a conjecture: for a given d, the strictly d-regular random (3, 2s)-SAT problem has an SAT-UNSAT (satisfiable-unsatisfiable) phase transition. Our experiments support this conjecture. Finally, our experiments also show that the parameter d is correlated with the intractability of the 3-SAT problem. Therefore, our research maybe helpful for generating random hard instances of the 3-CNF formula.

中文翻译：

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严格d-规则随机（3,2 s）-SAT问题的可满足性阈值的性质

甲ķ -cnf（合取范式）式是有规律的（K，S）-cnf一个如果发生的每一个变量š倍式中，其中ķ ⩾2和小号> 0是整数。常规（3，s）-CNF公式具有某些良好的结构特性，因此，对此类随机公式进行结构的概率分析比对随机3-CNF公式进行这种分析更容易。常规（3，s）-CNF公式的某些子类还具有与随机3-CNF公式不同的难处理性特征。为此，我们提出严格的d-正规（k，2 s）-cnf式，这是一个普通的（ķ，2个小号）-cnf公式其中d ⩾0为偶数，并且每个文字发生\（S - {d \超过2} \）或\（S + {d \在2} \）时间（从一个可变的文字X是X和¬ X，其中X是正的并且¬ X为负）。在本文中，我们提出了一个新的模型来生成严格的d-规则随机（k，2 s）-CNF公式，并着眼于严格的d-规则随机（3，2 s）-CNF公式。令F为严格d-规则随机（3，2 s）-CNF公式，使得2 s > d。我们证明存在一个实数s ₀，使得当s > s ₀时公式F极不可能满足，并给出了一个实数s ₀的数值解。结果得到模拟实验的支持，并且与d = 0情况下的现有结论相符。此外，我们有一个猜想：对于给定d，严格d-规则随机数（3，2 s）-SAT问题具有SAT-UNSAT（可满足-不可满足）相变。我们的实验支持这一推测。最后，我们的实验还表明，参数d与3-SAT问题的难处理性相关。因此，我们的研究可能有助于生成3-CNF公式的随机硬实例。