We extend the convergence law for sparse random graphs proven by Lynch to arbitrary relational languages. We consider a finite relational vocabulary |$\sigma $| and a first-order theory |$T$| for |$\sigma $| composed of symmetry and anti-reflexivity axioms. We define a binomial random model of finite |$\sigma $|-structures that satisfy |$T$| and show that first-order properties have well defined asymptotic probabilities when the expected number of tuples satisfying each relation in |$\sigma $| is linear. It is also shown that these limit probabilities are well behaved with respect to several parameters that represent the density of tuples in each relation |$R$| in the vocabulary |$\sigma $|⁠. An application of these results to the problem of random Boolean satisfiability is presented. We show that in a random |$k$|-CNF formula on |$n$| variables, where each possible clause occurs with probability |$\sim c/n^{k-1}$|⁠, independently any first-order property of |$k$|-CNF formulas that implies unsatisfiability does almost surely not hold as |$n$| tends to infinity.

中文翻译：

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稀疏随机关系结构上一阶句子的概率：在随机CNF公式的可定义性上的应用

我们将Lynch证明的稀疏随机图的收敛定律扩展到任意关系语言。我们考虑一个有限的关系词汇| $ \ sigma $ | 和一阶理论| $ T $ | 为| $ \ sigma $ | 由对称和抗反射公理组成。我们定义一个有限| $ \ sigma $ |的二项式随机模型。满足| $ T $ |的结构 并证明当| $ \ sigma $ |中满足每个关系的元组的预期数目时，一阶性质具有定义明确的渐近概率 是线性的。还表明，这些极限概率相对于代表每个关系| $ R $ |中元组的密度的几个参数表现良好。在词汇| $ \ $六西格玛|⁠。提出了这些结果在随机布尔可满足性问题上的应用。我们显示随机| $ k $ | | $ n $ |上的-CNF公式 变量，其中每个可能的子句以| $ \ sim c / n ^ {k-1} $ |⁠的概率出现，独立地| $ k $ |的任何一阶属性 -CNF公式暗示不满足，几乎肯定不会成立为| $ n $ | 趋于无穷大。