For a sequence of random structures with $n$-element domains over a relational signature, we define its first order (FO) complexity as a certain subset in the Banach space $\ell^{\infty}/c_0$. The well-known FO zero-one law and FO convergence law correspond to FO complexities equal to $\{0,1\}$ and a subset of $\mathbb{R}$, respectively. We present a hierarchy of FO complexity classes, introduce a stochastic FO reduction that allows to transfer complexity results between different random structures, and deduce using this tool several new logical limit laws for binomial random structures. Finally, we introduce a conditional distribution on graphs, subject to a FO sentence $\varphi$, that generalises certain well-known random graph models, show instances of this distribution for every complexity class, and prove that the set of all $\varphi$ validating 0--1 law is not recursively enumerable.

翻译：对于关系签名上具有$n$元定义域的一序列随机结构，我们将其一阶（FO）复杂度定义为巴拿赫空间$\ell^{\infty}/c_0$中的某个子集。著名的FO零一律和FO收敛律分别对应FO复杂度等于$\{0,1\}$和$\mathbb{R}$的子集的情形。我们提出了一个FO复杂度类层次结构，引入了一种随机FO约化方法，该方法允许在不同随机结构之间传递复杂度结果，并利用这一工具推导出二项随机结构的若干新逻辑极限律。最后，我们引入了一个受FO语句$\varphi$约束的图上的条件分布，该分布推广了某些著名的随机图模型，展示了每个复杂度类中该分布的实例，并证明了所有验证0-1律的$\varphi$的集合不是递归可枚举的。