This paper links sizes of model classes to the minimum lengths of their defining formulas, that is, to their description complexities. Limiting to models with a fixed domain of size n, we study description complexities with respect to the extension of propositional logic with the ability to count assignments. This logic, called GMLU, can alternatively be conceived as graded modal logic over Kripke models with the universal accessibility relation. While GMLU is expressively complete for defining multisets of assignments, we also investigate its fragments GMLU(d) that can count only up to the integer threshold d. We focus in particular on description complexities of equivalence classes of GMLU(d). We show that, in restriction to a poset of type realizations, the order of the equivalence classes based on size is identical to the order based on description complexities. This also demonstrates a monotone connection between Boltzmann entropies of model classes and description complexities. Furthermore, we characterize how the relation between domain size n and counting threshold d determines whether or not there exists a dominating class, which essentially means a model class with limit probability one. To obtain our results, we prove new estimates on r-associated Stirling numbers. As another crucial tool, we show that model classes split into two distinct cases in relation to their description complexity.

翻译：本文建立了模型类大小与其定义公式的最小长度（即描述复杂度）之间的联系。限于固定域大小为n的模型，我们研究了在扩展命题逻辑（具有计数赋值能力）下的描述复杂度。这种逻辑被称为GMLU，也可视为在全称可达关系克里普克模型上的分级模态逻辑。尽管GMLU在定义赋值多重集方面具有表达完备性，我们还研究了其子逻辑GMLU(d)，该逻辑仅能计数至整数阈值d。我们特别关注GMLU(d)等价类的描述复杂度。研究表明，在类型实现的偏序集限制下，基于大小的等价类序与基于描述复杂度的序相同。这同时揭示了模型类玻尔兹曼熵与描述复杂度之间的单调联系。此外，我们刻画了域大小n与计数阈值d之间的关系如何决定主导类（即极限概率为1的模型类）的存在性。为获得这些结果，我们证明了对r关联斯特林数的新估计。另一个关键工具是，我们发现模型类可根据其描述复杂度划分为两个截然不同的情形。