A monotone connection between model class size and description length

Abstract: 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.