  We demonstrate some novel links between entropy and description complexity, a
notion referring to the minimal formula length for specifying given properties.
Let MLU be the logic obtained by extending propositional logic with the
universal modality, and let GMLU be the corresponding extension with the
ability to count. In the finite, MLU is expressively complete for specifying
sets of variable assignments, while GMLU is expressively complete for
multisets. We show that for MLU, the model classes with maximal Boltzmann
entropy are the ones with maximal description complexity. Concerning GMLU, we
show that expected Boltzmann entropy is asymptotically equivalent to expected
description complexity multiplied by the number of proposition symbols
considered. To contrast these results, we show that this link breaks when we
move to considering first-order logic FO over vocabularies with higher-arity
relations. To establish the aforementioned result, we show that almost all
finite models require relatively large FO-formulas to define them. Our results
relate to links between Kolmogorov complexity and entropy, demonstrating a way
to conceive such results in the logic-based scenario where relational
structures are classified by formulas of different sizes.
