Relating description complexity to entropy

TL;DR

This paper explores the relationship between entropy and description complexity in logical frameworks, demonstrating that in certain logics, models with maximal entropy also have maximal description complexity, linking logical expressiveness with information measures.

Contribution

It establishes novel connections between entropy and description complexity within extended propositional logics and shows how these links break in higher-arity first-order logic.

Findings

Maximal entropy models have maximal description complexity in MLU.

Expected entropy is asymptotically proportional to description complexity in GMLU.

The link between entropy and complexity does not hold in higher-arity first-order logic.

Abstract

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.