Relating Description Complexity to Entropy

Authors Reijo Jaakkola , Antti Kuusisto , Miikka Vilander

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Author Details

Reijo Jaakkola
  • Tampere University, Finland
Antti Kuusisto
  • Tampere University, Finland
  • University of Helsinki, Finland
Miikka Vilander
  • Tampere University, Finland

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Reijo Jaakkola, Antti Kuusisto, and Miikka Vilander. Relating Description Complexity to Entropy. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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 prove 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Finite Model Theory
  • finite model theory
  • entropy
  • formula size
  • randomness
  • formula size game


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