Relating Description Complexity to Entropy

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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.

Original languageEnglish
Title of host publication40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023
EditorsPetra Berenbrink, Patricia Bouyer, Anuj Dawar, Mamadou Moustapha Kante
Number of pages18
ISBN (Electronic)9783959772662
Publication statusPublished - 1 Mar 2023
Publication typeA4 Article in conference proceedings
EventInternational Symposium on Theoretical Aspects of Computer Science - Hamburg, Germany
Duration: 7 Mar 20239 Mar 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


ConferenceInternational Symposium on Theoretical Aspects of Computer Science


  • entropy
  • finite model theory
  • formula size
  • formula size game
  • randomness

Publication forum classification

  • Publication forum level 1

ASJC Scopus subject areas

  • Software


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