Work

Relating description complexity to entropy

Year: 2024

Type: article

Abstract: We demonstrate novel links between entropy and description complexity, a notion referring to the minimal formula length for specifying given properties. Let PLC denote propositional logic with the ability to count assignments, and let PLC 1 be the fragment that counts only to one, essentially quantifying assignments. In the finite, PLC 1 is expressively complete for specifying sets of variable assignments, while PLC is expressively complete for multisets. We show that for both logics, the model classes with maximal Boltzmann entropy are the ones with maximal description complexity. Concerning PLC, we show that expected Boltzmann entropy is asymptotically equivalent to expected description complexity multiplied by the number of proposition symbols considered. For contrast, we prove this link breaks for first-order logic over vocabularies with higher-arity relations. Our results relate to links between Kolmogorov complexity and entropy, providing analogous results in the logic-based scenario with relational structures classified by formulas of different sizes.

Source: Journal of Computer and System Sciences

DOI: 10.1016/j.jcss.2024.103615

Authors: Reijo Jaakkola, Antti Kuusisto, Miikka Vilander

Institutions: Tampere University, University of Tampere, University of Helsinki, Helsinki Institute of Physics

Language: English

Topic: Computability, Logic, AI Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,520

Funder: Academy of Finland