Relating description complexity to entropy
| Autor: | Jaakkola, Reijo, Kuusisto, Antti, Vilander, Miikka |
|---|---|
| Přispěvatelé: | Berenbrink, Petra, Bouyer, Patricia, Dawar, Anuj, Kante, Mamadou Moustapha, Tampere University, Computing Sciences |
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: |
FOS: Computer and information sciences
G.2.0 F.4.1 formula size 03C13 (Primary) 94A17 68Q30 (Secondary) Computer Science - Logic in Computer Science formula size game Mathematics of computing → Discrete mathematics randomness Mathematics - Logic Theory of computation → Finite Model Theory Logic in Computer Science (cs.LO) finite model theory FOS: Mathematics 111 Mathematics Logic (math.LO) entropy |
| Popis: | 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. LIPIcs, Vol. 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023), pages 38:1-38:18 |
| Databáze: | OpenAIRE |
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