A Modal Characterization Theorem for a Probabilistic Fuzzy Description Logic

A Modal Characterization Theorem for a Probabilistic Fuzzy Description Logic

Paul Wild, Lutz Schröder, Dirk Pattinson, Barbara König

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence
Main track. Pages 1900-1906. https://doi.org/10.24963/ijcai.2019/263

The fuzzy modality probably is interpreted over probabilistic type spaces by taking expected truth values. The arising probabilistic fuzzy description logic is invariant under probabilistic bisimilarity; more informatively, it is non-expansive wrt. a suitable notion of behavioural distance. In the present paper, we provide a characterization of the expressive power of this logic based on this observation: We prove a probabilistic analogue of the classical van Benthem theorem, which states that modal logic is precisely the bisimulation-invariant fragment of first-order logic. Specifically, we show that every formula in probabilistic fuzzy first-order logic that is non-expansive wrt. behavioural distance can be approximated by concepts of bounded rank in probabilistic fuzzy description logic.
Keywords:
Knowledge Representation and Reasoning: Description Logics and Ontologies
Knowledge Representation and Reasoning: Reasoning about Knowlege and Belief
Knowledge Representation and Reasoning: Non-classical Logics for Knowledge Representation
Uncertainty in AI: Uncertainty in AI