# A Modal Characterization Theorem for a Probabilistic Fuzzy Description Logic

## TL;DR

This paper establishes a probabilistic analogue of the van Benthem theorem for fuzzy description logic, characterizing its expressive power through invariance under probabilistic bisimilarity and behavioral distance.

## Contribution

It proves that non-expansive probabilistic fuzzy first-order formulas can be approximated by bounded rank concepts in probabilistic fuzzy description logic, extending classical modal logic results.

## Key findings

- Probabilistic fuzzy description logic is invariant under probabilistic bisimilarity.
- Non-expansive formulas can be approximated by bounded rank concepts.
- The paper provides a probabilistic van Benthem theorem analogue.

## Abstract

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. For a modal logic perspective on the same result, see arXiv:1810.04722.