A van Benthem Theorem for Fuzzy Modal Logic

TL;DR

This paper extends the van Benthem theorem to fuzzy modal logic, showing that bisimulation-invariant fuzzy first-order formulas can be approximated by fuzzy modal formulas, thus characterizing fuzzy modal logic.

Contribution

It introduces a fuzzy version of the van Benthem theorem, linking bisimulation invariance to fuzzy modal logic in a novel way.

Findings

Fuzzy first-order formulas invariant under bisimulation are approximable by fuzzy modal formulas.

The paper establishes a precise correspondence between bisimulation invariance and fuzzy modal logic.

It broadens the theoretical understanding of fuzzy modal logic's expressive power.

Abstract

We present a fuzzy (or quantitative) version of the van Benthem theorem, which characterizes propositional modal logic as the bisimulation-invariant fragment of first-order logic. Specifically, we consider a first-order fuzzy predicate logic along with its modal fragment, and show that the fuzzy first-order formulas that are non-expansive w.r.t. the natural notion of bisimulation distance are exactly those that can be approximated by fuzzy modal formulas.