Quantitative Hennessy-Milner Theorems via Notions of Density

TL;DR

This paper extends the coalgebraic Hennessy-Milner theorem to a broader class of functors in metric spaces, including continuous probabilistic systems, by relaxing quantale restrictions and introducing notions of density and closure.

Contribution

It provides a more general quantitative coalgebraic Hennessy-Milner theorem applicable to functors in metric spaces, including continuous probabilistic systems, with relaxed quantale conditions.

Findings

First quantitative coalgebraic Hennessy-Milner theorem for functors in metric spaces.

Application to continuous probabilistic transition systems with Borel measures.

Introduction of notions of closure and density, extending Stone-Weierstrass theorem variants.

Abstract

The classical Hennessy-Milner theorem is an important tool in the analysis of concurrent processes; it guarantees that any two non-bisimilar states in finitely branching labelled transition systems can be distinguished by a modal formula. Numerous variants of this theorem have since been established for a wide range of logics and system types, including quantitative versions where lower bounds on behavioural distance (e.g.~in weighted, metric, or probabilistic transition systems) are witnessed by quantitative modal formulas. Both the qualitative and the quantitative versions have been accommodated within the framework of coalgebraic logic, with distances taking values in quantales, subject to certain restrictions, such as being so-called value quantales. While previous quantitative coalgebraic Hennessy-Milner theorems apply only to liftings of set functors to (pseudo-)metric spaces, in the present work we provide a quantitative coalgebraic Hennessy-Milner theorem that applies more widely to functors native to metric spaces; notably, we thus cover, for the first time, the well-known Hennessy-Milner theorem for continuous probabilistic transition systems, where transitions are given by Borel measures on metric spaces, as an instance. In the process, we also relax the restrictions imposed on the quantale, and additionally parametrize the technical account over notions of closure and, hence, density, providing associated variants of the Stone-Weierstrass theorem; this allows us to cover, for instance, behavioural ultrametrics.