A van Benthem Theorem for Quantitative Probabilistic Modal Logic

TL;DR

This paper extends the classical van Benthem theorem to probabilistic systems, showing that probabilistic modal logic characterizes the bisimulation-invariant fragment of probabilistic first-order logic through approximation.

Contribution

It provides a probabilistic analogue of the van Benthem theorem, demonstrating the density of probabilistic modal logic in the bisimulation-invariant fragment of probabilistic first-order logic.

Findings

Probabilistic modal logic is dense in the bisimulation-invariant fragment of probabilistic first-order logic.

Bisimulation-invariant first-order formulas can be approximated by modal formulas of bounded rank.

The result generalizes classical modal logic characterizations to probabilistic systems.

Abstract

In probabilistic transition systems, behavioural metrics provide a more fine-grained and stable measure of system equivalence than crisp notions of bisimilarity. They correlate strongly to quantitative probabilistic logics, and in fact the distance induced by a probabilistic modal logic taking values in the real unit interval has been shown to coincide with behavioural distance. For probabilistic systems, probabilistic modal logic thus plays an analogous role to that of Hennessy-Milner logic on classical labelled transition systems. In the quantitative setting, invariance of modal logic under bisimilarity becomes non-expansivity of formula evaluation w.r.t. behavioural distance. In the present paper, we provide a characterization of the expressive power of probabilistic modal 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 quantitative probabilistic modal logic lies dense in the bisimulation-invariant fragment, in the indicated sense of non-expansive formula evaluation, of quantitative probabilistic first-order logic; more precisely, bisimulation-invariant first-order formulas are approximable by modal formulas of bounded rank. For a description logic perspective on the same result, see arXiv:1906.00784.