On the Continuity of the Probabilistic Bisimilarity Distance

2026-06-25

AI Summary

This work addresses the discontinuity of probabilistic bisimulation distance under perturbations of transition probabilities, which limits its applicability in empirical models. We introduce, for the first time, a notion of robust probabilistic bisimulation and establish that it provides a necessary and sufficient condition for distance continuity. This characterization is then generalized to arbitrary pairs of states, yielding a complete account of continuity. Building on this theoretical foundation, we design a polynomial-time algorithm to decide continuity. Both theoretical analysis and experimental evaluation demonstrate that this decision procedure incurs only negligible overhead compared to computing the distance itself, thereby substantially enhancing practicality and scalability.

Abstract

The probabilistic bisimilarity distance provides a quantitative measure of behavioural difference for labelled Markov chains, but it may be discontinuous under perturbations of the transition probabilities. This lack of continuity undermines its applicability to empirically derived models, where transition probabilities are often approximations. Recently, we (CAV 2025) introduced robust probabilistic bisimilarity as a sufficient condition for continuity at distance zero. In this paper, we show that it is also a necessary condition, that is, two states are robustly probabilistic bisimilar if and only if their probabilistic bisimilarity distance is small for any small enough perturbation of the transition probabilities. We further extend robustness to non-bisimilar state pairs to establish a complete characterization for continuity of the probabilistic bisimilarity distance. Based on this characterization, we develop a polynomial time algorithm to decide continuity. Finally, we complement our theoretical contributions with an experimental evaluation demonstrating the proposed approach in practice. Our results show that the extra step of deciding continuity requires minimal additional cost when compared to computing the probabilistic bisimilarity distance.

Problem

• probabilistic bisimilarity distance
• continuity
• labelled Markov chains
• robustness
• transition probabilities

Innovation

• probabilistic bisimilarity distance
• continuity
• robust probabilistic bisimilarity
• polynomial-time algorithm
• labelled Markov chains