Logical, Metric, and Algorithmic Characterisations of Probabilistic Bisimulation

Many behavioural equivalences or preorders for probabilistic processes involve a lifting operation that turns a relation on states into a relation on distributions of states. We show that several existing proposals for lifting relations can be reconciled to be different presentations of essentially the same lifting operation. More interestingly, this lifting operation nicely corresponds to the Kantorovich metric, a fundamental concept used in mathematics to lift a metric on states to a metric on distributions of states, besides the fact the lifting operation is related to the maximum flow problem in optimisation theory. The lifting operation yields a neat notion of probabilistic bisimulation, for which we provide logical, metric, and algorithmic characterisations. Specifically, we extend the Hennessy-Milner logic and the modal mu-calculus with a new modality, resulting in an adequate and an expressive logic for probabilistic bisimilarity, respectively. The correspondence of the lifting operation and the Kantorovich metric leads to a natural characterisation of bisimulations as pseudometrics which are post-fixed points of a monotone function. We also present an “on the fly” algorithm to check if two states in a finitary system are related by probabilistic bisimilarity, exploiting the close relationship between the lifting operation and the maximum flow problem.

💡 Research Summary

The paper investigates probabilistic bisimulation for probabilistic labelled transition systems (pLTS) by focusing on a single, unified lifting operation that transforms a relation on states into a relation on probability distributions over states. The authors first present a concise definition of this lifting (denoted R†) as the smallest relation satisfying two natural closure properties: (i) point distributions are related exactly when their underlying states are related, and (ii) convex combinations of related distributions remain related. They then prove that this definition is equivalent to two alternative characterisations that appear in the literature: a decomposition‑based view (splitting distributions into weighted point masses) and a weight‑function view (a coupling w(s,t) that respects the original relation). The weight‑function view reveals a direct connection to the optimal transport problem: w is precisely a feasible flow in a transport network, and the existence of such a flow is equivalent to Δ R† Θ.

The authors further relate the lifting to the Kantorovich metric. Given any metric d on states, the Kantorovich lifting produces a metric on distributions by solving a minimum‑cost transport problem. They show that when d(s,t)=0 exactly for pairs in the original relation R, the induced Kantorovich distance is zero precisely on pairs of distributions related by R†. Consequently, probabilistic bisimulation can be characterised as the greatest fixed point of a monotone operator on pseudometrics: the operator maps a candidate distance d to a new distance that measures the maximal discrepancy between successor distributions, using the Kantorovich lifting of d. The bisimilarity pseudometric is the least (or greatest, depending on convention) fixed point where distance zero coincides with bisimilarity.

On the logical side, the paper extends classic Hennessy‑Milner logic (HML) and the modal μ‑calculus with a new “probabilistic choice” modality ⟨L_i∈I p_i·ϕ_i⟩. This modality expresses that a distribution can be decomposed into sub‑distributions each satisfying a formula ϕ_i with probability weight p_i. Adding this modality to HML yields a logic that is both adequate (two states satisfy exactly the same set of formulas iff they are bisimilar) and expressive (each state has a characteristic formula). Adding it to the modal μ‑calculus yields an even more powerful logic capable of describing infinite behaviours while retaining full expressiveness for probabilistic bisimilarity.

Algorithmically, the authors propose an “on‑the‑fly” decision procedure for checking whether two states in a finite pLTS are bisimilar. The algorithm maintains a candidate relation R and iteratively refines it: for each pair (s,t) in R and each action a, it checks whether every transition s ─a→Δ can be matched by some transition t ─a→Θ such that Δ and Θ are related by the lifted relation R†. The check of Δ R† Θ is reduced to solving a maximum‑flow problem on a bipartite network constructed from the support of Δ and Θ, using the weight‑function characterisation. If the flow fails, the pair (s,t) is removed from R. Because each flow computation is polynomial‑time, the whole algorithm runs in polynomial time with respect to the size of the state space and the number of transitions, and it explores only the part of the system needed to refute bisimilarity.

The paper situates its contributions within a broad body of related work: earlier definitions of probabilistic bisimulation, various logical characterisations, metric approaches based on Kantorovich distances, and existing global decision algorithms. By showing that all previously proposed lifting constructions are instances of the same operation, and by linking this operation to both optimal‑transport theory and fixed‑point metric theory, the authors provide a unified theoretical foundation. The three characterisations—logical, metric, and algorithmic—are shown to be mutually consistent and to complement each other, offering a comprehensive toolkit for reasoning about probabilistic concurrent systems. The work thus advances both the theory of probabilistic process equivalences and the practical methods for their verification.