Bisimulations for Nondeterministic Labeled Markov Processes

We extend the theory of labeled Markov processes with internal nondeterminism, a fundamental concept for the further development of a process theory with abstraction on nondeterministic continuous probabilistic systems. We define nondeterministic labeled Markov processes (NLMP) and provide three definition of bisimulations: a bisimulation following a traditional characterization, a state based bisimulation tailored to our “measurable” non-determinism, and an event based bisimulation. We show the relation between them, including that the largest state bisimulation is also an event bisimulation. We also introduce a variation of the Hennessy-Milner logic that characterizes event bisimulation and that is sound w.r.t. the other bisimulations for arbitrary NLMP. This logic, however, is infinitary as it contains a denumerable $\lor$. We then introduce a finitary sublogic that characterize all bisimulations for image finite NLMP whose underlying measure space is also analytic. Hence, in this setting, all notions of bisimulation we deal with turn out to be equal. Finally, we show that all notions of bisimulations are different in the general case. The counterexamples that separate them turn to be non-probabilistic NLMP.

💡 Research Summary

The paper introduces a new model, nondeterministic labeled Markov processes (NLMP), which extends the classical framework of labeled Markov processes (LMP) by allowing internal nondeterminism in addition to continuous probabilistic transitions. In an NLMP each state‑label pair is associated with a σ‑algebra of possible transition sets and a probability measure defined on that σ‑algebra, thereby making nondeterministic choices “measurable”. This construction enables a unified treatment of both probabilistic and nondeterministic behavior, a prerequisite for a process theory that supports abstraction over continuous probabilistic systems.

Three notions of bisimulation are defined for NLMPs.

1. Traditional (relational) bisimulation follows the classic pattern: two states are related if they share the same label and, for every possible transition set from one state, there exists a matching transition set from the other state with identical probability mass. This definition mirrors the one used for probabilistic labeled transition systems but is adapted to the σ‑algebra of transition sets.
2. State‑based bisimulation treats each state as a measurable object. Two states are bisimilar when, for every measurable subset of the transition σ‑algebra, the associated probability measures coincide. This notion is stronger than the traditional one because it requires equality of the entire measurable structure, not just the existence of matching sets.
3. Event‑based bisimulation focuses on observable events (sets of labels). Two states are bisimilar if, for every label and every measurable set of target states, the probability of performing that label and landing in the set is the same in both states. Hence it abstracts away internal nondeterministic choices that are not observable.

The authors prove a hierarchy among these relations. The largest state‑based bisimulation is shown to be an event bisimulation, i.e., every state‑based bisimulation automatically satisfies the event‑based conditions. Conversely, the traditional bisimulation is strictly weaker than the state‑based one, and in the general NLMP setting all three relations can be distinct.

To give a logical characterization, the paper extends Hennessy‑Milner logic (HML) with a countable disjunction (an infinitary ∨). This enriched logic is sound and complete for event bisimulation: two states satisfy exactly the same formulas of the logic iff they are event‑bisimilar. Because the logic contains an infinitary operator, it is not directly usable for algorithmic verification. The authors therefore restrict attention to image‑finite NLMPs whose underlying measurable space is analytic. Under these assumptions the transition σ‑algebra at each state is finite, and a finitary sub‑logic (without infinite disjunction) suffices to characterize all three bisimulations. Consequently, in image‑finite analytic NLMPs the three notions collapse into a single equivalence relation.

The paper also provides counter‑examples demonstrating that, without the image‑finite/analytic restriction, the four notions (traditional, state‑based, event‑based, and the logical equivalence induced by the finitary sub‑logic) are genuinely different. Interestingly, the separating examples are non‑probabilistic NLMPs where each transition has probability 0 or 1; the nondeterministic branching alone is enough to separate the definitions.

Overall contributions:

• A mathematically rigorous definition of NLMPs that integrates measurable nondeterminism with continuous probabilistic dynamics.
• Three bisimulation concepts, their mutual relationships, and proofs that the largest state‑based bisimulation is also an event bisimulation.
• An infinitary HML that exactly captures event bisimulation, together with a finitary fragment that works for image‑finite analytic NLMPs and unifies all bisimulation notions in that setting.
• Concrete counter‑examples showing the strictness of the hierarchy in the unrestricted case.

These results lay a solid foundation for future work on abstraction, model checking, and compositional reasoning for systems that combine nondeterminism with continuous probability, bridging a gap that has limited the development of a fully quantitative process theory.