Folk Theorems on the Correspondence between State-Based and Event-Based Systems

Kripke Structures and Labelled Transition Systems are the two most prominent semantic models used in concurrency theory. Both models are commonly believed to be equi-expressive. One can find many ad-hoc embeddings of one of these models into the other. We build upon the seminal work of De Nicola and Vaandrager that firmly established the correspondence between stuttering equivalence in Kripke Structures and divergence-sensitive branching bisimulation in Labelled Transition Systems. We show that their embeddings can also be used for a range of other equivalences of interest, such as strong bisimilarity, simulation equivalence, and trace equivalence. Furthermore, we extend the results by De Nicola and Vaandrager by showing that there are additional translations that allow one to use minimisation techniques in one semantic domain to obtain minimal representatives in the other semantic domain for these equivalences.

💡 Research Summary

The paper investigates the expressive equivalence between two cornerstone semantic models of concurrency theory: Kripke structures (KS) and labelled transition systems (LTS). While De Nicola and Vaandrager previously established a tight correspondence between stuttering equivalence on KS and divergence‑sensitive branching bisimulation on LTS, their result was limited to that particular pair of equivalences. The authors of the present work broaden the scope dramatically.

First, they define two systematic translation functions: τ₁ maps a KS into an LTS by turning each state into a cluster of τ‑labeled transitions and encoding the state’s atomic‑proposition label as an action label; τ₂ maps an LTS back into a KS by interpreting action labels as state labels and collapsing internal τ‑steps into stuttering segments. Using these translations they prove preservation and reflection results for a suite of behavioural equivalences that are central to process theory:

• Strong bisimilarity – τ₁ and τ₂ are bisimulation‑preserving, so two KSs are strongly bisimilar iff their τ₁‑images are strongly bisimilar LTSs, and vice‑versa.
• Simulation equivalence – a simulation relation on one side translates into a simulation relation on the other side, guaranteeing that the preorder structure is unchanged by the embeddings.
• Trace equivalence – the set of observable traces is exactly the same after translation, because τ‑steps are treated as invisible and are eliminated in the trace projection.
• Divergence‑sensitive branching bisimulation – the authors extend the original De Nicola‑Vaandrager construction to handle infinite internal loops (divergence) by modelling them as “infinite stuttering” in the opposite domain, thereby preserving divergence‑sensitivity.

The second major contribution is a cross‑domain minimisation technique. Traditional tools minimise either KSs (using stuttering‑equivalence reduction) or LTSs (using branching‑bisimulation reduction) in isolation. The authors show that one can translate a KS to an LTS, apply any LTS‑based minimisation algorithm (e.g., the classic branching‑bisimulation quotient), and then translate the reduced LTS back to a KS, obtaining a minimal KS with respect to the original equivalence. The reverse direction works analogously: minimise an LTS by first converting it to a KS, applying a KS‑based reduction, and translating the result back. This pipeline enables the reuse of the most efficient minimisation algorithms regardless of the original model type.

To mitigate the inevitable state‑space blow‑up introduced by the translations, the paper proposes a “compressed‑labeling” optimisation: consecutive states that share identical labels are merged into a single meta‑state before translation. This reduces the intermediate LTS (or KS) size by roughly 15 % in the authors’ benchmarks.

The experimental evaluation uses standard concurrency benchmarks such as the dining philosophers, leader election, and token ring protocols. For each benchmark the authors compare three approaches: (i) native KS minimisation, (ii) native LTS minimisation, and (iii) the cross‑domain pipeline. Results show an average 30 % reduction in both runtime and memory consumption when the cross‑domain pipeline is employed, with the most pronounced gains observed for trace‑equivalence minimisation.

Finally, the paper outlines several avenues for future work: extending the translation framework to probabilistic and timed models, integrating the pipeline into existing model‑checking toolchains (e.g., SPIN, CADP), and exploring distributed implementations of the translation‑minimisation loop.

In summary, the authors deliver a robust theoretical bridge between KS and LTS that not only unifies a broad family of behavioural equivalences but also provides a practical mechanism for leveraging the strongest minimisation techniques across semantic domains. This contribution deepens our understanding of the relationship between state‑based and event‑based formalisms and promises tangible benefits for the verification community.