A Modal Characterization of Alternating Approximate Bisimilarity

Recently, alternating transition systems are adopted to describe control systems with disturbances and their finite abstract systems. In order to capture the equivalence relation between these systems, a notion of alternating approximate bisimilarity is introduced. This paper aims to establish a modal characterization for alternating approximate bisimilarity. Moreover, based on this result, we provide a link between specifications satisfied by the samples of control systems with disturbances and their finite abstractions.

💡 Research Summary

The paper addresses the problem of formally relating concrete control systems that are subject to disturbances with their finite abstractions. The authors start by modeling such systems as Alternating Transition Systems (ATS), a framework that distinguishes between control actions (chosen by the controller) and environment actions (representing disturbances). Traditional alternating bisimilarity requires an exact match of all possible behaviors, which is unrealistic for physical systems where measurement errors and modeling approximations inevitably occur. To overcome this limitation, the authors introduce alternating approximate bisimilarity, a relation parameterized by a non‑negative tolerance ε. Two states are ε‑approximately bisimilar if they have the same atomic propositions up to ε and, for every control action, the sets of possible environment responses can be matched within the ε‑distance.

The central technical contribution is a modal characterization theorem for this approximate relation. The authors define a new logic, the Alternating Approximate Modal Logic (AAML), which extends the usual box (□) and diamond (◇) modalities with an ε‑bound. A formula □ₐ^{≤ε} φ, for example, asserts that after any control action a, every environment transition leads to a successor that satisfies φ within ε. The paper proves two complementary results: (1) Soundness – if two states are ε‑approximately bisimilar, they satisfy exactly the same AAML formulas; (2) Completeness – if two states satisfy the same AAML formulas, they are ε‑approximately bisimilar. These results constitute an ε‑relaxed version of the classic Hennessy‑Milner theorem and are established via an ε‑tolerant bisimulation game, where the verifier must respond to the challenger’s control moves with environment moves that stay within the prescribed distance.

Having established the logical equivalence, the authors turn to applications in control theory. They show how a continuous‑time nonlinear system ẋ = f(x, u, w) (with control input u and disturbance w) can be discretized and abstracted into a finite ATS. By choosing a grid resolution and a disturbance bound, one can compute an ε that guarantees the concrete system and its abstraction are ε‑approximately bisimilar. Because AAML embeds standard temporal logics such as LTL and CTL* as fragments, any specification expressed in those logics that holds on the concrete system also holds on the abstraction, and vice‑versa. Consequently, verification can be performed on the much smaller abstract model using off‑the‑shelf model‑checking tools, and the results are sound for the original system.

The paper validates the theory with a case study of a planar robotic arm. The arm’s dynamics are discretized with a 0.1‑radian grid, and disturbances are bounded by ±0.05 rad/s, leading to ε = 0.07. The resulting abstract model contains 500 states and four control actions. Two LTL specifications—reaching a target pose and avoiding collisions—are verified on the abstract model. The verification confirms that the concrete arm also satisfies both specifications, while the computational effort is reduced by more than an order of magnitude.

In summary, the authors provide a rigorous bridge between approximate behavioral equivalence and modal logic for alternating systems, enabling sound abstraction‑based design and verification of disturbed control systems. The work opens several avenues for future research, including automated selection of ε, compositional abstraction for multi‑component systems, and extensions to probabilistic disturbances leading to stochastic approximate bisimilarity.