O-Minimal Hybrid Reachability Games

In this paper, we consider reachability games over general hybrid systems, and distinguish between two possible observation frameworks for those games: either the precise dynamics of the system is seen by the players (this is the perfect observation framework), or only the starting point and the delays are known by the players (this is the partial observation framework). In the first more classical framework, we show that time-abstract bisimulation is not adequate for solving this problem, although it is sufficient in the case of timed automata . That is why we consider an other equivalence, namely the suffix equivalence based on the encoding of trajectories through words. We show that this suffix equivalence is in general a correct abstraction for games. We apply this result to o-minimal hybrid systems, and get decidability and computability results in this framework. For the second framework which assumes a partial observation of the dynamics of the system, we propose another abstraction, called the superword encoding, which is suitable to solve the games under that assumption. In that framework, we also provide decidability and computability results.

💡 Research Summary

The paper investigates reachability games played on general hybrid systems, focusing on how the amount of information available to the players influences the decidability of the winner‑determination problem. Two observation frameworks are distinguished. In the perfect observation setting, each player knows the exact continuous dynamics (the differential equations governing flows) as well as the discrete transition rules. The authors first show that the classical time‑abstract bisimulation—sufficient for timed automata—fails to be a sound abstraction for hybrid systems: two states that are bisimilar may admit different winning strategies because subtle differences in the continuous evolution can be exploited by a player. To overcome this, they introduce suffix equivalence, an equivalence relation based on encoding every execution as a finite word over a set of region‑type labels. Two states are equivalent if every possible continuation (suffix) of their word encodings yields the same set of admissible moves. This relation preserves the outcome of the game: a winning strategy from one state can be transferred to any state in the same suffix‑equivalence class.

The authors then restrict attention to o‑minimal hybrid systems, i.e., hybrid systems whose continuous dynamics are definable in an o‑minimal structure. O‑minimality guarantees a finite cell decomposition of the state space, and within each cell the dynamics behave in a tame, monotone way. This property makes the construction of suffix‑equivalence classes effective: each cell can be labelled, trajectories become words over a finite alphabet, and the suffix relation can be computed by a finite‑state procedure. Consequently, the reachability game under perfect observation becomes decidable and computable: an algorithm enumerates the finite quotient induced by suffix equivalence and solves the game on this abstract graph using standard fixed‑point techniques.

In the partial observation framework, players only know the initial point and the elapsed delays; they cannot observe the exact continuous state during the play. The suffix‑equivalence abstraction is too fine for this setting because the hidden continuous information may differ while the observable data remain the same. To address this, the paper proposes the superword encoding. For a given delay, all possible continuous evolutions compatible with the observable data are merged into a single “super‑symbol”, and a play is encoded as a word over these super‑symbols. Two configurations are equivalent if they generate the same superword. This abstraction respects the limited information available to the players and again yields a finite quotient for o‑minimal systems. The authors prove that the superword abstraction is sound: the winner of the original game coincides with the winner on the abstract superword graph. Hence, under partial observation as well, the reachability game is decidable and effectively solvable.

The paper’s contributions can be summarized as follows:

1. Negative result for time‑abstract bisimulation on general hybrid systems, highlighting the need for finer abstractions.
2. Definition of suffix equivalence and proof of its correctness for perfect‑observation games.
3. Application to o‑minimal hybrid systems, providing a constructive method to compute the finite quotient and establishing decidability.
4. Introduction of superword encoding for partial‑observation games, together with a correctness proof and an algorithmic solution.
5. Complexity discussion indicating that, although the abstract state space may be exponential in the number of cells and labels, the procedures are fully algorithmic and amenable to implementation.

Overall, the work extends the theory of hybrid games beyond the timed‑automata world, showing that with appropriate logical and geometric tools (o‑minimality) one can obtain robust decision procedures even when players have limited knowledge of the system’s continuous evolution. This opens avenues for applying formal game‑theoretic verification to realistic cyber‑physical systems where exact state observation is often impossible.