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

This paper investigates reachability games played on hybrid systems, distinguishing two observation frameworks: perfect observation, where players have full knowledge of the system’s continuous dynamics, and partial observation, where players only know the initial state and elapsed delays. The authors first demonstrate that the standard time‑abstract bisimulation, which suffices for timed automata, is inadequate for general hybrid systems: they exhibit a hybrid system in which two time‑abstract bisimilar states differ in winning status. To overcome this limitation, they introduce a trajectory‑to‑word encoding called suffix encoding. Each continuous trajectory is partitioned into time intervals, each interval mapped to a symbol, yielding a word; the set of all suffixes of these words characterises a state. Two states sharing the same suffix set are said to be suffix‑equivalent, and the paper proves that suffix‑equivalence preserves winning/losing status in perfect‑observation games, providing a strictly finer abstraction than time‑abstract bisimulation.

The authors then focus on o‑minimal hybrid systems, a class of structures where every definable set can be decomposed into finitely many “cells”. Leveraging the finiteness of cells, they show that the suffix sets are effectively computable and that the resulting equivalence relation has finite index. Consequently, the control problem (i.e., determining the set of winning states and synthesising a winning controller) becomes decidable, and concrete algorithms are presented for computing winning regions and extracting strategies.

For the partial‑observation framework, the suffix encoding alone is insufficient because players cannot observe the exact continuous evolution. The paper therefore proposes a second encoding, superword encoding, which aggregates all possible suffixes that could arise from a given observable information (initial point and elapsed time). Two states with identical superwords are superword‑equivalent, and this relation is shown to preserve winning status under partial observation. Again, within o‑minimal structures the superword sets are finite and effectively computable, yielding decidability and algorithmic synthesis results analogous to the perfect‑observation case.

A substantial part of the work is devoted to comparing these approaches with prior literature on hybrid games (e.g., HHM99, dAHM01). The authors argue that earlier models treated time elapse as a discrete player action and relied on classical bisimulation, which either restricts the admissible strategies or fails to capture the necessary information for control synthesis. By explicitly modelling continuous dynamics and introducing richer word‑based abstractions, the present paper achieves a more faithful representation of hybrid control problems and provides the first general decidability results for both observation regimes in the o‑minimal setting.

The paper concludes by outlining future directions, such as extending the techniques to non‑o‑minimal structures, handling richer objectives (safety, resource constraints), and integrating the methods into practical verification tools. Overall, the work makes a significant theoretical contribution by identifying the precise abstraction needed for hybrid reachability games and by delivering concrete, decidable procedures for controller synthesis in a broad and expressive class of hybrid systems.