On Reachability for Hybrid Automata over Bounded Time

This paper investigates the time-bounded version of the reachability problem for hybrid automata. This problem asks whether a given hybrid automaton can reach a given target location within T time units, where T is a constant rational value. We show that, in contrast to the classical (unbounded) reachability problem, the timed-bounded version is decidable for rectangular hybrid automata provided only non-negative rates are allowed. This class of systems is of practical interest and subsumes, among others, the class of stopwatch automata. We also show that the problem becomes undecidable if either diagonal constraints or both negative and positive rates are allowed.

💡 Research Summary

The paper studies the time‑bounded reachability problem for hybrid automata, asking whether a given hybrid automaton can reach a target location within a fixed rational time bound T. While classical (unbounded) reachability is undecidable for many hybrid models, the authors show that imposing a time bound dramatically changes the decidability landscape.

The main positive result is that for rectangular hybrid automata (RHA) with only non‑negative rates, time‑bounded reachability is decidable. The proof proceeds by establishing a uniform bound K(H,T) on the number of discrete transitions that any T‑time‑bounded run can contain. This bound depends linearly on the number of variables, the maximal absolute rate appearing in the automaton, and the time bound T. Consequently, any successful run can be compressed to a finite sequence of at most K(H,T) steps. The authors then encode the existence of such a bounded‑length run as a formula in the first‑order theory of the reals (existential linear arithmetic). Since the existential theory of the reals is decidable in EXPSPACE, the time‑bounded reachability problem for non‑negative RHA is decidable.

To simplify the reduction, the paper shows that one can, without loss of generality, assume (i) guards contain no strict inequalities, (ii) all resets are deterministic (either reset to a constant or left unchanged), and (iii) each guard is either trivially true or a conjunction of equalities of the form x₁=1∧…∧x_k=1, with corresponding resets setting the primed variables to 0. These syntactic normalisations preserve the existence of T‑bounded runs while making the logical encoding straightforward.

On the negative side, the authors prove two undecidability results. First, if both positive and negative rates are allowed, the time‑bounded reachability problem becomes undecidable. The proof reduces the halting problem of a two‑counter machine to time‑bounded reachability by encoding counter values as continuous variables and using negative rates to simulate decrements. Second, if diagonal constraints (e.g., x−y≤c) are permitted in guards, even with only non‑negative rates, the problem is undecidable. Diagonal constraints enable direct comparison of counters, allowing a faithful simulation of a two‑counter machine within a bounded time frame. Both reductions show that a T‑bounded run exists if and only if the corresponding two‑counter machine halts, establishing undecidability.

The paper also discusses the relevance of these results to practical models. The class of non‑negative RHA includes stopwatch automata, a widely used subclass for modeling timers and clocks. While unbounded reachability for stopwatch automata is known to be undecidable, the time‑bounded variant is now shown to be decidable, opening the door to automated verification tools based on SMT solvers for real arithmetic.

In summary, the work delineates a clear frontier: time‑bounded reachability is decidable for rectangular hybrid automata with only non‑negative rates, but becomes undecidable as soon as either negative rates or diagonal guards are introduced. This contributes both a theoretical understanding of hybrid system verification under time constraints and practical guidance for designing models amenable to automated analysis.