Reachability and liveness in parametric timed automata

Abstract

We study timed systems in which some timing features are unknown parameters. Parametric timed automata (PTAs) are a classical formalism for such systems but for which most interesting problems are undecidable. Notably, the parametric reachability emptiness problem, i.e., the emptiness of the parameter valuations set allowing to reach some given discrete state, is undecidable. Lower-bound/upper-bound parametric timed automata (L/U-PTAs) achieve decidability for reachability properties by enforcing a separation of parameters used as upper bounds in the automaton constraints, and those used as lower bounds. In this paper, we first study reachability. We exhibit a subclass of PTAs (namely integer-points PTAs) with bounded rational-valued parameters for which the parametric reachability emptiness problem is decidable. Using this class, we present further results improving the boundary between decidability and undecidability for PTAs and their subclasses such as L/U-PTAs. We then study liveness. We prove that: (1) deciding the existence of at least one parameter valuation for which there exists an infinite run in an L/U-PTA is PSpace-complete; (2) the existence of a parameter valuation such that the system has a deadlock is however undecidable; (3) the problem of the existence of a valuation for which a run remains in a given set of locations exhibits a very thin border between decidability and undecidability.

Configuring Timing Parameters to Ensure Execution-Time Opacity in Timed Automata

“…In fact, we show that, while the ∃-ET-opacity p-emptiness problem is decidable for L/U-PTAs (Theorem 2), the full ET-opacity p-emptiness problem becomes undecidable for this same class. This confirms (after previous works in [17,28,11,12]) that L/U-PTAs stand at the frontier between decidability and undecidability.…”

“…Lower/upper parametric timed automaton While most decision problems are undecidable for the general class of PTAs (see [5] for a survey), lower/upper parametric timed automata (L/U-PTAs) [27] is the most well-known subclass of PTAs with some decidability results: for example, reachabilityemptiness ("the emptiness of the valuations set for which a given location is reachable"), which is undecidable for PTAs [3], becomes decidable for L/U-PTAs [27]. Various other results were studied for this subclass (e.g., [17,28,12]). Definition 6 (Lower/upper parametric timed automaton [27]).…”

“…Decidability We now show that the ∃-ET-opacity p-emptiness problem is decidable for L/U-PTAs. Despite early positive results for L/U-PTAs [27,17], more recent results (notably [28,11,12]) mostly proved undecidable properties of L/U-PTAs, and therefore this positive result is welcome.…”