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.
Highlights
Timed automata (TAs) [AD94] are a powerful formalism that extend finite-state automata with clocks that can be compared with integer constants in locations (“invariants”) and along transitions (“guards”); some clocks can be reset to 0 along transitions
This can be the case when the equivalence relation refers to the equality of the sets of parameter valuations such that some reachability property is satisfied, because boundedness imposes constraints on parameters that cannot be expressed in an L/U-Parametric timed automata (PTA) (e. g., enforcing upper bounds on upper-bound parameters): a consequence is that undecidability results for bounded L/U-PTAs cannot be automatically extended to L/U-PTAs; decidability results for L/U-PTAs cannot be automatically extended to bounded L/U-PTAs
Despite the vast number of undecidability results linked to the formalism of parametric timed automata, and to which we contribute in this paper, we exhibited a new subclass of PTAs for which the EF-emptiness problem is decidable
Summary
Timed automata (TAs) [AD94] are a powerful formalism that extend finite-state automata with clocks (real-valued variables evolving linearly) that can be compared with integer constants in locations (“invariants”) and along transitions (“guards”); some clocks can be reset to 0 along transitions. Many interesting problems for TAs (including the reachability of a location [AD94]) are decidable. The classical definition of TAs is not tailored to verify systems only partially specified, especially when the value of some timing constants is not yet known. Key words and phrases: Parametric timed automata, L/U-PTA, reachability, liveness, deadlock-freeness. ∗ This manuscript is an extended version of [ALR16a, AL17] Key words and phrases: Parametric timed automata, L/U-PTA, reachability, liveness, deadlock-freeness. ∗ This manuscript is an extended version of [ALR16a, AL17]
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