Extension of (un)decidability results to rational-valued parameters

Determine whether the decidability and undecidability status of execution-time opacity problems for one-clock parametric timed automata over dense time persists when parameters are rational-valued rather than integer-valued, and identify proof techniques that can accommodate rational-valued parameters without relying on integer-only Presburger arithmetic.

Background

The paper establishes several (un)decidability results for execution-time opacity in one-clock parametric timed automata under the assumption that parameters are integer-valued, leveraging reductions to Presburger arithmetic and its variants. While the proposed PET constructions apply to rational-valued parameters, the core (un)decidability proofs depend critically on integer arithmetic.

Consequently, the authors explicitly note that it is unclear how to carry over these results to rational-valued parameters, highlighting the need for alternative proof techniques that move beyond integer-specific reasoning. This directly motivates an open problem concerning the extension of the established results to the rational-parameter setting.