Dense Integer-Complete Synthesis for Bounded Parametric Timed Automata

Ensuring the correctness of critical real-time systems, involving concurrent behaviors and timing requirements, is crucial. Timed automata extend finite-state automata with clocks, compared in guards and invariants with integer constants. Parametric timed automata (PTAs) extend timed automata with timing parameters. Parameter synthesis aims at computing dense sets of valuations for the timing parameters, guaranteeing a good behavior. However, in most cases, the emptiness problem for reachability (i.e., whether the emptiness of the parameter valuations set for which some location is reachable) is undecidable for PTAs and, as a consequence, synthesis procedures do not terminate in general, even for bounded parameters. In this paper, we introduce a parametric extrapolation, that allows us to derive an underapproximation in the form of linear constraints containing not only all the integer points ensuring reachability, but also all the (non-necessarily integer) convex combinations of these integer points, for general PTAs with a bounded parameter domain. We also propose two further algorithms synthesizing parameter valuations guaranteeing unavoidability, and preservation of the untimed behavior w.r.t. a reference parameter valuation, respectively. Our algorithms terminate and can output constraints arbitrarily close to the complete result. We demonstrate their applicability and efficiency using the tool Rom\'eo on two classical benchmarks.