🤖 AI Summary
The undecidability of reachability parameter synthesis for bounded parametric timed automata (PTAs) poses a fundamental challenge. To address this, we propose a symbolic verification method based on parameter extrapolation, yielding the first dense lower approximation—via convex hulls—of the integer reachability set. Our approach integrates abstract interpretation with convex polyhedral analysis, enabling terminating synthesis for three critical properties: reachability, inevitability, and non-timed-behavior preservation. The synthesized parameter valuations converge arbitrarily closely to the exact solution. We implement and evaluate the method within the Romeo–IMITATOR toolchain, demonstrating its efficiency and high precision across multiple benchmarks. All algorithms are provably terminating and produce dense, practically usable parameter sets—i.e., sets containing infinitely many integer-valued parameter instantiations satisfying the specification.
📝 Abstract
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., 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 symbolic sets of valuations 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 sets of valuations arbitrarily close to the complete result. We demonstrate their applicability and efficiency using the tools Rom'eo and IMITATOR on several benchmarks.
