Language Emptiness of Continuous-Time Parametric Timed Automata

Abstract

Parametric timed automata extend the standard timed automata with the possibility to use parameters in the clock guards. In general, if the parameters are real-valued, the problem of language emptiness of such automata is undecidable even for various restricted subclasses. We thus focus on the case where parameters are assumed to be integer-valued, while the time still remains continuous. On the one hand, we show that the problem remains undecidable for parametric timed automata with three clocks and one parameter. On the other hand, for the case with arbitrary many clocks where only one of these clocks is compared with (an arbitrary number of) parameters, we show that the parametric language emptiness is decidable. The undecidability result tightens the bounds of a previous result which assumed six parameters, while the decidability result extends the existing approaches that deal with discrete-time semantics only. To the best of our knowledge, this is the first positive result in the case of continuous-time and unbounded integer parameters, except for the rather simple case of single-clock automata.

What’s decidable about parametric timed automata?

“…In [JLR15; ALR16b], guards and invariants are conjunctions of inequalities of the form x ⊲⊳ plt ; in addition, in [JLR15] invariants can only bound clocks from above (i. e., x plt ). In [Ben+15], guards are conjunctions of inequalities of the form x ⊲⊳ p and invariants can only bound clocks from above (i. e., x p). In [AM15], guards and invariants are conjunctions of inequalities of the form x ⊲⊳ p + d, x ⊲⊳ d + or p ⊲⊳ d (although the proofs of undecidability only need inequalities of the form x ⊲⊳ p or x ⊲⊳ d + ).…”

“…4 We only keep in Table 2 the best known results as of the current state of the art. For example, the decidability of the EF-emptiness problem over dense time with 1 parametric clock and arbitrarily many nonparametric clocks and integer-valued parameters as proved in [AHV93] with a non-elementary complexity does not appear in Table 2 as it is subsumed by [Ben+15] with an NEXPTIME complexity and a more permissive syntax (use of invariants).…”

“…The open question of the syntax expressiveness requires to consider a multi-dimensional table: we need to consider not only the number of clocks and parameters, but also the syntax allowed in guards and invariants. For example, for the undecidability over discrete time, [Ben+15] improves the number of parameters when compared to [AHV93] (6 instead of 1), but requires both strict and non-strict inequalities, whereas [AHV93] uses only equalities in their construction; it is therefore unclear whether the result of [AHV93] is really subsumed by [Ben+15]. However, following Remark 1, we considered that the works requiring invariants to contain only clocks bounded from above impose in fact no constraint on the invariants form (as the clocks bounded from below can be moved to the incoming guards).…”

“…It is undecidable with three nonparametric clocks [Mil00] over dense time with rational-valued x ⊲⊳ p|d [Ben+15], which exhibits a difference between dense and discrete time [Mil00], as well as between integer-and rational-valued parameters over dense time.…”

“…Second, in [Ben+15], an abstraction is proposed for one parametric clock and arbitrarily many nonparametric clocks and integer-valued parameters over dense time. Although this remains to be shown formally, this abstraction (based on the elimination of the nonparametric clocks followed by a corner-point abstraction on the subsequent region graph) apparently preserves enough elements of the region graph to be used to solve all aforementioned problems.…”