Language Emptiness of Continuous-Time Parametric Timed Automata

TL;DR

This paper investigates the decidability of language emptiness in continuous-time parametric timed automata with integer parameters, establishing both undecidability and decidability results depending on the number of clocks and parameters involved.

Contribution

It provides the first decidability result for continuous-time automata with multiple clocks and unbounded integer parameters, and refines the bounds of previous undecidability results.

Findings

Undecidability for three clocks and one parameter.

Decidability when only one clock is compared with multiple parameters.

Extends previous results from discrete to continuous time.

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.