Improved Undecidability Results for Reachability Games on Recursive Timed Automata

TL;DR

This paper investigates the decidability of reachability games on recursive timed automata, showing that even with time bounds, the problem remains undecidable for automata with three or more clocks, and explores a related automaton class.

Contribution

It extends the understanding of decidability boundaries in recursive timed automata and introduces results for recursive stopwatch automata.

Findings

Undecidability persists for recursive timed automata with three or more clocks under time bounds.

Decidability frontier characterized for recursive stopwatch automata.

Time-bounded restrictions do not recover decidability for certain recursive timed automata.

Abstract

We study reachability games on recursive timed automata (RTA) that generalize Alur-Dill timed automata with recursive procedure invocation mechanism similar to recursive state machines. It is known that deciding the winner in reachability games on RTA is undecidable for automata with two or more clocks, while the problem is decidable for automata with only one clock. Ouaknine and Worrell recently proposed a time-bounded theory of real-time verification by claiming that restriction to bounded-time recovers decidability for several key decision problem related to real-time verification. We revisited games on recursive timed automata with time-bounded restriction in the hope of recovering decidability. However, we found that the problem still remains undecidable for recursive timed automata with three or more clocks. Using similar proof techniques we characterize a decidability frontier for a generalization of RTA to recursive stopwatch automata.