On The Reachability Problem for Recursive Hybrid Automata with One and Two Players

Motivated by the success of bounded model checking framework for finite state machines, Ouaknine and Worrell 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. In support of this theory, the list of undecidable problems recently shown decidable under time-bounded restriction is rather impressive: language inclusion for timed automata, emptiness problem for alternating timed automata, and emptiness problem for rectangular hybrid automata. The objective of our study was to recover decidability for general recursive timed automata—and perhaps for recursive hybrid automata—under time-bounded restriction in order to provide an appealing verification framework for powerful modeling environments such as Stateflow/Simulink. Unfortunately, however, we answer this question in negative by showing that time-bounded reachability problem stays undecidable for recursive timed automata with five or more clocks. While the bad news continues even when one considers well-behaved subclasses of recursive hybrid automata, we recover decidability by considering recursive hybrid automata with bounded context using a pass-by-reference mechanism, or by restricting the number of variables to two, with rates in {0, 1}.