Classes of Timed Automata and the Undecidability of Universality

AbstractUniversality for deterministic Timed Automata (TA) is PSPACE-complete but becomes highly undecidable when unrestricted nondeterminism is allowed. More precisely, universality for nondeterministic TA is Π11-hard and it is still open whether it is π11-complete. It is interesting to note that the entire arithmetical hierarchy is contained in this computability gap between determinism and nondeterminism. In this paper we consider three types of syntactical restrictions to nondeterministic TA, which may contribute to a better understanding of the universality problem for TA. For the first two types, which are of independent interest, the universality problem is shown to be Π11-complete. For the third one, universality is Π10-complete, which is the same as saying that the complementary problem is complete in the recursively enumerable class. We also show that all the restrictions define proper subclasses of the class of timed languages defined by nondeterministic TA; and establish the relationships between the classes