This paper is concerned with the language inclusion problem for timed automata: given timed automata A and B, is every word accepted by B also accepted by A? Alur and Dill [AD94] showed that the language inclusion problem is decidable if A has no clocks and undecidable if A has two clocks (with no restriction on B). However, the status of the problem when A has one clock is not determined by [AD94]. In this paper we close this gap for timed automata over infinite words by showing that the one-clock language inclusion problem is undecidable. For timed automata over finite words, building on our earlier paper [OW04], we show that the one-clock language inclusion problem is decidable with non-primitive recursive complexity. This reveals a surprising divergence between the theory of timed automata over finite words and over infinite words. Finally, we show that if epsilon-transitions or non-singular postconditions are allowed, then the one-clock language inclusion problem is undecidable over both finite and infinite words.
Proceedings of ICALP 05, LNCS 3580, 2005. 12 pages.