## Decidability and complexity results for timed
automata via channel machines

*Parosh A. Abdulla*, *Johann Deneux*,
*Joël
Ouaknine*, and *James Worrell*
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.

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Springer-Verlag.

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