Decidability and complexity results for timed automata via channel machines

This paper is concerned with the language inclusion problem for timed automata: given timed automata ${\mathcal A}$ and ${\mathcal B}$, is every word accepted by ${\mathcal B}$ also accepted by ${\mathcal A}$? Alur and Dill [3] showed that the language inclusion problem is decidable if ${\mathcal A}$ has no clocks and undecidable if ${\mathcal A}$ has two clocks (with no restriction on ${\mathcal B}$). However, the status of the problem when ${\mathcal A}$ has one clock is not determined by [3]. 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 [20], 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 ε-transitions or non-singular postconditions are allowed, then the one-clock language inclusion problem is undecidable over both finite and infinite words.