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.