Universality and Language Inclusion for Open and Closed Timed Automata

The algorithmic analysis of timed automata is fundamentally limited by the undecidability of the universality problem. For this reason and others, there has been considerable interest in restricted classes of timed automata. In this paper we study the universality problem for two prominent such subclasses: open and closed timed automata. This problem is described as open in [6, 8] in the case of open timed automata. We show here that the problem is undecidable for open timed automata over strongly monotonic time (no two events are allowed to occur at the same time), and decidable over weakly monotonic time. For closed timed automata, we show that the problem is undecidable regardless of the monotonicity assumptions on time. As a corollary, we settle the various language inclusion problems over these classes of timed automata.