The Containment Problem for Unambiguous Register Automata and Unambiguous Timed Automata

Abstract

We investigate the complexity of the containment problem “Does $L(\mathcal {A})\subseteq L({\mathscr{B}})$ hold?” for register automata and timed automata, where ${\mathscr{B}}$ is assumed to be unambiguous and $\mathcal {A}$ is arbitrary. We prove that the problem is decidable in the case of register automata over $(\mathbb N,=)$ , in the case of register automata over $(\mathbb Q,<)$ when ${\mathscr{B}}$ has a single register, and in the case of timed automata when ${\mathscr{B}}$ has a single clock. We give a 2-EXPSPACE algorithm in the first case, whose complexity is a single exponential in the case that ${\mathscr{B}}$ has a bounded number of registers. In the other cases, we give an EXPSPACE algorithm.