## A Theory of Timed Automata

Rajeev Alur and David L. Dill

We propose {\it timed (finite) automata\/} to model the behavior of real-time systems over time. Our definition provides a simple, and yet powerful, way to annotate state-transition graphs with timing constraints using finitely many real-valued {\it clocks\/}. A timed automaton accepts {\it timed words\/} --- infinite sequences in which a real-valued time of occurrence is associated with each symbol. We study timed automata from the perspective of formal language theory: we consider closure properties, decision problems, and subclasses. We consider both nondeterministic and deterministic transition structures, and both \Buchi\ and Muller acceptance conditions. We show that nondeterministic timed automata are closed under union and intersection, but not under complementation, whereas deterministic timed Muller automata are closed under all Boolean operations. The main construction of the paper is an (PSPACE) algorithm for checking the emptiness of the language of a (nondeterministic) timed automaton. We also prove that the universality problem and the language inclusion problem are solvable only for the deterministic automata: both problems are undecidable ($\Pi_1^1$-hard) in the nondeterministic case and PSPACE-complete in the deterministic case. Finally, we discuss the application of this theory to automatic verification of real-time requirements of finite-state systems.

Theoretical Computer Science 126:183-235, 1994. Preliminary versions appeared in Automata, Languages, and Programming: Proceedings of the 17th ICALP, LNCS 443, 1990, and Real Time: Theory in Practice, LNCS 600, 1991.