A Theory of Timed Automata

Alur, R. and D.L. Dill, A theory of timed automata, Theoretical Computer Science 126 (1994) 183-235. We propose timed (j&e) 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 clocks. A timed automaton accepts 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 Biichi 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 (II i-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. *Preliminary versions of this paper appear in the Proc. 17th Internat. Colloq. on Automata, Languages, and Programming (1990), and in the Proc. of the REX workshop “Real-Time: Theory in Practice” (1991). **Current address: AT&T Bell Laboratories, 600 Mountain Avenue, Room 2D-144, Murray Hill, NJ 07974. *** Supported by the National Science Foundation under grant MIP-8858807, and by the United States Navy, Office of the Chief of Naval Research under grant N00014-91-J-1901. This publication does not necessarily reflect the position or the policy of the US Government, and no official endorsement of this work should be inferred. 0304-3975/94/$07.00