# Minimizable timed automata

• Jan Springintveld
• Frits Vaandrager
Selected Presentations State Charts
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1135)

## Abstract

State minimization plays a fundamental role in both classical automata theory and in the theory of reactive systems. Many algorithms and results are based on the fact that for each finite automaton there exists an equivalent minimum state automaton that can be effectively computed and that is unique up to isomorphism.

Timed safety automata (TSA's) [5], finite automata with clocks, have been used extensively for the specification and verification of real-time systems. However, there does not always exist a unique minimum state TSA that is equivalent to a given TSA. This problem occurs irrespective of the selected notions of state (including or excluding clock values) and equivalence on states (language equivalence, bisimulation equivalence, etc.).

Henzinger, Kopke and Wong-Toi [4] convincingly showed that if states do not include clock values, state minimization for timed automata is neither useful nor interesting. In this paper, we discuss state minimization for states that do include clock values, i.e., at the semantic level, and work in bisimulation equivalence. In this setting, a timed automaton is minimal when there does not exist a pair of bisimilar but distinct states in the transition system induced by the timed automaton.

We present a new model of minimizable timed automata (MTA's), a variant of the TSA model, and prove that
1. 1.

The MTA and TSA model are equally expressive in the sense that for each MTA there exists a bisimilar TSA and for each TSA there exists a bisimilar MTA.

2. 2.

For each MTA there exists a bisimilar minimal MTA that can be effectively computed and that is unique up to isomorphism.

## Keywords

Transition System Operational Semantic Finite Automaton Propositional Variable Relevance Formula
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
R. Alur, C. Courcoubetis, N. Halbwachs, D. Dill, and H. Wong-Toi. Minimization of timed transition systems. In W.R. Cleaveland, editor, Proceedings CONCUR 92, Stony Brook, NY, USA, volume 630 of Lecture Notes in Computer Science, pages 340–354. Springer-Verlag, 1992.Google Scholar
2. 2.
R. Alur, C. Courcoubetis, and T.A. Henzinger. The observational power of clocks. In B. Jonsson and J. Parrow, editors, Proceedings CONCUR 94, Uppsala, Sweden, volume 836 of Lecture Notes in Computer Science, pages 162–177. Springer-Verlag, 1994.Google Scholar
3. 3.
R. Alur and D.L. Dill. A theory of timed automata. Theoretical Computer Science, 126:183–235, 1994.Google Scholar
4. 4.
T.A. Henzinger, P.W. Kopke, and H. Wong-Toi. The expressive power of clocks. In Proceedings 22nd ICALP, volume 944 of Lecture Notes in Computer Science, pages 417–428. Springer-Verlag, 1995.Google Scholar
5. 5.
T.A. Henzinger, X. Nicollin, J. Sifakis, and S. Yovine. Symbolic model checking for real-time systems. Information and Computation, 111:193–244, 1994.Google Scholar
6. 6.
Z. Kohavi. Switching and Finite Automata Theory. McGraw-Hill, Inc., 1970.Google Scholar
7. 7.
K.G. Larsen, F. Laroussinie, and C. Weise. From timed automata to logic — and back. In Proceedings 20th International Symposium on Mathematical Foundations of Computer Science (MFCS'95), Prague, Czech Republic, volume 969 of Lecture Notes in Computer Science. Springer-Verlag, 1995.Google Scholar
8. 8.
R. Milner. Communication and Concurrency. Prentice-Hall International, Englewood Cliffs, 1989.Google Scholar
9. 9.
K. Cerāns. Algorithmic Problems in Analysis of Real Time System Specifications. Dr.sc.comp. thesis, University of Latvia, Rīga, 1992.Google Scholar
10. 10.
K. Cerāns. Decidability of bisimulation equivalences for parallel timer processes. In G. v. Bochmann and D.K. Probst, editors, Proceedings of the 4th International Workshop on Computer Aided Verification, Montreal, Canada, volume 663 of Lecture Notes in Computer Science, pages 302–315. Springer-Verlag, 1992.Google Scholar
11. 11.
M. Yannakalds and D. Lee. An efficient algorithm for minimizing real-time transition systems. In C. Courcoubetis, editor, Proceedings of the 5th International Conference on Computer Aided Verification, Elounda, Greece, volume 697 of Lecture Notes in Computer Science, pages 210–224. Springer-Verlag, 1993.Google Scholar