Accepting Zeno words: a way toward timed refinements

Abstract. Timed models were introduced to describe the behaviors of real-time systems and they were usually required to produce only executions with divergent sequences of times. However, when some physical phenomena are represented by convergent executions, Zeno words appear in a natural way. Moreover, time can progress if such an infinite execution can be followed by other ones.Therefore, in a first part, we extend the definition of timed automata, allowing to generate sequences of infinite convergent executions, while keeping good properties for the verification of systems: emptiness is still decidable.In a second part, we define a new notion of refinement for timed systems, in which actions are replaced by recognizable Zeno (timed) languages. We study the properties of these timed refinements and we prove that the class of transfinite timed languages is the closure of the usual one (languages accepted by Muller or Büchi timed automata) under refinement.

[1]  J. Richard Büchi,et al.  The monadic second order theory of ω1 , 1973 .

[2]  Rajeev Alur,et al.  A Theory of Timed Automata , 1994, Theor. Comput. Sci..

[3]  Béatrice Bérard,et al.  Untiming Timed Languages , 1995, Inf. Process. Lett..

[4]  Michael R. Hansen,et al.  Finite Divergence , 1995, Theor. Comput. Sci..

[5]  Paul Caspi,et al.  A Kleene theorem for timed automata , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[6]  Thomas A. Henzinger,et al.  Temporal proof methodologies for real-time systems , 1991, POPL '91.

[7]  R. McNaughton.  Review: J. Richard Buchi, Weak Second-Order Arithmetic and Finite Automata; J. Richard Buchi, On a Decision Method in Restricted second Order Arithmetic , 1963, Journal of Symbolic Logic.

[8]  J. R. Büchi,et al.  The monadic second order theory of all countable ordinals , 1973 .

[9]  J. R. Büchi.  On a Decision Method in Restricted Second Order Arithmetic , 1990 .

[10]  Mihalis Yannakakis,et al.  Minimum and maximum delay problems in real-time systems , 1991, Formal Methods Syst. Des..

[11]  Yaacov Choueka.  Finite Automata, Definable Sets, and Regular Expressions over omega^n-Tapes , 1978, J. Comput. Syst. Sci..

[12]  Jonathan S. Ostroff,et al.  Temporal logic for real-time systems , 1989 .

[13]  Nancy A. Lynch,et al.  Using mappings to prove timing properties , 1990, PODC '90.

[14]  Paul Gastin,et al.  On the Power of Non-Observable Actions in Timed Automata , 1996, STACS.

[15]  Thomas A. Henzinger,et al.  Symbolic Model Checking for Real-Time Systems , 1994, Inf. Comput..

[16]  Paul Gastin,et al.  Removing epsilon-Transitions in Timed Automata , 1997, STACS.

[17]  Nicolas Bedon,et al.  Finite Automata and Ordinals , 1996, Theor. Comput. Sci..