Finitary Languages

  • Krishnendu Chatterjee
  • Nathanaël Fijalkow
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6638)


The class of ω-regular languages provides a robust specification language in verification. Every ω-regular condition can be decomposed into a safety part and a liveness part. The liveness part ensures that something good happens “eventually”. Finitary liveness was proposed by Alur and Henzinger as a stronger formulation of liveness [2]. It requires that there exists an unknown, fixed bound b such that something good happens within b transitions. In this work we consider automata with finitary acceptance conditions defined by finitary Büchi, parity and Streett languages. We give their topological complexity of acceptance conditions, and present a regular-expression characterization of the languages they express. We provide a classification of finitary and classical automata with respect to the expressive power, and give optimal algorithms for classical decisions questions on finitary automata. We (a) show that the finitary languages are Σ2 0-complete; (b) present a complete picture of the expressive power of various classes of automata with finitary and infinitary acceptance conditions; (c) show that the languages defined by finitary parity automata exactly characterize the star-free fragment of ωB-regular languages [4]; and (d) show that emptiness is NLOGSPACE-complete and universality as well as language inclusion are PSPACE-complete for finitary automata.


Expressive Power Regular Language Finitary Automaton Acceptance Condition Topological Complexity 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alpern, B., Schneider, F.B.: Defining Liveness. Information Processing Letters 21(4), 181–185 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alur, R., Henzinger, T.A.: Finitary Fairness. ACM Transactions on Programming Languages and Systems 20(6), 1171–1194 (1998)CrossRefGoogle Scholar
  3. 3.
    Bojanczyk, M.: Beyond omega-regular languages. In: International Symposium on Theoretical Aspects of Computer Science, STACS, pp. 11–16 (2010)Google Scholar
  4. 4.
    Bojanczyk, M., Colcombet, T.: Bounds in ω-Regularity. In: Proceedings of the 21st Annual IEEE Symposium on Logic in Computer Science, LICS 2006, pp. 285–296. IEEE Computer Society, Los Alamitos (2006)Google Scholar
  5. 5.
    Bojanczyk, M., Torunczyk, S.: Deterministic Automata and Extensions of Weak MSO. In: International Conference on the Foundations of Software Technology and Theoretical Computer Science, FSTTCS, pp. 73–84 (2009)Google Scholar
  6. 6.
    Büchi, J.R.: On a decision method in restricted second-order arithmetic. In: Proceedings of the 1st International Congress of Logic, Methodology, and Philosophy of Science, CLMPS 1960, pp. 1–11. Stanford University Press, Stanford (1962)Google Scholar
  7. 7.
    Chatterjee, K., Fijalkow, N.: Finitary languages. CoRR abs/1101.1727 (2011)Google Scholar
  8. 8.
    Chatterjee, K., Henzinger, T.A., Horn, F.: Finitary Winning in ω-regular Games. ACM Transactions on Computational Logic 11(1) (2009)Google Scholar
  9. 9.
    Choueka, Y.: Theories of automata on ω-tapes: A simplified approach. Journal of Computer and System Sciences 8, 117–141 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gurevich, Y., Harrington, L.: Trees, Automata, and Games. In: Proceedings of the 14th Annual ACM Symposium on Theory of Computing, STOC 1982, pp. 60–65. ACM Press, New York (1982)Google Scholar
  11. 11.
    Hummel, S., Skrzypczak, M., Toruńczyk, S.: On the Topological Complexity of MSO+U and Related Automata Models. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 429–440. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Manna, Z., Pnueli, A.: The Temporal Logic of Reactive and Concurrent Systems: Specification. Springer, Heidelberg (1992)CrossRefzbMATHGoogle Scholar
  13. 13.
    Pnueli, A., Rosner, R.: On the Synthesis of a Reactive Module. In: Proceedings of the 16th Annual ACM Symposium on Principles of Programming Languages, POPL 1989, pp. 179–190 (1989)Google Scholar
  14. 14.
    Ramadge, P.J., Wonham, W.M.: Supervisory control of a class of discrete-event processes. SIAM Journal on Control and Optimization 25(1), 206–230 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Safra, S.: Exponential Determinization for ω-Automata with Strong-Fairness Acceptance Condition. In: Annual ACM Symposium on Theory of Computing, STOC. ACM Press, New York (1992)Google Scholar
  16. 16.
    Thomas, W.: Languages, Automata, and Logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. Beyond Words, vol. 3, pp. 389–455. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  17. 17.
    Wadge, W.W.: Reducibility and Determinateness of Baire Spaces. PhD thesis, UC Berkeley (1984)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Krishnendu Chatterjee
    • 1
  • Nathanaël Fijalkow
    • 1
    • 2
  1. 1.IST (Institute of Science and Technology)Austria
  2. 2.ÉNS Cachan (École Normale Supérieure de Cachan)France

Personalised recommendations