Structural complexity of ω-automata

  • Sriram C. Krishnan
  • Anuj Puri
  • Robert K. Brayton
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 900)

Abstract

In this paper we relate expressiveness of ω-automata to their complexity. Expressiveness is related to the different subclasses of the ω-regular languages that are accepted by automata that arise by restrictions on the acceptance conditions used. For example, different subclasses of the ω-regular languages arise from identifying the ω-languages with different classes and levels in the Borel hierarchy. Within the class of ω-regular languages, Wagner and Kaminski identified a strict hierarchy of languages induced by restricting the number of pairs allowed in a deterministic Rabin automaton (DRA).

Complexity relates to the smallest size automaton required to realize an ω-regular language. Safra shows that there are ω-regular languages for which deterministic Streett automata (DSA) are exponentially smaller than nondeterministic Buchi automata; in contrast, we show that for every DSA or DRA whose language is in class Gδ, there exists a DBA of size linear in the original automaton. We show in particular that the language of a DRA is in class Gδ if and only if the language can be realized as a DBA on the same transition structure as the DRA. We present a simple construction to transform a DRA with h pairs and n states to an equivalent DRA with O(n.hk) states and k pairs (i.e. DR(n,h)DR(n.hk, k)), where k is the Rabin Index (RI) of the language-the minimum number of pairs required to realize the language as a DRA. We also present a construction to translate a DSA into a minimum-pair DRA, achieving a transformation DS(n,h)DR(n.hk, r), where k is the Streett index (SI), and r the RI of the language.

We prove that it is NP-hard to determine the RI (SI) of a language specified by a DRA (DSA). However, for a DRA (DSA) with h pairs, determining whether the RI (SI) is h, or any constant c, is in polynomial time.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. Carton. Chain Automata. In IFIP 13th World Computer Congress, pages 451–458, August 1994.Google Scholar
  2. 2.
    E. S. Chang, Z. Manna, and A. Pnueli. The Safety-Progress Classification. In F. L. Bauer, W. Bauer, and H. Schwichtenberg, editors, Logic and Algebra of Specification, pages 143–202, 1993.Google Scholar
  3. 3.
    E. A. Emerson and C. L. Lei. Modalities for Model Checking: Branching Time Logic Strikes Back. Science of Computer Programming, 8(3):275–306, June 1987.Google Scholar
  4. 4.
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York, 1979.Google Scholar
  5. 5.
    M. Kaminski. A Classification of ω-regular languages. Theoretical Computer Science, 36:217–229, 1985.Google Scholar
  6. 6.
    S. C. Krishnan, A. Puri, and R. K. Brayton. Deterministic ω-automata vis-a-vis Deterministic Buchi Automata. In Algorithms and Computation, volume 834 of LNCS, pages 378–386. Springer-Verlag, 1994.Google Scholar
  7. 7.
    R. P. Kurshan. Automata-Theoretic Verification of Coordinating Processes. Princeton University Press, 1994. To appear.Google Scholar
  8. 8.
    Z. Manna and A. Pnueli. The Temporal Logic of Reactive and Concurrent Systems. Springer Verlag, 1992.Google Scholar
  9. 9.
    Kenneth L. McMillan. Personal communication, August 1994.Google Scholar
  10. 10.
    Shmuel Safra. Complexity of Automata on Infinite Objects. PhD thesis, The Weizmann Institute of Science, Rehovot, Israel, March 1989.Google Scholar
  11. 11.
    L. Staiger. Research in the theory of ω-languages. Journal of Information Processing and Cybernetics, 23:415–439, 1987.Google Scholar
  12. 12.
    W. Thomas. Automata on Infinite Objects. In J. van Leeuwen, editor, Formal Models and Semantics, volume B of Handbook of Theoretical Computer Science, pages 133–191. Elsevier Science, 1990.Google Scholar
  13. 13.
    K. Wagner. On ω-Regular Sets. Information and Control, 43:123–177, 1979.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Sriram C. Krishnan
    • 1
  • Anuj Puri
    • 1
  • Robert K. Brayton
    • 1
  1. 1.Department of EECSUniversity of CaliforniaBerkeley

Personalised recommendations