Skip to main content
SpringerLink
Log in

Level Two of the Quantifier Alternation Hierarchy Over Infinite Words

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

The study of various decision problems for logic fragments has a long history in computer science. This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment. The alphabetic topology was introduced as part of an effective characterization of the fragment Σ2 over infinite words. Here, Σ2 consists of the first-order formulas with two blocks of quantifiers, starting with an existential quantifier. Its Boolean closure is \(\mathbb {B}{\Sigma }_{2}\). Our first main result is an effective characterization of the Boolean closure of the alphabetic topology, that is, given an ω-regular language L, it is decidable whether L is a Boolean combination of open sets in the alphabetic topology. This is then used for transferring Place and Zeitoun’s recent decidability result for \(\mathbb {B}{\Sigma }_{2}\) from finite to infinite words.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. During the preparation of this submission, we learned that Pierron, Place and Zeitoun [10] independently found another proof for the decidability of \(\mathbb {B}{\Sigma }_{2}\) over infinite words. For documenting the independency of the two proofs, we also include the technical report [7] of this paper in the list of references.

References

  1. Brzozowski, J.A., Knast, R.: The dot-depth hierarchy of star-free languages is infinite. J. Comput. Syst. Sci. 16(1), 37–55 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. Diekert, V., Gastin, P., Kufleitner, M.: A survey on small fragments of first-order logic over finite words. Int. J. Found Comput. Sci. 19(3), 513–548 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Diekert, V., Kufleitner, M.: Fragments of first-order logic over infinite words. Theory Comput. Syst. 48(3), 486–516 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Eilenberg, S.: Automata Languages, and Machines, volume B. Academic Press (1976)

  5. Kallas, J., Kufleitner, M., Lauser, A.: First-order fragments with successor over infinite words. In: STACS 2011, Proceedings, Volume 9 of LIPIcs, pp 356–367. Dagstuhl Publishing (2011)

  6. Kozen, D.: Lower bounds for natural proof systems. In: Proceedings of the 18th Annual Symposium on Foundations of Computer Science, FOCS’77, pp 254–266. IEEE Computer Society Press, Providence, Rhode Island (1977)

  7. Kufleitner, M., Walter, T.: Level two of the quantifier alternation hierarchy over infinite words. arXiv:1509.06207 (2015)

  8. Kufleitner, M., Walter, T.: One quantifier alternation in first-order logic with modular predicates. RAIRO-Theor. Inf. Appl. 49(1), 1–22 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Perrin, D., Pin, J.: Infinite Words, volume 141 of Pure and Applied Mathematics. Elsevier (2004)

  10. Pierron, T., Place, T., Zeitoun, M.: Quantifier alternation for infinite words. arXiv:1511.09011 (2015)

  11. Pin, J.: A variety theorem without complementation. In: Russian Mathematics (Iz. VUZ), vol. 39, pp 80–90 (1995)

  12. Pin, J.: Syntactic semigroups. In: Handbook of Formal Languages, vol. 1, pp 679–746. Springer (1997)

  13. Place, T., Zeitoun, M.: Going higher in the first-order quantifier alternation hierarchy on words. In: ICALP 2014, Proceedings, Part II, vol. 8573 of LNCS, pp 342–353. Springer (2014)

  14. Ramsey, F.P.: On a problem of formal logic. Proc. London Math. Soc. 30, 264–286 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  15. Schwarz, S., Staiger, L.: Topologies Refining the Cantor Topology on X ω. In: IFIP TCS 2010, Proceedings, Volume 323 of 1 Advances in Information and Communication Technology, pp 271–285. Springer (2010)

  16. Staiger, L., Wagner, K.W.: Automatentheoretische und automatenfreie Charakterisierungen topologischer Klassen Regulärer Folgenmengen. Elektron. Inform.-verarb Kybernetik 10, 379–392 (1974)

    MATH  Google Scholar 

  17. Straubing, H.: Finite semigroup varieties of the form \(\mathbb {V}\mathbb {D}\). J Pure Appl. Algebra 36(1), 53–94 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  18. Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser (1994)

  19. Thomas, W.: Classifying regular events in symbolic logic. J. Comput. Syst. Sci. 25, 360–376 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  20. Thomas, W.: Automata on infinite objects. In: Handbook of Theoretical Computer Science, Chapter 4, pp 133–191. Elsevier (1990)

  21. Wilke, T.: Locally Threshold Testable Languages of Infinite Words. In: STACS ’93, Proceedings, Volume 665 of LNCS, pp 607–616. Springer (1993)

Download references

Author information

Authors and Affiliations

  1. FMI, Universität Stuttgart, Stuttgart, Germany

    Manfred Kufleitner & Tobias Walter

Authors
  1. Manfred Kufleitner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tobias Walter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manfred Kufleitner.

Additional information

This work was supported by the German Research Foundation (DFG) under grants DI 435/5-2 and DI 435/6-1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kufleitner, M., Walter, T. Level Two of the Quantifier Alternation Hierarchy Over Infinite Words. Theory Comput Syst 62, 467–480 (2018). https://doi.org/10.1007/s00224-017-9801-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-017-9801-x

Keywords

Search

Navigation