Skip to main content
SpringerLink
Log in

Constructing the Maximum Prefix-Closed Subset for a Set of –ω-Words Defined by a –ω-Regular Expression

  • Published:
Cybernetics and Systems Analysis Aims and scope

Abstract

In this paper, we present a method of constructing the maximum prefix-closed subset of a set of –ω-words R defined by a –ω-regular expression. This method is based on constructing a labeled graph called a graph of elementary extensions whose vertices are some –ω-regular subsets of the set R. With every graph vertex, we associate a linear equation over sets of –ω-words. Thus, the graph of elementary extensions determines a system of linear equations. As a result of solving this system of equations, for each vertex of the graph, we obtain the maximum prefix-closed with respect to R subset of the set of –ω-words corresponding to this vertex. The union of such subsets that correspond to all initial vertices of the graph, that is, the vertices constructing the graph starts with, is a maximum prefix-closed subset of the given set R . A method of constructing such a graph is proposed, which we called a method of incomplete intersections. We also present a method to solve the system of equations determined by the graph of elementary extensions.

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

Similar content being viewed by others

References

  1. A. N. Chebotarev, “Synthesis of ω-automata specified in the first order logical languages LP and LF,” Cybern. Syst. Analysis, Vol. 54, No. 4, 527–540 (2018). https://doi.org/10.1007/s10559-018-0054-8.

    Article  MATH  Google Scholar 

  2. P. Thiemann and M. Sulzmann, “From ω-regular expressions to Büchi automata via partial derivatives,” in: A. H. Dediu, E. Formenti, C. Martín-Vide, and B. Truthe (eds.), Language and Automata Theory and Applications, LATA 2015, Lecture Notes in Computer Science, Vol. 8977, Springer, Cham (2015), pp. 287–298. https://doi.org/10.1007/978-3-319-15579-1_22.

  3. C. Löding, A. Tollkötter, “Transformations between regular expressions and ω-automata,” in: P. Faliszewski, A. Muscholl and R. Niedermeier (eds.), Leibniz International Proceedings in Informatics (LIPIcs), Proc. 41th Intern. Symp. on Mathematical Foundation of Computer Science (MFCS 2016) (Krakow, Poland, Aug 22–26, 2016), Vol. 58, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2016), pp. 88:1–88:13. https://doi.org/10.4230/LIPIcs.MFCS.2016.88.

  4. A. N. Chebotarev, “Constructing a – ω-regular expression specified by a graph of elementary extensions,” Cybern. Syst. Analysis, Vol. 58, No. 2, 165–170 (2022). https://doi.org/10.1007/s10559-022-00447-0.

    Article  MATH  Google Scholar 

  5. W. Thomas, “Star-free regular sets of ω-sequences,” Inf. Control., Vol. 42, No. 2, 148–156 (1979). https://doi.org/10.1016/S0019-9958(79)90629-6.

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Diekert and P. Gastin, “First-order definable languages,” in: J. Flum, E. Grädel, and T. Wilke (eds.), Logic and Automata: History and Perspectives, Amsterdam University Press (2007), pp. 261–306.

  7. D. Perrin and J.-E. Pin, Infinite Words: Automata, Semigroups, Logic and Games, Pure and Applied Mathematics Series, Vol. 141, Elsevier, Amsterdam (2004).

    MATH  Google Scholar 

  8. A. N. Chebotarev, “Intersection of ω-regular expressions,” Cybern. Syst. Analysis, Vol. 57, No. 5, 676–684 (2021). https://doi.org/10.1007/s10559-021-00393-3.

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Calbrix, M. Nivat, and A. Podelski, “Ultimately periodic words of rational ω-languages,” in: M. G. Main, A. C. Melton, M. W. Mislove, D. Schmidt, and S. D. Brookes (eds.), MFPS 1993, Lecture Notes in Computer Science, Vol. 802, Springer, Heidelberg (1994), pp. 554–566.

    Google Scholar 

  10. D. Angluin and D. Fisman, “Learning regular omega languages,” Theor. Comput. Sci., Vol. 650, 57–72 (2016). https://doi.org/10.1016/j.tcs.2016.07.031.

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Staiger, “On ω-power languages,” in: G. Păun and A. Salomaa (eds.), New Trends in Formal Languages, Lecture Notes in Computer Science, Vol. 1218, Springer, Berlin–Heidelberg, (1997), pp. 377–394. https://doi.org/10.1007/3-540-62844-4_27.

Download references

Author information

Authors and Affiliations

  1. V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    A. N. Chebotarev

Authors
  1. A. N. Chebotarev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. N. Chebotarev.

Additional information

Translated from Kibernetyka ta Systemnyi Analiz, No. 6, November–December, 2023, pp. 19–29.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chebotarev, A.N. Constructing the Maximum Prefix-Closed Subset for a Set of –ω-Words Defined by a –ω-Regular Expression. Cybern Syst Anal 59, 880–889 (2023). https://doi.org/10.1007/s10559-023-00623-w

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10559-023-00623-w

Keywords

Search

Navigation