Skip to main content
SpringerLink
Log in

The Boolean Algebra of Piecewise Testable Languages

  • Conference paper
  • First Online:
Book cover Pursuit of the Universal (CiE 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9709))

Included in the following conference series:

  • 683 Accesses

Abstract

We characterize up to isomorphism the Boolean algebra (BA, for short) of regular piecewise testable languages and show the decidability of classes of regular languages related to this characterization. This BA turns out isomorphic to several other natural BAs of regular languages, in particular to the BA of regular aperiodic languages.

A. Konovalov and V. Selivanov supported by RFBR project 13-01-00015a.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Gehrke, M., Grigorieff, S., Pin, J.É.: Duality and equational theory of regular languages. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 246–257. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  2. Goncharov, S.S.: Countable Boolean Algebras and Decidability. Plenum, New York (1996)

    MATH  Google Scholar 

  3. Hanf, W.: The boolean algebra of logic. Bull. Amer. Math. Soc. 20(4), 456–502 (1975)

    MathSciNet  MATH  Google Scholar 

  4. Ketonen, J.: The structure of countable Boolean algebras. Ann. Math. 108, 41–89 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Konovalov, A.: Boolean algebras of regular quasi-aperiodic languages. In: Brattka, V., Diener, H., Spreen, D. (eds.) Logic, Computation, Hierarchies, pp. 191–204. Ontos Publishing de Gruiter, Boston (2014)

    Google Scholar 

  6. Selivanov, V., Konovalov, A.: Boolean algebras of regular \(\omega \)-languages. In: Dediu, A.-H., Martín-Vide, C., Truthe, B. (eds.) LATA 2013. LNCS, vol. 7810, pp. 504–515. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  7. Lempp, S., Peretyat’kin, M., Solomon, R.: The Lindenbaum algebra of the theory of the class of all finite models. J. Math. Log. 2(2), 145–225 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Marini, C., Sorbi, A., Simi, G., Sorrentino, M.: A note on algebras of languages. Theor. Comput. Sci. 412, 6531–6536 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Pin, J.-E.: Private communication

    Google Scholar 

  10. Pippenger, N.: Regular languages and stone duality. Theor. Comput. Syst. 30(2), 121–134 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Schmitz, H.: Some forbidden patterns in automata for dot-depth one languages. Ph.D. thesis, Technical report, 220, University of Würzburg, Department of Computer Science (1999)

    Google Scholar 

  12. Selivanov, V.L.: Universal Boolean algebras with applications. In: Abstracts of International Conference in Algebra, Novosibirsk, p. 127 (1991) (in Russian)

    Google Scholar 

  13. Selivanov, V.L.: Hierarchies, numerations, index sets. In: Handwritten Notes, p. 290 (1992)

    Google Scholar 

  14. Selivanov, V.L.: Positive structures. In: Cooper, S.B., Goncharov, S.S. (eds.) Computability and Models, Perspectives East and West, pp. 321–350. Kluwer Academic/Plenum Publishers, New York (2003)

    Chapter  Google Scholar 

  15. Selivanov, V.L.: Hierarchies and reducibilities on regular languages related to modulo counting. RAIRO Theor. Inform. Appl. 41, 95–132 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Selivanov, V., Konovalov, A.: Boolean algebras of regular languages. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 386–396. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  17. Stern, J.: Characterizations of some classes of regular events. Theoret. Comput. Sci. 35, 17–42 (1985)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  19. Szilard, A., Yu, S., Zhang, K., Shallit, J.: Characterizing regular languages with polynomial densities. In: Havel, I.M., Koubek, V. (eds.) MFCS 1992. LNCS, vol. 629, pp. 494–503. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  20. Thomas, W.: Languages, automata and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Language theory, vol. B, pp. 133–191. Springer, Heidelberg (1996)

    Google Scholar 

  21. Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) A Chapter of Handbook of Formal Languages. Springer, Heidelberg (1997)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A.P. Ershov Institute of Informatics Systems, Siberian Division Russian Academy of Sciences, Novosibirsk State University, Novosibirsk, Russia

    Anton Konovalov & Victor Selivanov

Authors
  1. Anton Konovalov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Victor Selivanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor Selivanov .

Editor information

Editors and Affiliations

  1. Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. Université Paris-Diderot - Paris 7, Paris, France

    Laurent Bienvenu

  3. University of South Florida, Tampa, Florida, USA

    Nataša Jonoska

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Konovalov, A., Selivanov, V. (2016). The Boolean Algebra of Piecewise Testable Languages. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds) Pursuit of the Universal. CiE 2016. Lecture Notes in Computer Science(), vol 9709. Springer, Cham. https://doi.org/10.1007/978-3-319-40189-8_30

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-40189-8_30

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-40188-1

  • Online ISBN: 978-3-319-40189-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation