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The Boolean Algebra of Piecewise Testable Languages

  • Anton Konovalov
  • Victor SelivanovEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9709)

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.

Keywords

Boolean algebra Frechét ideal Regular language Aperiodic language Piecewise testable language 

References

  1. 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)CrossRefGoogle Scholar
  2. 2.
    Goncharov, S.S.: Countable Boolean Algebras and Decidability. Plenum, New York (1996)zbMATHGoogle Scholar
  3. 3.
    Hanf, W.: The boolean algebra of logic. Bull. Amer. Math. Soc. 20(4), 456–502 (1975)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ketonen, J.: The structure of countable Boolean algebras. Ann. Math. 108, 41–89 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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. 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)CrossRefGoogle Scholar
  7. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Marini, C., Sorbi, A., Simi, G., Sorrentino, M.: A note on algebras of languages. Theor. Comput. Sci. 412, 6531–6536 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pin, J.-E.: Private communicationGoogle Scholar
  10. 10.
    Pippenger, N.: Regular languages and stone duality. Theor. Comput. Syst. 30(2), 121–134 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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. 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. 13.
    Selivanov, V.L.: Hierarchies, numerations, index sets. In: Handwritten Notes, p. 290 (1992)Google Scholar
  14. 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)CrossRefGoogle Scholar
  15. 15.
    Selivanov, V.L.: Hierarchies and reducibilities on regular languages related to modulo counting. RAIRO Theor. Inform. Appl. 41, 95–132 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)CrossRefGoogle Scholar
  17. 17.
    Stern, J.: Characterizations of some classes of regular events. Theoret. Comput. Sci. 35, 17–42 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Straubing, H.: Finite Automata, Formal Logic and Circuit Complexity. Birkhäuser, Boston (1994)CrossRefzbMATHGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 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. 21.
    Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) A Chapter of Handbook of Formal Languages. Springer, Heidelberg (1997)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.A.P. Ershov Institute of Informatics Systems, Siberian Division Russian Academy of SciencesNovosibirsk State UniversityNovosibirskRussia

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