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A positive extension of Eilenberg’s variety theorem for non-regular languages


In this paper we go further with the study initiated by Behle, Krebs and Reifferscheid (in: Proceedings CAI 2011, Lecture Notes in Computer Science, vol 6742, pp 97–114, 2011), who gave an Eilenberg-type theorem for non-regular languages via typed monoids. We provide a new extension of that result, inspired by the one carried out by Pin in the regular case in 1995, who considered classes of languages not necessarily closed under complement. We introduce the so-called positively typed monoids, and give a correspondence between varieties of such algebraic structures and positive varieties of possibly non-regular languages. We also prove a similar result for classes of languages with weaker closure properties.

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  1. Ballester-Bolinches, A., Pin, J.É., Soler-Escrivà, X.: Formations of finite monoids and formal languages: Eilenberg’s variety theorem revisited. Forum Math. 26(6), 1737–1761 (2014)

    MathSciNet  Article  Google Scholar 

  2. Behle, C., Krebs, A., Reifferscheid, S.: Typed monoids—an Eilenberg-like theorem for non regular languages. In: Proceedings CAI 2011. Lecture Notes in Computer Science, vol. 6742, pp. 97–114 (2011)

  3. Cadilhac, M., Krebs, A., McKenzie, P.: The algebraic theory of Parikh automata. Theory Comput. Syst. 62, 1241–1268 (2018)

    MathSciNet  Article  Google Scholar 

  4. Cano, A., Jurvanen, E.: Varieties of languages and frontier check. In: Proceedings of 13th International Conference on Automata and Formal Languages AFL2011, pp. 153–167 (2011)

  5. Cano Gómez, A., Steinby, M.: Generalized contexts and n-ary syntactic semigroups of tree languages. Asian Eur. J. Math. 4, 49–79 (2011)

    MathSciNet  Article  Google Scholar 

  6. Chaubard, L., Pin, J.É., Straubing, H.: First order formulas with modular predicates. In: Proceedings of the 21st Annual IEEE Symposium on Logic in Computer Science (LICS 2006), pp. 211–220. IEEE (2006)

  7. Cano Gómez, A.: Semigroupes ordonnés et opérations sur les langages rationnels. Ph.D. Thesis, Université Paris 7 and Departamento de Sistemas Informáticos y Computación, Universidad Politécnica de Valencia (2003)

  8. Cano Gómez, A., Pin, J.É.: Shuffle on positive varieties of languages. Theor. Comput. Sci. 312, 433–461 (2004)

    MathSciNet  Article  Google Scholar 

  9. Eilenberg, S.: Automata, Languages and Machines, vol. B. Academic Press, New York (1976)

    MATH  Google Scholar 

  10. Klaedtke, F., Rueß, H.: Monadic second-order logics with cardinalities. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) Automata, Languages and Programming. ICALP 2003. Lect. Notes Comput. Sci., vol. 2719, pp. 681–696. Springer, Heidelberg (2003)

  11. Krebs, A., Lange, K.J., Reifferscheid, S.: Characterizing \(TC^0\) in terms of infinite groups. Theory Comput. Syst. 40(4), 303–325 (2007)

    MathSciNet  Article  Google Scholar 

  12. Pin, J.É.: Syntactic semigroups. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1 (chap. 10). Springer, Berlin (1997)

  13. Pin, J.É.: Varieties of Formal Languages. Plenum Pub. Corp, New York (1986)

    Book  Google Scholar 

  14. Pin, J.É.: A variety theorem without complementation. Russ. Math. (Iz. VUZ) 39, 74–83 (1995)

    Google Scholar 

  15. Pin, J.É., Straubing, H.: Some results on C-varieties. Theor. Inform. Appl. 39, 239–262 (2005)

    MathSciNet  Article  Google Scholar 

  16. Polák, L.: A classification of rational languages by semilattice-ordered monoids. Arch. Math. (Brno) 40, 395–406 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Sakarovitch, J.: An algebraic framework for the study of the syntactic monoids application to the group languages. In: Mazurkiewic, A. (ed.) MFCS, pp. 510–516. Springer, Heidelberg (1976)

    Google Scholar 

  18. Salamanca, J.: Unveiling Eilenberg-type Correspondences: Birkhoff’s theorem for (finite) algebras + duality. arXiv:1702.02822 (2017)

  19. Steinby, M.: A theory of tree language varieties. In: Nivat, M., Podelski, A. (eds.) Tree Automata and Languages, pp. 57–81. North-Holland, Amsterdam (1992)

    MATH  Google Scholar 

  20. Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkh’auser, Boston (1994)

    Book  Google Scholar 

  21. Straubing, H.: On logical descriptions of regular languages. In: Rajsbaum, S. (ed.) LATIN 2002. Lect. Notes Comput. Sci., vol. 2286, pp. 528–538. Springer, Berlin (2002)

  22. Urbat, H., Admek, J., Chen, L., Milius, S.: Eilenberg theorems for free. In: Larsen, K.M., Bodlaender, H.L., Raskin, J.F. (eds.) MFCS 2017, vol. 83, pp. 43:1–43:15. LIPIcs, Leibnitz (2017). arXiv:1602.05831

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We thank the anonymous referees for their careful reading of the paper and for many helpful comments and suggestions which have certainly improved the final version.

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Correspondence to A. Martínez-Pastor.

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The third author is supported by Proyecto PGC2018-096872-B-100-AR, Agencia Estatal de Investigación (Spain), and by Proyecto Prometeo/2017/057, Generalitat Valenciana (Spain).

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Cano, A., Cantero, J. & Martínez-Pastor, A. A positive extension of Eilenberg’s variety theorem for non-regular languages. AAECC 32, 553–573 (2021).

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  • Monoids
  • Varieties
  • Formal languages

Mathematics Subject Classification

  • 68Q70
  • 68Q45
  • 20M07
  • 20M35