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Bridges for concatenation hierarchies

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Automata, Languages and Programming (ICALP 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1443))

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Abstract

In the seventies, several classification schemes for the rational languages were proposed, based on the alternate use of certain operators (union, complementation, product and star). Some thirty years later, al-though much progress has been done, several of the original problems are still open. Furthermore, their significance has grown considerably over the years, on account of the successive discoveries of surprising links with other fields, like non commutative algebra, finite model theory, structural complexity and topology. In this article, we solve positively a question raised in 1985 about concatenation hierarchies of rational languages, which are constructed by alternating boolean operations and concate-nation products. We establish a simple algebraic connection between the Straubing-Thérien hierarchy, whose basis is the trivial variety, and the group hierarchy, whose basis is the variety of group languages. Thanks to a recent result of Almeida and Steinberg, this reduces the decidability problem for the group hierarchy to a property stronger than decidability for the Straubing-Thérien hierarchy.

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References

  1. J. Almeida and B. Steinberg, On the decidability of iterated semidirect products with applications to complexity, preprint.

    Google Scholar 

  2. M. Arfi, Polynomial operations and rational languages, 4th STACS, Lect. Notes in Comp. Sci. 247, Springer, (1987), 198–206.

    Google Scholar 

  3. M. Arfi, Opérations polynomiales et hiérarchies de concaténation, Theoret. Comput. Sci. 91, (1991), 71–84.

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Borchert, D. Kuske, F. Stephan, On existentially first-order definable languages and their relation to NP. Proceedings of ICALP 1998, Lect. Notes in Comp. Sci., Springer Verlag, Berlin, Heidelberg, New York, (1998), this volume.

    Google Scholar 

  5. J. A. Brzozowski, Hierarchies of aperiodic languages, RAIRO Inform. Théor. 10, (1976), 33–49.

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Eilenberg, Automata, languages and machines, Vol. B, Academic Press, New York, 1976.

    MATH  Google Scholar 

  8. K. Henckell and J. Rhodes, The theorem of Knast, the PG = BG and Type II Conjectures, in J. Rhodes (ed.) Monoids and Semigroups with Applications, Word Scientific, (1991), 453–463.

    Google Scholar 

  9. R. Knast, A semigroup characterization of dot-depth one languages, RAIRO Inform. Théor. 17, (1983), 321–330.

    MATH  MathSciNet  Google Scholar 

  10. R. Knast, Some theorems on graph congruences, RAIRO Inform. Théor. 17, (1983), 331–342.

    MathSciNet  Google Scholar 

  11. S. W. Margolis and J.E. Pin, Product of group languages, FCT Conference, Lect. Notes in Comp. Sci. 199, (1985), 285–299.

    Article  MATH  MathSciNet  Google Scholar 

  12. J.-E. Pin, Hiérarchies de concaténation, RAIRO Informatique Théorique 18, (1984), 23–46.

    MATH  MathSciNet  Google Scholar 

  13. J.-E. Pin, Variétés de langages formels, Masson, Paris, 1984; English translation: Varieties of formal languages, Plenum, New-York, 1986.

    MATH  Google Scholar 

  14. J.-E. Pin, Logic on words, Bulletin of the European Association of Theoretical Computer Science 54, (1994), 145–165.

    MATH  Google Scholar 

  15. J.-E. Pin, Finite semigroups and recognizable languages: an introduction, in NATO Advanced Study Institute Semigroups, Formal Languages and Groups, J. Fountain (ed.), Kluwer academic publishers, (1995), 1–32.

    Google Scholar 

  16. J.-E. Pin, BG = PG, a success story, in NATO Advanced Study Institute Semi-groups, Formal Languages and Groups, J. Fountain (ed.), Kluwer academic publishers, (1995), 33–47.

    Google Scholar 

  17. J.-E. Pin, A variety theorem without complementation, Izvestiya VUZ Matematika 39 (1995) 80–90. English version, Russian Mathem. (Iz. VUZ) 39 (1995) 74–83.

    MATH  MathSciNet  Google Scholar 

  18. J.-E. Pin, Logic, semigroups and automata on words, Annals of Mathematics and Artificial Intelligence, (1996), 16, 343–384.

    Article  MATH  MathSciNet  Google Scholar 

  19. J.-E. Pin, Polynomial closure of group languages and open sets of the Hall topology, Theoretical Computer Science 169, (1996), 185–200.

    Article  MATH  MathSciNet  Google Scholar 

  20. J.-E. Pin, Syntactic semigroups, in Handbook of language theory, G. Rozenberg et A. Salomaa eds., vol. 1, ch. 10, pp. 679–746, Springer (1997).

    Google Scholar 

  21. J.-E. Pin and H. Straubing, 1981, Monoids of upper triangular matrices, Colloquia Mathematica Societatis Janos Bolyai 39, Semigroups, Szeged, 259–272.

    Google Scholar 

  22. J.-E. Pin and P. Weil, Polynomial closure and unambiguous product, Theory Comput. Systems 30, (1997), 1–39.

    Article  MathSciNet  Google Scholar 

  23. Ch. Reutenauer, Sur les variétés de langages et de monoÏdes, Lect. Notes in Comp. Sci. 67, (1979) 260–265.

    MATH  MathSciNet  Google Scholar 

  24. M.P. Schützenberger, On finite monoids having only trivial subgroups, Information and Control 8, (1965), 190–194.

    Article  MATH  MathSciNet  Google Scholar 

  25. I. Simon, Piecewise testable events, Proc. 2nd GI Conf., Lect. Notes in Comp. Sci. 33, Springer Verlag, Berlin, Heidelberg, New York, (1975), 214–222.

    Google Scholar 

  26. I. Simon, The product of rational languages, Proceedings of ICALP 1993, Lect. Notes in Comp. Sci. 700, Springer Verlag, Berlin, Heidelberg, New York, (1993), 430–444.

    Google Scholar 

  27. H. Straubing, Families of recognizable sets corresponding to certain varieties of finite monoids, J. Pure Appl. Algebra, 15, (1979), 305–318.

    Article  MATH  MathSciNet  Google Scholar 

  28. H. Straubing, A generalization of the Schützenberger product of finite monoids, Theoret. Comp. Sci. 13, (1981), 137–150.

    Article  MATH  MathSciNet  Google Scholar 

  29. H. Straubing, Finite semigroups varieties of the form U * D, J. Pure Appl. Algebra 36, (1985), 53–94.

    Article  MATH  MathSciNet  Google Scholar 

  30. H. Straubing, Semigroups and languages of dot-depth two, Theoret. Comp. Sci. 58, (1988), 361–378.

    Article  MATH  MathSciNet  Google Scholar 

  31. H. Straubing, The wreath product and its application, in Formal properties of finite automata and applications, J.-E. Pin (ed.), Lect. Notes in Comp. Sci. 386, Springer Verlag, Berlin, Heidelberg, New York, (1989), 15–24.

    Google Scholar 

  32. H. Straubing and P. Weil, On a conjecture concerning dot-depth two languages, Theoret. Comp. Sci. 104, (1992), 161–183.

    Article  MATH  MathSciNet  Google Scholar 

  33. D. Thérien, Classification of finite monoids: the language approach, Theoret. Comp. Sci. 14, (1981), 195–208.

    Article  MATH  Google Scholar 

  34. W. Thomas, Classifying regular events in symbolic logic, J. Comput. Syst. Sci 25, (1982), 360–375.

    Article  MATH  Google Scholar 

  35. P. Weil, Closure of varieties of languages under products with counter, J. Comput. System Sci. 45, (1992), 316–339.

    Article  MATH  MathSciNet  Google Scholar 

  36. P. Weil, Some results on the dot-depth hierarchy, Semigroup Forum 46, (1993), 352–370.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. LIAFA, CNRS, Université Paris VII, 2 Place Jussieu, 75251, Paris Cedex 05, France

    Jean-éric Pin

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  1. Jean-éric Pin
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Kim G. Larsen Sven Skyum Glynn Winskel

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© 1998 Springer-Verlag Berlin Heidelberg

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Pin, Jé. (1998). Bridges for concatenation hierarchies. In: Larsen, K.G., Skyum, S., Winskel, G. (eds) Automata, Languages and Programming. ICALP 1998. Lecture Notes in Computer Science, vol 1443. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055073

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  • DOI: https://doi.org/10.1007/BFb0055073

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64781-2

  • Online ISBN: 978-3-540-68681-1

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