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Positive varieties and infinite words

  • Jean -éric Pin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1380)

Abstract

Carrying on the work of Arnold, Pécuchet and Perrin, Wilke has obtained a counterpart of Eilenberg's variety theorem for finite and infinite words. In this paper, we extend this theory for classes of languages that are closed under union and intersection, but not necessarily under complement. As an example, we give a purely algebraic characterization of various classes of recognizable sets defined by topological properties (open, closed, F and F δ ) or by combinatorial properties

Keywords

Finite Automaton Variety Theorem Positive Variety Finite Union Infinite Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. Arnold, A syntactic congruence for rational Ω-languages, Theoret. Comput. Sci. 39, (1985) 333–335.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J. R. Büchi, Weak second-order arithmetic and finite automata, Z. Math. Logik und Grundl. Math. 6, (1960) 66–92.zbMATHGoogle Scholar
  3. 3.
    J. R. Büchi, On a decision method in restricted second-order arithmetic, in Proc. 1960 Int. Congr. for Logic, Methodology and Philosophy of Science, Stanford Univ. Press, Standford, (1962) 1–11.Google Scholar
  4. 4.
    S. Eilenberg, Automata, languages and machines, Vol. B, Academic Press, New York (1976).Google Scholar
  5. 5.
    R. McNaughton, Testing and generating infinite sequences by a finite automaton Information and Control 9, (1966) 521–530.zbMATHMathSciNetGoogle Scholar
  6. 6.
    D. Muller, Infinite sequences and finite machines, in Switching Theory and Logical Design, Proc. Fourth Annual Symp. IEEE, (1963) 3–16.Google Scholar
  7. 7.
    J.-P. Pécuchet, Variétés de semigroupes et mots infinis, in B. Monien and G. Vidal-Naquet eds., STACS 86, Lecture Notes in Computer Science 210, Springer, (1986) 180–191.Google Scholar
  8. 8.
    J.-P. Pécuchet, étude syntaxique des parties reconnaissables de mots infinis, in Proc. 13th ICALP, (L. Kott ed.) Lecture Notes in Computer Science 226, Springer, Berlin, (1986) 294–303.Google Scholar
  9. 9.
    D. Perrin, Variétés de semigroupes et mots infinis, C.R. Acad. Sci. Paris 295, (1982) 595–598.zbMATHMathSciNetGoogle Scholar
  10. 10.
    D. Perrin, Recent results on automata and infinite words, in Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 176, Springer, Berlin, (1984) 134–148.Google Scholar
  11. 11.
    D. Perrin, An introduction to automata on infinite words, in Automata on Infinite Words (Nivat, M. ed.), Lecture Notes in Computer Science 192, Springer, Berlin, (1984) 2–17.Google Scholar
  12. 12.
    D. Perrin and J.-é. Pin, First order logic and star-free sets, J. Comput. System Sci. 32, (1986), 393–406.CrossRefMathSciNetGoogle Scholar
  13. 13.
    D. Perrin and J.-é. Pin, Semigroups and automata on infinite words, in NATO Advanced Study Institute Semigroups, Formal Languages and Groups, J. Fountain (ed.), Kluwer academic publishers, (1995), 49–72.Google Scholar
  14. 14.
    D. Perrin and J.-é. Pin, Mots infinis, to appear (LITP report 97-04), (1997). Accessible on the web: http://liafa.jussieu.fr/~jep.Google Scholar
  15. 15.
    J.-é. Pin, Variétés de langages formels, Masson, Paris (1984); English translation: Varieties of formal languages, Plenum, New-York (1986).Google Scholar
  16. 16.
    J.-é. 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
  17. 17.
    J.-é. Pin, A variety theorem without complementation, Russian Mathematics (Iz. VUZ) 39 (1995), 80–90.zbMATHMathSciNetGoogle Scholar
  18. 18.
    J.-é. Pin, A negative answer to a question of Wilke on varieties of Ω-languages, Information Processing Letters, (1995), 197–200.Google Scholar
  19. 19.
    J.-é. Pin, Logic, Semigroups and Automata on Words, Annals of Mathematics and Artificial Intelligence 16 (1996), 343–384.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    J.-é. Pin, Syntactic semigroups, in Handbook of language theory, G. Rozenberg and A. Salomaa (éd.), Springer Verlag, 1997.Google Scholar
  21. 21.
    J.-é. Pin and P. Weil, A Reiterman theorem for pseudovarieties of finite first-order structures, Algebra Universalis 35 (1996), 577–595.CrossRefMathSciNetGoogle Scholar
  22. 22.
    J.-é. Pin and P. Weil, Polynomial closure and unambiguous product, Theory Comput. Systems 30 (1997), 1–39.CrossRefMathSciNetGoogle Scholar
  23. 23.
    I. Simon, Piecewise testable events, Proc. 2nd GI Conf., Lecture Notes in Computer Science 33, Springer, Berlin, (1975) 214–222.Google Scholar
  24. 24.
    W. Thomas, Automata on infinite objects, in Handbook of Theoretical Computer Science, vol B, Formal models and semantics, Elsevier, (1990) 135–191.Google Scholar
  25. 25.
    T. Wilke, An Eilenberg theorem for ∞-languages, in Automata, Languages and Programming, Lecture Notes in Computer Science 510, Springer Verlag, Berlin, Heidelberg, New York, (1991), 588–599.Google Scholar
  26. 26.
    T. Wilke, An algebraic theory for regular languages of finite and infinite words, Int. J. Alg. Comput. 3, (1993), 447–489.zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    T. Wilke, Locally threshold testable languages of infinite words, in STACS 93, P. Enjalbert, A. Finkel, K.W. Wagner (Eds.), Lecture Notes in Computer Science 665, Springer, Berlin, (1993) 607–616.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Jean -éric Pin
    • 1
  1. 1.LIAFA, CNRS and Université Paris VIIParis Cedex 05France

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