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On recognizable languages in divisibility monoids

  • Manfred Droste
  • Dietrich Kuske
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1684)

Abstract

Kleene’s theorem on recognizable languages in free monoids is considered to be of eminent importance in theoretical computer science. It has been generalized into various directions, including trace and rational monoids. Here, we investigate divisibility monoids which are defined by and capture algebraic properties sufficient to obtain a characterization of the recognizable languages by certain rational expressions as known from trace theory. The proofs rely on Ramsey’s theorem, distributive lattice theory and on Hashigushi’s rank function generalized to our divisibility monoids. We obtain Ochma’nski’s theorem on recognizable languages in free partially commutative monoids as a consequence.

Keywords

Distributive Lattice Closure Property Label Sequence Irreducible Element Free Monoid 
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.
    R. M. Amadio and P.-L. Curien. Domains and Lambda Calculi. Cambridge Tracts in Theoretical Computer Science. 46. Cambridge: Cambridge University Press, 1998.zbMATHGoogle Scholar
  2. 2.
    G. Berry and J.-J. Levy. Minimal and optimal computations of recursive programs. J. ACM, 26:148–175, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    G. Birkhoff. Lattice Theory. Colloquium Publications vol. 25. American Mathematical Society, Providence, 1940; third edition, seventh printing from 1993.Google Scholar
  4. 4.
    G. Boudol. Computational semantics of term rewriting. In M. Nivat and J.C. Reynolds, editors, Algebraic Methods in Semantics, pages 169–236. Cambridge University Press, 1985.Google Scholar
  5. 5.
    J.R. Büchi. On a decision method in restricted second order arithmetics. In E. Nagel et al., editor, Proc. Intern. Congress on Logic, Methodology and Philosophy of Science, pages 1–11. Stanford University Press, Stanford, 1960.Google Scholar
  6. 6.
    V. Diekert and Y. Mètivier. Partial commutation and traces. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 3. Springer, 1997.Google Scholar
  7. 7.
    V. Diekert and G. Rozenberg. The Book of Traces. World Scientific Publ. Co., 1995.Google Scholar
  8. 8.
    M. Droste. Recognizable languages in concurrency monoids. Theoretical Comp. Science, 150:77–109, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. Droste and D. Kuske. On recognizable languages in left divisibility monoids. Technical Report MATH-AL-9-1998, TU Dresden, 1998.Google Scholar
  10. 10.
    C. Duboc. Commutations dans les monoïdes libres: un cadre thèorique pour l’ètude du parallelisme. Thèse, Facultè des Sciences de l’Universitè de Rouen, 1986.Google Scholar
  11. 11.
    K. Hashigushi. Recognizable closures and submonoids of free partially commutative monoids. Theoretical Comp. Science, 86:233–241, 1991.CrossRefGoogle Scholar
  12. 12.
    D. Kuske. On rational and on left divisibility monoids. Technical Report MATHAL-3-1999, TU Dresden, 1999.Google Scholar
  13. 13.
    E. Ochmański. Regular behaviour of concurrent systems. Bull. Europ. Assoc. for Theor. Comp. Science, 27:56–67, 1985.Google Scholar
  14. 14.
    P. Panangaden and E.W. Stark. Computations, residuals and the power of indeterminacy. In Automata, Languages and Programming, Lecture Notes in Comp. Science vol. 317, pages 439–454. Springer, 1988.Google Scholar
  15. 15.
    F.P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264–286, 1930.CrossRefGoogle Scholar
  16. 16.
    J. Sakarovitch. Easy multiplications. I. The realm of Kleene’s Theorem. Information and Computation, 74:173–197, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    M.P. Schützenberger. On the definition of a family of automata. Inf. Control, 4:245–270, 1961.zbMATHCrossRefGoogle Scholar
  18. 18.
    E.W. Stark. Concurrent transition systems. Theoretical Comp. Science, 64:221–269, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    G. Winskel. Event structures. In W. Brauer, W. Reisig, and G. Rozenberg, editors, Petri nets: Applications and Relationships to Other Models of Concurrency, Lecture Notes in Comp. Science vol. 255, pages 325–392. Springer, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Manfred Droste
    • 1
  • Dietrich Kuske
    • 1
  1. 1.Institut für AlgebraTechnische Universität DresdenDresdenGermany

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