Star-Free Open Languages and Aperiodic Loops

  • Martin Beaudry
  • François Lemieux
  • Denis Thérien3
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2010)


It is known that recognition of regular languages by finite monoids can be generalized to context-free languages and finite groupoids, which are finite sets closed under a binary operation. A loop is a groupoid with a neutral element and in which each element has a left and a right inverse. It has been shown that finite loops recognize exactly those regular languages that are open in the group topology. In this paper, we study the class of aperiodic loops, which are those loops that contain no nontrivial group. We show that this class is stable under various definitions, and we prove some closure properties. We also prove that aperiodic loops recognize only star-free open languages and give some examples. Finally, we show that the wreath product principle can be applied to groupoids, and we use it to prove a decomposition theorem for recognizers of regular open languages.


Wreath Product Regular Language Closure Property Tree Language Open Language 
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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  1. 1.
    M Ajtai, 1 1-formulae on finite structures, Annals of Pure and Applied Logic, 24 pp.1–48, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A.A. Albert, Quasigroups. I, Trans. Amer. Math. Soc., Vol. 54 (1943) pp.507–519.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A.A. Albert, Quasigroups. II, Trans. Amer. Math. Soc., Vol. 55 (1944) pp.401–419.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    D.A. Barrington, Bounded-Width Polynomial-Size Branching Programs Recognize Exactly those Languages in NC1, JCSS 38,1 (1989), pp. 150–164.zbMATHMathSciNetGoogle Scholar
  5. 5.
    D. Barrington and D. Thérien, Finite Monoids and the Fine Structure of NC1, JACM 35,4 (1988), pp. 941–952.CrossRefGoogle Scholar
  6. 6.
    P.W. Beam, S. A. Cook, and H. J. Hoover, Log Depth Circuits for Division and Related Problems, in Proc. of the 25th IEEE Symp. on the Foundations of Computer Science (1984), pp. 1–6.Google Scholar
  7. 7.
    M. Beaudry, Languages recognized by finite aperiodic groupoids, Theoretical Computer Science, vol. 209, 1998, pp. 299–317.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Beaudry, Finite idempotent groupoids and regular languages, Theoretical Informatics and Applications, vol. 32, 1998, pp. 127–140.MathSciNetGoogle Scholar
  9. 9.
    M. Beaudry, F. Lemieux, and D. Thérien, Finite loops recognize exactly the regular open languages, in Proc. of the 24th International Colloquium on Automata, Languages and Programming, Springer Lecture Notes in Comp. Sci. 1256 (1997), pp. 110–120.Google Scholar
  10. 10.
    F. Bédard, F. Lemieux and P. McKenzie, Extensions to Barrington’s M-program model, TCS 107 (1993), pp. 31–61.zbMATHCrossRefGoogle Scholar
  11. 11.
    R.H. Bruck, Contributions to the Theory of Loops, Trans. AMS, (60) 1946 pp.245–354.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    R.H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1966.Google Scholar
  13. 13.
    H. Caussinus and F. Lemieux, The Complexity of Computing over Quasigroups, In the Proceedings of the 14th annual FST&TCS Conference, LNCS 1256, Springer-Verlag 1994, pp.36–47.Google Scholar
  14. 14.
    O. Chein, H. O. Pfugfelder, and J. D. H. Smith, Quasigroups and Loops: Theory and Applications, Helderman Verlag Berlin, 1990.Google Scholar
  15. 15.
    S.A. Cook, A Taxonomy of Problems with Fast Parallel Algorithms, Information and Computation 64 (1985), pp. 2–22.zbMATHGoogle Scholar
  16. 16.
    S. Eilenberg, Automata, Languages and Machines, Academic Press, Vol. B, (1976).Google Scholar
  17. 17.
    M.L. Furst, J.B. Saxe and M. Sipser, Parity, Circuits, and the Polynomial-Time Hierarchy, Proc. of the 22nd IEEE Symp. on the Foundations of Computer Science (1981), pp. 260–270. Journal version Math. Systems Theory 17 (1984), pp. 13-27.Google Scholar
  18. 18.
    F. Lemieux, Complexité, langages hors-contextes et structures algebriques non-associatives, Masters Thesis, Université de Montréal, 1990.Google Scholar
  19. 19.
    F. Lemieux, Finite Groupoids and their Applications to Computational Complexity, Ph.D. Thesis, McGill University, May 1996.Google Scholar
  20. 20.
    C. Moore, F. Lemieux, D. Thérien, J. Berman, and A. Drisko, Circuits and Ex-pressions with Nonassociative Gates, JCSS 60 (2000) pp.368–394.zbMATHGoogle Scholar
  21. 21.
    A. Muscholl, Characterizations of LOG, LOGDCFL and NP based on groupoid programs, Manuscript, 1992.Google Scholar
  22. 22.
    H. O. Pfugfelder, Quasigroups and Loops: Introduction, Heldermann Verlag, 1990.Google Scholar
  23. 23.
    J.-E. Pin, Variétés de languages formels, Masson (1984). Also Varieties of Formal Languages, Plenum Press, New York, 1986.Google Scholar
  24. 24.
    J.-E. Pin, On Reversible Automata, in Proceedings of the first LATIN Conference, Sao-Paulo, Notes in Computer Science 583, Springer Verlag, 1992, 401–416Google Scholar
  25. 25.
    J.-E. Pin, Polynomial closure of group languages and open set of the Hall topology, Theoretical Computer Science 169 (1996) 185–200zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    J.E. Savage, The complexity of computing, Wiley, 1976.Google Scholar
  27. 27.
    M.P. Schützenberger, On Finite Monoids having only trivial subgroups, Information and Control 8 (1965), pp. 190–194.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    I. Simon, Piecewise testable Events, Proc. 2nd GI Conf., LNCS 33, Springer, pp. 214–222, 1975.Google Scholar
  29. 29.
    H. Venkateswaran, Circuit definitions of nondeterministic complexity classes, Proceedings of the 8th annual FST&TCS Conference, 1988.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Martin Beaudry
    • 1
  • François Lemieux
    • 2
  • Denis Thérien3
    • 3
  1. 1.Département de mathématiques et d’informatiqueUniversité de SherbrookeSherbrooke (Qc)Canada
  2. 2.Département d’informatique et de mathématiqueUniversité du Québec à ChicoutimiChicoutimi (Qc)Canada
  3. 3.School of Computer ScienceMcGill UniversityMontréal (Qc)Canada

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