Advertisement

The complexity of computing over quasigroups

  • Hervé Caussinus
  • François Lemieux
Complexity Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 880)

Abstract

In [7] the notions of recognition by semigroups and by programs over semigroups were extended to groupoids. As a consequence of this transformation, the induced classes of languages became CFL instead of REG, in the first case, and SAC1 instead of NC1 in the second case. In this paper, we investigate the classes obtained when the groupoids are restricted to be quasigroups (i.e. the multiplication table forms a latin square). We prove that languages recognized by quasigroups are regular and that programs over quasigroups characterize NC1. We introduce the notions of linear recognition by semigroups and by programs over semigroups. This leads to a new characterization of the linear context-free languages and NL. Here again, when quasigroups are used, only languages in REG and NC1 can be obtained. We also consider the problem of evaluating a well-parenthesized expression over a finite loop (a quasigroup with an identity). This problem is in NC1 for any finite loop, and we give algebraic conditions for its completeness. In particular, we prove that it is sufficient that the loop be noasolvable, extending a well-known theorem of Barrington.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A.A. Albert, Quasigroups. I, Trans. Amer. Math. Soc., Vol. 54 (1943) pp.507–519.Google Scholar
  2. 2.
    A.A. Albert, Quasigroups. II, Trans. Amer. Math. Soc., Vol. 55 (1944) pp.401–419.Google Scholar
  3. 3.
    D.A. Barrington, Bounded-Width Polynomial-Size Branching Programs Recognize Exactly Those Languages in NC1, JCSS 38, 1 (1989), pp. 150–164.Google Scholar
  4. 4.
    D.A. Barrington H. Straubing and D. Thérien, Non-Uniform Automata Over Groups, Information and Computation 89 (1990), pp. 109–132.Google Scholar
  5. 5.
    D. Barrington and D. Thérien, Finite Monoids and the Fine Structure of NC 1, JACM 35, 4 (1988), pp. 941–952.Google Scholar
  6. 6.
    M. Beaudry and P. McKenzie, Circuits, Matrices and Nonassociative Computation, Proc. of the 7th Structure in Complexity Theory Conference, (1992), pp. 94–106.Google Scholar
  7. 7.
    F. Bédard, F. Lemieux and P. McKenzie, Extensions to Barrington's M-program model, TCS 107 (1993), pp. 31–61.Google Scholar
  8. 8.
    R.H. Bruck, Contributions to the Theory of Loops, Trans. AMS, (60) 1946 pp.245–354.Google Scholar
  9. 9.
    R.H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1966.Google Scholar
  10. 10.
    S.R. Buss, The Boolean Formula Value Problem is in ALOGTIME, Proc. of the 19th ACM Symp. on the Theory of Computing (1987), pp. 123–131.Google Scholar
  11. 11.
    A.K. Chandra, S. Fortune and R. Lipton, Unbounded Fan-in Circuits and Associative Functions, Proc. of the 15th ACM Symp. on the Theory of Computing (1983), pp. 52–60.Google Scholar
  12. 12.
    S.A. Cook, A Taxonomy of Problems with Fast Parallel Algorithms, Information and Control 64 (1985), pp. 2–22.Google Scholar
  13. 13.
    J. Dénes and A.D. Keedwell, Latin Squares and their Applications, English University Press, 1974.Google Scholar
  14. 14.
    W.D. Maurer and J. Rhodes, A Property of Finite Simple Non-abelian Groups, Proc. AMS 16 (1965), 552–554.Google Scholar
  15. 15.
    G.L. Miller, On the n log n Isomorphic Technique, Proc. of the 10th ACM Symp. on the Theory of Computing (1978), pp. 51–58.Google Scholar
  16. 16.
    A. Muscholl, Characterizations of LOG, LOGDCFL and NP based on groupoid programs, Manuscript, 1992.Google Scholar
  17. 17.
    H.O. Pfugfelder, Quasigroups and Loops: Introduction, Heldermann Verlag, 1990.Google Scholar
  18. 18.
    J.-E. Pin, Variétés de languages formels, Masson, 1984. Also Varieties of Formal Languages, Plenum Press, New York, 1986.Google Scholar
  19. 19.
    M.P. Schützenberger, On Finite Monoids having only trivial subgroups, Information and Control 8 (1965), pp. 190–194.Google Scholar
  20. 20.
    H. Straubing, Representing Functions by Words over Finite Semigroups, Université de Montréal, Technical Report #838, 1992.Google Scholar
  21. 21.
    D. Thérien, Classification of Finite Monoids: The Language Approach, TCS 14 (1981), pp. 195–208.Google Scholar
  22. 22.
    H. Venkateswaran, Properties that Characterize LOGCFL, Proc. of the 19th ACM Symp. on the Theory of Computing (1987), pp. 141–150.Google Scholar
  23. 23.
    M.J. Wolf, Nondeterministic Circuits, Space Complexity and Quasigroups, TCS 125 (1994), pp. 295–314.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Hervé Caussinus
    • 1
  • François Lemieux
    • 2
  1. 1.Département d'informatique et de recherche opérationnelleUniversité de MontrealMontréalCanada
  2. 2.School of Computer ScienceMcGill UniversityMontréalCanada

Personalised recommendations