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.
This research was supported by FCAR (Québec) and by NSERC (Canada).
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A.A. Albert, Quasigroups. I, Trans. Amer. Math. Soc., Vol. 54 (1943) pp.507–519.
A.A. Albert, Quasigroups. II, Trans. Amer. Math. Soc., Vol. 55 (1944) pp.401–419.
D.A. Barrington, Bounded-Width Polynomial-Size Branching Programs Recognize Exactly Those Languages in NC1, JCSS 38, 1 (1989), pp. 150–164.
D.A. Barrington H. Straubing and D. Thérien, Non-Uniform Automata Over Groups, Information and Computation 89 (1990), pp. 109–132.
D. Barrington and D. Thérien, Finite Monoids and the Fine Structure of NC 1, JACM 35, 4 (1988), pp. 941–952.
M. Beaudry and P. McKenzie, Circuits, Matrices and Nonassociative Computation, Proc. of the 7th Structure in Complexity Theory Conference, (1992), pp. 94–106.
F. Bédard, F. Lemieux and P. McKenzie, Extensions to Barrington's M-program model, TCS 107 (1993), pp. 31–61.
R.H. Bruck, Contributions to the Theory of Loops, Trans. AMS, (60) 1946 pp.245–354.
R.H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1966.
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.
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.
S.A. Cook, A Taxonomy of Problems with Fast Parallel Algorithms, Information and Control 64 (1985), pp. 2–22.
J. Dénes and A.D. Keedwell, Latin Squares and their Applications, English University Press, 1974.
W.D. Maurer and J. Rhodes, A Property of Finite Simple Non-abelian Groups, Proc. AMS 16 (1965), 552–554.
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.
A. Muscholl, Characterizations of LOG, LOGDCFL and NP based on groupoid programs, Manuscript, 1992.
H.O. Pfugfelder, Quasigroups and Loops: Introduction, Heldermann Verlag, 1990.
J.-E. Pin, Variétés de languages formels, Masson, 1984. Also Varieties of Formal Languages, Plenum Press, New York, 1986.
M.P. Schützenberger, On Finite Monoids having only trivial subgroups, Information and Control 8 (1965), pp. 190–194.
H. Straubing, Representing Functions by Words over Finite Semigroups, Université de Montréal, Technical Report #838, 1992.
D. Thérien, Classification of Finite Monoids: The Language Approach, TCS 14 (1981), pp. 195–208.
H. Venkateswaran, Properties that Characterize LOGCFL, Proc. of the 19th ACM Symp. on the Theory of Computing (1987), pp. 141–150.
M.J. Wolf, Nondeterministic Circuits, Space Complexity and Quasigroups, TCS 125 (1994), pp. 295–314.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Caussinus, H., Lemieux, F. (1994). The complexity of computing over quasigroups. In: Thiagarajan, P.S. (eds) Foundation of Software Technology and Theoretical Computer Science. FSTTCS 1994. Lecture Notes in Computer Science, vol 880. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58715-2_112
Download citation
DOI: https://doi.org/10.1007/3-540-58715-2_112
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58715-6
Online ISBN: 978-3-540-49054-8
eBook Packages: Springer Book Archive