Abstract
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.
Work supported by NSERC (Canada) and FCAR (Québec). The last author is also supported by the von Humboldt foundation
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M Ajtai, ∑ 1 1-formulae on finite structures, Annals of Pure and Applied Logic, 24 pp.1–48, 1983.
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. Barrington and D. Thérien, Finite Monoids and the Fine Structure of NC1, JACM 35,4 (1988), pp. 941–952.
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.
M. Beaudry, Languages recognized by finite aperiodic groupoids, Theoretical Computer Science, vol. 209, 1998, pp. 299–317.
M. Beaudry, Finite idempotent groupoids and regular languages, Theoretical Informatics and Applications, vol. 32, 1998, pp. 127–140.
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.
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.
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.
O. Chein, H. O. Pfugfelder, and J. D. H. Smith, Quasigroups and Loops: Theory and Applications, Helderman Verlag Berlin, 1990.
S.A. Cook, A Taxonomy of Problems with Fast Parallel Algorithms, Information and Computation 64 (1985), pp. 2–22.
S. Eilenberg, Automata, Languages and Machines, Academic Press, Vol. B, (1976).
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.
F. Lemieux, Complexité, langages hors-contextes et structures algebriques non-associatives, Masters Thesis, Université de Montréal, 1990.
F. Lemieux, Finite Groupoids and their Applications to Computational Complexity, Ph.D. Thesis, McGill University, May 1996.
C. Moore, F. Lemieux, D. Thérien, J. Berman, and A. Drisko, Circuits and Ex-pressions with Nonassociative Gates, JCSS 60 (2000) pp.368–394.
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.
J.-E. Pin, On Reversible Automata, in Proceedings of the first LATIN Conference, Sao-Paulo, Notes in Computer Science 583, Springer Verlag, 1992, 401–416
J.-E. Pin, Polynomial closure of group languages and open set of the Hall topology, Theoretical Computer Science 169 (1996) 185–200
J.E. Savage, The complexity of computing, Wiley, 1976.
M.P. Schützenberger, On Finite Monoids having only trivial subgroups, Information and Control 8 (1965), pp. 190–194.
I. Simon, Piecewise testable Events, Proc. 2nd GI Conf., LNCS 33, Springer, pp. 214–222, 1975.
H. Venkateswaran, Circuit definitions of nondeterministic complexity classes, Proceedings of the 8th annual FST&TCS Conference, 1988.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Beaudry, M., Lemieux, F., Thérien3, D. (2001). Star-Free Open Languages and Aperiodic Loops. In: Ferreira, A., Reichel, H. (eds) STACS 2001. STACS 2001. Lecture Notes in Computer Science, vol 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44693-1_8
Download citation
DOI: https://doi.org/10.1007/3-540-44693-1_8
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41695-1
Online ISBN: 978-3-540-44693-4
eBook Packages: Springer Book Archive