Abstract
It has been known for a short time that a class of recurrent neural networks has universal computational abilities. These networks can be viewed as iterated piecewise-linear maps in a high-dimensional space. In this paper, we show that similar systems in dimension two are also capable of universal computations. On the contrary, it is necessary to resort to more complex systems (e.g., iterated piecewise-monotone maps) in order to retain this capability in dimension one.
This work was partially supported by the Programme de Recherches Coordonnées C3 of the CNRS and the Ministère de la Recherche et de la Technologie.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
F. Bothelho and M. Garzon. On dynamical properties of neural networks. Complex Systems, 5(4):401–403, 1991.
P. Collet and J.P. Eckmann. Iterated maps on the interval as dynamical systems, volume I of Progress in Physics. Birkhäuser, Boston, 1980.
M. Garzon and S.P. Franklin. Neural computability II. In Proc. 3rd Int. Joint Conf. on Neural Networks, Wash. D.C., volume 1, pages 631–637, 1989.
M. L. Minsky. Computation: Finite and Infinite Machines. Prentice Hall, Engelwood Cliffs, 1967.
C. Moore. Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity, 4:199–230, 1991.
C. Preston. Iterates of piecewise monotone mappings on an interval, volume 1347 of Lecture Notes in Mathematics. Springer-Verlag, 1988.
D. Richardson. Tessellation with local transformations. Journal of Computer and System Sciences, 6:373–388, 1972.
Y. V. Rogozhin. Seven universal Turing machines. Mat. Issled., 69:76–90, 1982. (Russian).
H. T. Siegelman and E. D. Sontag. Neural nets are universal computing devices. SYCON Report 91-08, Rutgers University, May 1991.
H. T. Siegelman and E. D. Sontag. On the computational power of neural nets. In Proc. Fifth ACM Workshop on Computational Learning Theory, July 1992.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cosnard, M., Garzon, M., Koiran, P. (1993). Computability properties of low-dimensional dynamical systems. In: Enjalbert, P., Finkel, A., Wagner, K.W. (eds) STACS 93. STACS 1993. Lecture Notes in Computer Science, vol 665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56503-5_37
Download citation
DOI: https://doi.org/10.1007/3-540-56503-5_37
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56503-1
Online ISBN: 978-3-540-47574-3
eBook Packages: Springer Book Archive