Computability properties of low-dimensional dynamical systems

  • Michel Cosnard
  • Max Garzon
  • Pascal Koiran
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 665)


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.


Cellular Automaton Turing Machine Recurrent Neural Network Universal Turing Machine Pushdown Automaton 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Bothelho and M. Garzon. On dynamical properties of neural networks. Complex Systems, 5(4):401–403, 1991.Google Scholar
  2. 2.
    P. Collet and J.P. Eckmann. Iterated maps on the interval as dynamical systems, volume I of Progress in Physics. Birkhäuser, Boston, 1980.Google Scholar
  3. 3.
    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.Google Scholar
  4. 4.
    M. L. Minsky. Computation: Finite and Infinite Machines. Prentice Hall, Engelwood Cliffs, 1967.Google Scholar
  5. 5.
    C. Moore. Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity, 4:199–230, 1991.Google Scholar
  6. 6.
    C. Preston. Iterates of piecewise monotone mappings on an interval, volume 1347 of Lecture Notes in Mathematics. Springer-Verlag, 1988.Google Scholar
  7. 7.
    D. Richardson. Tessellation with local transformations. Journal of Computer and System Sciences, 6:373–388, 1972.Google Scholar
  8. 8.
    Y. V. Rogozhin. Seven universal Turing machines. Mat. Issled., 69:76–90, 1982. (Russian).Google Scholar
  9. 9.
    H. T. Siegelman and E. D. Sontag. Neural nets are universal computing devices. SYCON Report 91-08, Rutgers University, May 1991.Google Scholar
  10. 10.
    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.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Michel Cosnard
    • 1
  • Max Garzon
    • 1
  • Pascal Koiran
    • 1
  1. 1.Unité de Recherche Associée 1398 du CNRS Ecole Normale Supérieure de LyonLIP-IMAGLyon Cedex 07France

Personalised recommendations