Turing Universality in Dynamical Systems

  • Jean-Charles Delvenne
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3988)


A computer is classically formalized as a universal Turing machine. However over the years a lot of research has focused on the computational properties of dynamical systems other than Turing machines, such cellular automata, artificial neural networks, mirrors systems, etc.

In this talk we review some of the definitions that have been proposed for Turing universality of various systems, and the attempts to understand the relation between dynamical and computational properties of a system.


Cellular Automaton Turing Machine Countable Family Computational Property Reachability Problem 
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.


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  1. 1.
    Asarin, E., Bouajjani, A.: Perturbed turing machines and hybrid systems. In: Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science (LICS-2001), June 16–19, 2001, pp. 269–278. IEEE Computer Society Press, Los Alamitos (2001)CrossRefGoogle Scholar
  2. 2.
    Asarin, E., Maler, O., Pnueli, A.: Reachability analysis of dynamical systems having piecewise-constant derivatives. Theoretical Computer Science 138(1), 35–65 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society 21, 1–46 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bournez, O., Cosnard, M.: On the computational power of dynamical systems and hybrid systems. Theoretical Computer Science 168, 417–459 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Davis, M.D.: A note on universal Turing machines. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies, pp. 167–175. Princeton University Press, Princeton (1956)Google Scholar
  6. 6.
    Delvenne, J.-Ch., Kůrka, P., Blondel, V.D.: Computational universality in symbolic dynamical systems. Fundamenta Informaticae 71, 1–28 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Deutsch, D.: Quantum theory, the Church-Turing principle and the universal quantum computer. Proceedings of the Royal Society of London Ser. A A400, 97–117 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gács, P.: Reliable cellular automata with self-organization. In: 38th Annual Symposium on Foundations of Computer Science, Miami Beach, Florida, October 20–22, pp. 90–99. IEEE, Los Alamitos (1997)CrossRefGoogle Scholar
  9. 9.
    Koiran, P., Cosnard, M., Garzon, M.: Computability with low-dimensional dynamical systems. Theoretical Computer Science 132(1–2), 113–128 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Koiran, P., Moore, C.: Closed-form analytic maps in one and two dimensions can simulate universal Turing machines. Theoretical Computer Science 210(1), 217–223 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Langton, C.G.: Computation at the edge of chaos. Physica. D 42, 12–37 (1990)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Maass, W., Orponen, P.: On the effect of analog noise in discrete-time analog computations. Neural Computation 10(5), 1071–1095 (1998)CrossRefGoogle Scholar
  13. 13.
    Mitchell, M., Hraber, P.T., Crutchfield, J.P.: Dynamic computation, and the “edge of chaos”: A re-examination. In: Cowan, G., Pines, D., Melzner, D. (eds.) Complexity: Metaphors, Models, and Reality. Santa Fe Institute Proceedings, vol. 19, pp. 497–513. Addison-Wesley, Reading (1994)Google Scholar
  14. 14.
    Moore, Cr.: Unpredictability and undecidability in dynamical systems. Physical Review Letters 64(20), 2354–2357 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Moore, C.: Generalized shifts: Unpredictability and undecidability in dynamical systems. Nonlinearity 4, 199–230 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Moore, C.: Dynamical recognizers: real-time language recognition by analog computers. Theoretical Computer Science 201, 99–136 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Moore, C.: Recursion theory on the reals and continuous-time computation. Theoretical Computer Science 162(1), 23–44 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ollinger, N.: The intrinsic universality problem of one-dimensional cellular automata. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 632–641. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  19. 19.
    Pour-El, M.B., Richards, J.I.: Computability and noncomputability in classical analysis. Transactions of the American Mathematical Society 275, 539–560 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Siegelmann, H.T.: Neural Networks and Analog Computation: Beyond the Turing Limit. In: Progress in Theoretical Computer Science. Springer, Heidelberg (1999)Google Scholar
  21. 21.
    Sutner, K.: Almost periodic configurations on linear cellular automata. Fundamenta Informaticae 58(3–4), 223–240 (2003)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Koiran, P., Blondel, V., Bournez, O., Tsitsiklis, J.: The stability of saturated linear dynamical systems is undecidable. Journal of Computer and System Sciences 62, 442–462 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  24. 24.
    Wolfram, S.: A new kind of science. Wolfram Media, Inc, Champaign, IL (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jean-Charles Delvenne
    • 1
  1. 1.Department of Mathematical EngineeringUniversité catholique de LouvainLouvain-la-NeuveBelgium

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