Robust Simulations of Turing Machines with Analytic Maps and Flows

  • Daniel S. Graça
  • Manuel L. Campagnolo
  • Jorge Buescu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3526)


In this paper, we show that closed-form analytic maps and flows can simulate Turing machines in an error-robust manner. The maps and ODEs defining the flows are explicitly obtained and the simulation is performed in real time.


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  1. 1.
    Moore, C.: Unpredictability and undecidability in dynamical systems. Phys. Rev. Lett. 64, 2354–2357 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Koiran, P., Cosnard, M., Garzon, M.: Computability with low-dimensional dynamical systems. Theoret. Comput. Sci. 132, 113–128 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Siegelmann, H.T., Sontag, E.D.: On the computational power of neural networks. J. Comput. System Sci. 50, 132–150 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Moore, C.: Finite-dimensional analog computers: Flows, maps, and recurrent neural networks. In: Calude, C., Casti, J., Dinneen, M. (eds.) 1st International Conference on Unconventional Models of Computation - UMC 1998, pp. 59–71. Springer, Heidelberg (1998)Google Scholar
  5. 5.
    Casey, M.: The dynamics of discrete-time computation, with application to recurrent neural networks and finite state machine extraction. Neural Comp. 8, 1135–1178 (1996)CrossRefGoogle Scholar
  6. 6.
    Casey, M.: Correction to proof that recurrent neural networks can robustly recognize only regular languages. Neural Comp. 10, 1067–1069 (1998)CrossRefGoogle Scholar
  7. 7.
    Maass, W., Orponen, P.: On the effect of analog noise in discrete-time analog computations. Neural Comput. 10, 1071–1095 (1998)CrossRefGoogle Scholar
  8. 8.
    Pour-El, M.B., Richards, J.I.: The wave equation with computable initial data such that its unique solution is not computble. Adv. Math. 39, 215–239 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Pour-El, M.B., Zhong, N.: The wave equation with computable initial data whose unique solution is nowhere computable. Math. Log. Quart. 43, 499–509 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Weihrauch, K., Zhong, N.: Is wave propagation computable or can wave computers beat the Turing machine? Proc. London Math. Soc. 85, 312–332 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Koiran, P., Moore, C.: Closed-form analytic maps in one and two dimensions can simulate universal Turing machines. Theoret. Comput. Sci. 210, 217–223 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Branicky, M.S.: Universal computation and other capabilities of hybrid and continuous dynamical systems. Theoret. Comput. Sci. 138, 67–100 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Campagnolo, M.L., Moore, C., Costa, J.F.: An analog characterization of the Grzegorczyk hierarchy. J. Complexity 18, 977–1000 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Mycka, J., Costa, J.F.: Real recursive functions and their hierarchy. J. Complexity 20, 835–857 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Campagnolo, M.L., Moore, C., Costa, J.F.: Iteration, inequalities, and differentiability in analog computers. J. Complexity 16, 642–660 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Campagnolo, M.L.: The complexity of real recursive functions. In: Calude, C.S., Dinneen, M.J., Peper, F. (eds.) UMC 2002. LNCS, vol. 2509, pp. 1–14. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  17. 17.
    Hirsch, M.W., Smale, S.: Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, London (1974)zbMATHGoogle Scholar
  18. 18.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer, Heidelberg (1983)Google Scholar
  19. 19.
    Viana, M.: Dynamical systems: moving into the next century. In: Engquist, B., Schmid, W. (eds.) Mathematics Unlimited - 2001 and Beyond, pp. 1167–1178. Springer, Heidelberg (2001)Google Scholar
  20. 20.
    Pilyugin, S.Y.: Shadowing in Dynamical Systems. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  21. 21.
    Grebogi, C., Poon, L., Sauer, T., Yorke, J., Auerbach, D.: Shadowability of chaotic dynamical systems. In: Handbook of Dynamical Systems, vol. 2, pp. 313–344. Elsevier, Amsterdam (2002)CrossRefGoogle Scholar
  22. 22.
    Atkinson, K.E.: An Introduction to Numerical Analysis, 2nd edn. John Wiley & Sons, Chichester (1989)zbMATHGoogle Scholar
  23. 23.
    Graça, D.S., Costa, J.F.: Analog computers and recursive functions over the reals. J. Complexity 19, 644–664 (2003)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Daniel S. Graça
    • 1
    • 2
  • Manuel L. Campagnolo
    • 2
    • 3
  • Jorge Buescu
    • 4
  1. 1.Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade do AlgarveFaroPortugal
  2. 2.Center for Logic and Computation, Departamento de Matemática, Instituto Superior TécnicoUniversidade Técnica de LisboaLisboaPortugal
  3. 3.Departamento de Matemática, Instituto Superior de AgronomiaUniversidade Técnica de LisboaLisboaPortugal
  4. 4.Center for Mathematical Analysis, Geometry and Dynamical Systems, Departamento de Matemática, Instituto Superior TécnicoUniversidade Técnica de LisboaLisboaPortugal

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