Abstract
We investigate the computational power of continuous-time neural networks with Hopfield-type units. We prove that polynomial-size networks with saturated-linear response functions are at least as powerful as polynomially space-bounded Turing machines.
Part of this work was done during the author's visit to the Technical University of Graz, Austria.
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References
Anderson, J. A., Rosenfeld, E. (eds.) Neurocomputing: Foundations of Research. The MIT Press, Cambridge, MA, 1988.
Asarin, E., Maler, O. On some relations between dynamical systems and transition systems. Proc. 21st Internat. Colloq. on Automata, Languages, and Programming (Jerusalem, Israel, July 1994), 59–72. Springer-Verlag, Berlin, 1994.
Balcázar, J. L., Díaz, J., Gabarrö, J. On characterizations of the class PSPACE/poly. Theoret. Comput. Sci. 52 (1987), 251–267.
Balcázar, J. L., Diaz, J., Gabarrö, J. Structural Complexity I. Springer-Verlag, Berlin, 1988 (2nd Ed. 1995).
Branicky, M. Analog computation with continuous ODES. Proc. Workshop on Physics and Computation 1991 (Dallas, Texas, Nov. 1991), 265–274. IEEE Computer Society Press, Los Alamitos, CA, 1994.
Branicky, M. Universal computation and other capabilities of hybrid and continuous dynamical systems. Theoret. Comput. Sci. 138 (1995), 67–100.
Fogelman, F., Mejia, C., Goles, E., Martinez, S. Energy functions in neural networks with continuous local functions. Complex Systems 3 (1989), 269–293.
Fogelman, F., Goles, E., Weisbuch, G. Transient length in sequential iterations of threshold functions. Discr. Appl. Math. 6 (1983), 95–98.
Char, B. W, Geddes, K. O., Gonnet, G. H., Leong, B. L., Monagan, M. B., Watt, S. M. First Leaves: A Tutorial Introduction to Maple V. Springer-Verlag, New York, NY, 1992.
Hirsch. M. W., Smale, S. Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, San Diego, CA, 1974.
Hopfield, J. J. Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. USA 79 (1982), 2554–2558. Reprinted in [1], pp. 460–464.
Hopfield, J. J. Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Nat. Acad. Sci. USA 81 (1984), 3088–3092. Reprinted in [1], pp. 579–583.
Karp, R. M., Lipton, R. J. Turing machines that take advice. L'Enseignement Mathématique 28 (1982), 191–209.
Kilian, J., Siegelmann, H. T. The dynamic universality of sigmoidal neural networks. Information and Computation 128 (1996), 48–56.
Koiran, P. Dynamics of discrete time, continuous state Hopfield networks. Neural Computation 6 (1994), 459–468.
Lepley, M., Miller, G. Computational power for networks of threshold devices in an asynchronous environment. Unpublished manuscript, Dept. of Mathematics, Massachusetts Inst. of Technology, 1983.
Marcus, C. M., Westervelt, R. M. Dynamics of iterated-map neural networks. Phys. Rev. A 40 (1989), 501–504.
MATLAB Reference Guide. MathWorks Inc., Natick, MA, 1992.
Moore, C. Unpredictability and undecidability in physical systems. Phys. Rev. Lett. 64 (1990), 2354–2357.
Moore, C. Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity 4 (1991), 199–230.
Orponen, P. The computational power of discrete Hopfield nets with hidden units. Neural Computation 8 (1996), 403–415.
Orponen, P. Computing with truly asynchronous threshold logic networks. Theoret. Comput. Sci. 174 (1997), 97–121.
Orponen, P. A survey of continuous-time computation theory. Advances in Algorithms, Languages, and Complexity (eds. D.-Z. Du, K.-I Ko), 209–224. Kluwer Academic Publishers, Dordrecht, 1997
Pour-El, M. B., Richards, I. Computability in Analysis and Physics. Springer-Verlag, Berlin, 1989.
Rubel, L. A. Digital simulation of analog computation and Church's Thesis. J. Symb. Logic 54 (1989), 1011–1017.
Siegelmann, H. T., Sontag, E. D. On the computational power of neural nets. J. Comput. System Sciences 50 (1995), 132–150.
Vergis, A., Steiglitz, K., Dickinson, B. The complexity of analog computation. Math. and Computers in Simulation 28 (1986), 91–113.
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Orponen, P. (1997). The computational power of continuous time neural networks. In: Plášil, F., Jeffery, K.G. (eds) SOFSEM'97: Theory and Practice of Informatics. SOFSEM 1997. Lecture Notes in Computer Science, vol 1338. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63774-5_99
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DOI: https://doi.org/10.1007/3-540-63774-5_99
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