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Computability with low-dimensional dynamical systems. (English) Zbl 0821.68053

Summary: 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.

MSC:

68Q05 Models of computation (Turing machines, etc.) (MSC2010)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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