Analytic one-dimensional maps and two-dimensional ordinary differential equations can robustly simulate Turing machines

09/30/2021

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by   Daniel S. Graça, et al.

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In this paper, we analyze the problem of finding the minimum dimension n such that a closed-form analytic map/ordinary differential equation can simulate a Turing machine over ℝ^n in a way that is robust to perturbations. We show that one-dimensional closed-form analytic maps are sufficient to robustly simulate Turing machines; but the minimum dimension for the closed-form analytic ordinary differential equations to robustly simulate Turing machines is two, under some reasonable assumptions. We also show that any Turing machine can be simulated by a two-dimensional C^∞ ordinary differential equation on the compact sphere 𝕊^2.