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

TL;DR

This paper demonstrates that one-dimensional analytic maps and two-dimensional smooth ODEs can robustly simulate Turing machines, establishing minimal dimensions for such computational representations.

Contribution

It proves the minimal dimensions needed for robust simulation of Turing machines by analytic maps and ODEs, including the sufficiency of one-dimensional maps and two-dimensional ODEs.

Findings

One-dimensional analytic maps can robustly simulate Turing machines.

Two-dimensional ODEs can also robustly simulate Turing machines.

Any Turing machine can be simulated by a smooth ODE on the sphere.

Abstract

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 $\mathbb{R}^{n}$ 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 $C^{\infty}$ ordinary differential equation on the compact sphere $\mathbb{S}^{2}$.