Robustness of non-computability

TL;DR

This paper introduces a framework to analyze the robustness of non-computability results in continuous systems, examining whether super-Turing capabilities can be physically realistic and resilient to perturbations.

Contribution

It presents a novel framework for assessing the robustness of non-computability in continuous systems, applied to wave equations, differentiation, and basins of attraction.

Findings

Certain non-computability results are robust under perturbations.

The framework identifies conditions for robustness in physical systems.

Super-Turing capabilities may be theoretically possible with robustness considerations.

Abstract

Turing computability is the standard computability paradigm which captures the computational power of digital computers. To understand whether one can create physically realistic devices which have super-Turing power, one needs to understand whether it is possible to obtain systems with super-Turing capabilities which also have other desirable properties such as robustness to perturbations. In this paper we introduce a framework for analyzing whether a non-computability result is robust over continuous spaces. Then we use this framework to study the degree of robustness of several non-computability results which involve the wave equation, differentiation, and basins of attraction.