The Complexity of Proving Chaoticity and the Church-Turing Thesis

TL;DR

This paper explores the deep connection between chaos in dynamical systems and computational complexity, suggesting that proving chaos is as hard as solving the most difficult mathematical problems, and physical systems might perform incomputable computations.

Contribution

It establishes a theoretical link between chaos proof complexity and computational hardness, proposing that physical systems could inherently perform incomputable tasks.

Findings

Proving chaoticity is as hard as solving the most difficult mathematical problems.

Physical systems might compute incomputable functions through measurement.

Theoretical implications for the limits of physical computation.

Abstract

Proving the chaoticity of some dynamical systems is equivalent to solving the hardest problems in mathematics. Conversely, one argues that it is not unconceivable that classical physical systems may "compute the hard or even the incomputable" by measuring observables which correspond to computationally hard or even incomputable problems.