Arithmetical Hierarchy of the Besicovitch-Stability of Noisy Tilings

TL;DR

This paper investigates the algorithmic complexity of the Besicovitch stability of noisy subshifts of finite type, establishing its undecidability and placing it within the arithmetical hierarchy.

Contribution

It demonstrates the undecidability of the stability problem and determines its position in the arithmetical hierarchy, providing new insights into the complexity of noisy tilings.

Findings

The stability problem is undecidable.

It is $ ext{Pi}_2$-hard in the arithmetical hierarchy.

The problem has a $ ext{Pi}_4$ upper bound.

Abstract

The purpose of this article is to study the algorithmic complexity of the Besicovitch stability of noisy subshifts of finite type, a notion studied in a previous article. First, we exhibit an unstable aperiodic tiling, and then see how it can serve as a building block to implement several reductions from classical undecidable problems on Turing machines. It will follow that the question of stability of subshifts of finite type is undecidable, and the strongest lower bound we obtain in the arithmetical hierarchy is $\Pi_2$-hardness. Lastly, we prove that this decision problem, which requires to quantify over an uncountable set of probability measures, has a $\Pi_4$ upper bound.