Turing degrees of multidimensional SFTs

TL;DR

This paper explores the computability and Turing degree structures of 2-dimensional subshifts of finite type, establishing how their degrees relate to those of arbitrary subsets of infinite binary sequences.

Contribution

It demonstrates that for any subset of infinite binary sequences, there exists a 2D SFT with a Turing degree structure closely related to that subset, providing a comprehensive analysis of Turing degrees in SFTs.

Findings

Existence of SFTs with Turing degrees matching any given subset of
0,1
0 sequences.

If the subset contains a recursive member, the SFT shares the same Turing degrees.

SFTs with only non-recursive members have members with comparable but different degrees.

Abstract

In this paper we are interested in computability aspects of subshifts and in particular Turing degrees of 2-dimensional SFTs (i.e. tilings). To be more precise, we prove that given any \pizu subset $P$ of $\{0,1\}^\NN$ there is a SFT $X$ such that $P\times\ZZ^2$ is recursively homeomorphic to $X\setminus U$ where $U$ is a computable set of points. As a consequence, if $P$ contains a recursive member, $P$ and $X$ have the exact same set of Turing degrees. On the other hand, we prove that if $X$ contains only non-recursive members, some of its members always have different but comparable degrees. This gives a fairly complete study of Turing degrees of SFTs.