# The relationship between word complexity and computational complexity in subshifts

## TL;DR

This paper explores how the complexity of words in subshifts relates to the computational complexity of points within them, revealing conditions under which certain Turing spectra can be realized by subshifts with specific growth rates.

## Contribution

It characterizes the Turing spectra realizable by subshifts of various complexity growth rates, connecting word complexity with computational degrees.

## Key findings

- Turing spectrum can be realized via linear complexity subshift if it is a union of a finite set and cones.
- Turing spectrum can be realized via exponential complexity subshift if it contains a cone.
- Subshifts with intermediate complexity growth can realize spectra containing degree 0 or unions of cones.

## Abstract

We prove several results about the relationship between the word complexity function of a subshift and the set of Turing degrees of points of the subshift, which we call the Turing spectrum. Among other results, we show that a Turing spectrum can be realized via a subshift of linear complexity if and only if it consists of the union of a finite set and a finite number of cones, that a Turing spectrum can be realized via a subshift of exponential complexity (i.e. positive entropy) if and only if it contains a cone, and that every Turing spectrum which either contains degree 0 or is a union of cones is realizable by subshifts with a wide range of 'intermediate' complexity growth rates between linear and exponential.