# Nonlowness independent from frequent mind changes

## TL;DR

This paper proves that nonlowness and frequent mind changes are independent measures of computably enumerable degree strength by constructing specific sets with both properties.

## Contribution

It provides the first direct construction of low and nonlow c.e. sets with arbitrary mind change frequency, confirming their independence.

## Key findings

- Nonlowness and frequent mind changes are independent measures.
- Constructed low and nonlow c.e. sets with arbitrary mind change counts.
- First direct construction of a nonlow low2 array computable set.

## Abstract

It was recently shown that the computably enumerable (c.e.) degrees that embed the critical triple (Downey, Greenberg, Weber 2007) and the M3 lattice structure (Downey, Greenberg 2015) are exactly those that change their minds sufficiently often. Therefore the embeddability strength of a c.e. degree has much to do with the degree's mind change frequency. Nonlowness is another common measure of degree strength, with nonlow degrees expected to compute more degrees than low ones. We ask if nonlowness and frequent mind changes are independent measures of strength. Downey and Greenberg (2015) claimed this to be true without proof, so we present one here. We prove the claim by building low and nonlow c.e. sets with an arbitrary number of mind changes. We base our proof on our direct construction of a nonlow low2 array computable set. Such sets were always known to exist, but also never constructed directly in any publication.