Levels of uniformity

TL;DR

This paper introduces a hierarchy of degree structures between Medvedev and Muchnik lattices to quantify the uniformity of reductions, explores their algebraic properties, and compares their logical theories.

Contribution

It presents a new hierarchy of degree structures for measuring non-uniformity in reductions and analyzes their algebraic and logical properties.

Findings

Defined the notion of uniformity for Muchnik reductions

Studied the propositional theories of the new structures

Compared elementary equivalence of the structures

Abstract

We introduce a hierarchy of degree structures between the Medvedev and Muchnik lattices which allow varying amounts of non-uniformity. We use these structures to introduce the notion of the uniformity of a Muchnik reduction, which expresses how uniform a reduction is. We study this notion for several well-known reductions from algorithmic randomness. Furthermore, since our new structures are Brouwer algebras, we study their propositional theories. Finally, we study if our new structures are elementarily equivalent to each other.