A Survey of Muchnik and Medvedev degrees

TL;DR

This paper provides a comprehensive survey of the theory of Muchnik and Medvedev degrees, focusing on their structure, properties, and key results, especially for b1^0_1 subsets of Baire space.

Contribution

It offers a detailed overview of the degrees of b1^0_1 sets, including proofs of major results, some of which are more complete than previous presentations.

Findings

Characterization of Muchnik and Medvedev degrees

Analysis of degrees of b1^0_1 subsets of Baire space

Presentation of key proofs in the theory

Abstract

We survey the theory of Muchnik (weak) and Medvedev (strong) degrees of subsets of ${}^\omega\omega$ with particular attention to the degrees of $\Pi^0_1$ subsets of ${}^\omega2$. Later sections present proofs, some more complete than others, of the major results of the subject.