Well Quasiorders and Hierarchy Theory

TL;DR

This paper explores the application of well quasiorders (WQOs) to hierarchy and reducibility classifications across Descriptive Set Theory, Computability, and Automata Theory, highlighting extensions to functions and open research questions.

Contribution

It surveys existing results on WQOs in hierarchy theory and discusses open problems and future research directions in the field.

Findings

WQOs are crucial for classifying hierarchies in various theoretical fields.

Extensions of hierarchies to functions require more complex WQOs.

The paper identifies open problems and potential research avenues.

Abstract

We discuss some applications of WQOs to several fields were hierarchies and reducibilities are the principal classification tools, notably to Descriptive Set Theory, Computability theory and Automata Theory. While the classical hierarchies of sets usually degenerate to structures very close to ordinals, the extension of them to functions requires more complicated WQOs, and the same applies to reducibilities. We survey some results obtained so far and discuss open problems and possible research directions.