# A survey on difference hierarchies of regular languages

## TL;DR

This survey explores the structure and properties of difference hierarchies within regular languages, highlighting decidability results and classical techniques adapted from descriptive set theory.

## Contribution

It provides a comprehensive overview of difference hierarchies in regular languages, including new decidability results for specific subclasses.

## Key findings

- Decidability results for difference hierarchies based on shuffle ideals
- Decidability for strongly cyclic regular languages
- Decidability for polynomial closure of group languages

## Abstract

Difference hierarchies were originally introduced by Hausdorff and they play an important role in descriptive set theory. In this survey paper, we study difference hierarchies of regular languages. The first sections describe standard techniques on difference hierarchies, mostly due to Hausdorff. We illustrate these techniques by giving decidability results on the difference hierarchies based on shuffle ideals, strongly cyclic regular languages and the polynomial closure of group languages.