Wadge Degrees of Classes of omega-Regular k-Partitions

TL;DR

This paper develops a comprehensive theory of Wadge degrees for omega-regular k-partitions, enabling topological classification, decidability results, and complexity analysis for automata-recognizable infinite word classes.

Contribution

It introduces a new framework for analyzing the Wadge degrees of omega-regular k-partitions, completing topological classifications and establishing decidability and complexity results.

Findings

Characterized the structure of Wadge degrees for omega-regular k-partitions

Proved the decidability of related classification problems

Analyzed the complexity of these problems

Abstract

We develop a theory of k-partitions of the set of infinite words recognizable by classes of finite automata. The theory enables to complete proofs of existing results about topological classifications of the (aperiodic) omega-regular k-partitions and provides tools for dealing with other similar questions. In particular, we characterize the structure of Wadge degrees of (aperiodic) omega-regular $k$-partitions, prove the decidability of many related problems, and discuss their complexity.