Complexity Aspects of the Extension of Wagner's Hierarchy to $k$-Partitions

TL;DR

This paper constructs a quadratic algorithm to decide a preorder relation on iterated posets of $\omega$-languages and discusses the size of the representation of regular $\omega$-languages and suggests a more compact way to represent them.

Abstract

It is known that the Wadge reducibility of regular $ω$-languages is efficiently decidable (Krishnan et al., 1995), (Wilke, Yoo, 1995). In this paper we study analogous problem for regular k-partitions of $ω$-languages. In the series of previous papers (Selivanov, 2011), (Alaev, Selivanov, 2021), (Selivanov, 2012) there was a partial progress towards obtaining an efficient algorithm for deciding the Wadge reducibility in this setting as well. In this paper we finalize this line of research providing a quadratic algorithm (in RAM model). For this we construct a quadratic algorithm to decide a preorder relation on iterated posets. Additionally, we discuss the size of the representation of regular $ω$-languages and suggest a more compact way to represent them. The algorithm we provide is efficient for the more compact representation as well.

Theorems & Definitions (14)

• Lemma 1
• Theorem 2
• proof
• Claim 1
• proof : Proof of the claim
• Theorem 3
• proof
• Lemma 4
• Lemma 5: bhkk08
• proof : Proof of Lemma \ref{['lem:number-of-cycles']}