Wadge degrees of $Δ^0_2$ omega-powers

TL;DR

This work determines the Wadge (Wagner) complexity of ω-powers $L^{inity}$ where $L$ is regular, within the Δ$^{0}_{2}$ realm. It provides a constructive method: for each $n$ and each Wagner/Lavrentieff class among $D_n({f riangle}^{0}_{1})$, $reve D_n({f riangle}^{0}_{1})$, and $D_{2n+1}({f riangle}^{0}_{1})rac{reve D_{2n+1}({f riangle}^{0}_{1})}{ }$, there exists a regular language $L$ with $L^{inity}$ complete for that class; similarly for $D_0({f riangle}^{0}_{1})rac{reve D_0({f riangle}^{0}_{1})}{ }$. The proof uses an inductive construction of the difference hierarchy via transformations $L ightarrow L_0, L ightarrow L_1, L ightarrow L_2$ and propagation lemmas to ascend levels, yielding new complete ω-powers and clarifying the Wadge-Wagner landscape for non-self-dual $oldsymbol{ riangle}^{0}_{2}$ regular ω-powers. These results connect the complexity of ω-powers with the finite-difference structure of open sets and highlight open questions about transfinite levels.

Abstract

We provide, for each natural number $n$ and each class among $D_n(Σ^0_1)$, $\bar D_n(Σ^0_1)$ and $D_{2n+1}(Σ^0_1)\oplus\bar D_{2n+1}(Σ^0_1)$, a regular language whose associated omega-power is complete for this class.

Theorems & Definitions (7)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Lemma 3.1
• Remark 3.2
• Lemma 3.3
• Lemma 3.4