On Some Sets of Dictionaries Whose omega-Powers Have a Given Complexity

TL;DR

This paper investigates the descriptive set-theoretic complexity of sets of dictionaries whose omega-powers have specific topological complexities, establishing new relations and bounds among these complexity classes.

Contribution

It introduces a new relation showing that the set of dictionaries with Borel omega-powers is more complex than those with simpler classes, and improves lower bounds on their complexity.

Findings

W(Δ₁¹) is more complex than W(Σ₂⁰)

Lower bounds on the complexity of W(Δ₁¹) are improved

Complexity increases with k for W(Πₖ⁰) and W(Σₖ⁰) sets

Abstract

A dictionary is a set of finite words over some finite alphabet X. The omega-power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [Omega-powers and descriptive set theory, JSL 2005] the complexity of the set of dictionaries whose associated omega-powers have a given complexity. In particular, he considered the sets $W({\bf\Si}^0_{k})$ (respectively, $W({\bf\Pi}^0_{k})$, $W({\bf\Delta}_1^1)$) of dictionaries $V \subseteq 2^\star$ whose omega-powers are ${\bf\Si}^0_{k}$-sets (respectively, ${\bf\Pi}^0_{k}$-sets, Borel sets). In this paper we first establish a new relation between the sets $W({\bf\Sigma}^0_{2})$ and $W({\bf\Delta}_1^1)$, showing that the set $W({\bf\Delta}_1^1)$ is "more complex" than the set $W({\bf\Sigma}^0_{2})$. As an application we improve the lower bound on the complexity of $W({\bf\Delta}_1^1)$ given by Lecomte. Then we prove that, for every integer $k\geq 2$, (respectively, $k\geq 3$) the set of dictionaries $W({\bf\Pi}^0_{k+1})$ (respectively, $W({\bf\Si}^0_{k+1})$) is "more complex" than the set of dictionaries $W({\bf\Pi}^0_{k})$ (respectively, $W({\bf\Si}^0_{k})$) .