On some sets of dictionaries whose ω -powers have a given

Abstract

A dictionary is a set of finite words over some finite alphabet X. The ω-power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V . Lecomte studied in [10] the complexity of the set of dictionaries whose associated ω-powers have a given complexity. In particular, he considered the setsk -sets, Borel sets). In this paper we first establish a new relation between the sets W(Σ 0 2 ) and W(Δ 1 1 ), showing that the set W(Δ 1 1 ) is "more complex" than the set W(Σ 0 2 ). As an application we improve the lower bound on the complexity of W(Δ 1 1 ) given by Lecomte, showing that W(Δ 1 1 ) is in Σ 1 2 (2 2 )\ Π 0 2 . Then we prove that, for every integer k ≥ 2 (respectively, k ≥ 3), the set of dictionaries W(Π 0 k+1 ) (respectively, W(Σ 0 k+1 )) is "more complex" than the set of dictionaries W(Π 0 k ) (respectively, W(Σ 0 k )) .

Descriptive Set Theory and $ω$-Powers of Finitary Languages

“…) is "more complex" than the set L Π 0 k (respectively, L Σ 0 k ), with respect to the Wadge reducibility. The following result is proved in [Lec05,Fin10].…”

“…A consequence of Theorem 20 is that these sets are Π 1 1 -hard if ξ ≥ 3 (see Corollary 6.4 in [FL09]). It is proved in [Fin10] that for every integer k ≥ 2 (respectively, k ≥ 3) the set L Π 0 k+1 (respectively, L Σ 0 k+1…”