Relative computability of Q_r for I1–I4 modulo 32

Determine the relative computability, modulo Kleene’s quantifier 32, among the partial subfunctionals Q_r that decide non-emptiness for the following families: I1 (closed subsets of Baire space NN, alternatively Gδ subsets of [0,1], R, or 2^N), I2 (Fσ subsets of [0,1], R, or 2^N, alternatively σ-compact subsets of NN), I3 (countable subsets), and I4 (countable closed subsets). Specifically, establish which Q_r functionals compute one another (given 32) beyond what is implied by set-theoretic domain inclusions.

Background

The authors introduce a general scheme: for a family I of subsets of a space, Q_r is the partial subfunctional of 33 that decides whether X∈I is empty or not. They list four concrete families I1–I4 spanning closed/Gδ, Fσ/σ-compact, countable, and countable closed sets, respectively.

Beyond trivial implications from inclusions among these domains, the paper states that no results are currently known about how these Q_r relate computationally (modulo 32), highlighting a gap in our understanding of the landscape of partial subfunctionals of 33. The authors observe that Q_r1 interacts strongly with the Suslin functional, while Q_r2 is lame, suggesting rich structure yet to be charted.