Computability of Qc from 22b and 32

Determine whether the functional Qc, which decides non-emptiness of closed subsets of Cantor space 2^N (i.e., Qc(X)=1 if X is a non-empty closed subset of 2^N and Qc(X)=0 if X is empty), is computable from the combination of the functional 22b and Kleene’s quantifier 32 within Kleene’s S1–S9 framework. Here, 22b is the partial functional that is defined exactly on subsets X of 2^N with at most one element and returns 1 if X is non-empty and 0 otherwise, and 32 is the type-2 quantifier returning 0 exactly when (∃n ∈ N) f(n)=0 for an input f: N→N.

Background

Section 4 formulates open problems about partial subfunctionals of Kleene’s 33 that generalize Qc and 22. The authors single out a crucial question about whether Qc can be computed from 22b together with Kleene’s quantifier 32, noting they were unable to resolve it in the present work.

The paper defines Qc as the non-emptiness tester for closed subsets of 2^N and 22b as the non-emptiness tester restricted to sets with at most one element. Throughout the paper, 32 is assumed available as it is readily obtained from discontinuous functions via Grilliot’s trick. The authors conjecture that Qc is not computable from 22b and 32, but leave the problem open.