Index sets and parametric reductions

Abstract.

We investigate the index sets associated with the degree structures of computable sets under the parameterized reducibilities introduced by the authors. We solve a question of Peter Cholakand the first author by proving the fundamental index sets associated with a computable set A, {e : W _e ≤ _q ^u A} for q∈ {m, T} are Σ₄ ⁰ complete. We also show hat FPT(≤ _q ⁿ), that is {e : W _e computable and ≡ _q ⁿ?}, is Σ₄ ⁰ complete. We also look at computable presentability of these classes.