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 : We≤quA} for q∈ {m, T} are Σ40 complete. We also show hat FPT(≤qn), that is {e : We computable and ≡qn?}, is Σ40 complete. We also look at computable presentability of these classes.