 Definition 3.8: Given a set \(A \subseteq \mathbb{N}\), we define the class of \(A\)-partial recursive functions \(\mathbf{PartREC}^A\) to be the smallest class of partial functions containing the initial functions \(I_A = \{\mathbf{0},s,\pi^i_k,\chi_A(x)\}\) and closed under the functionals