 Since \(\mu y R(\vec{x},y) \simeq \mu y \chi_{\neg R}(\vec{x},y)\), it is easy to see that the class \(\mathbf{PartREC}\) can also be obtained by closing the primitive recursive functions under the operation defined by (\ref{unboundedminrel}). One consequence of  Theorem 2.3  is thus that it is indeed possible to define any \(k\)-ary partial recursive function \(f(\vec{x})\) by a single application of unbounded search applied to the relation \(T_k(e,\vec{x},s)\) for an appropriate choice of \(e\). More generally, the Normal Form Theorem illustrates how any such function may be defined from a single relation \(T_k(e,\vec{x},s)\) wherein the value of \(e\) serves as a description of the manner in which \(f(\vec{x})\) is defined in terms of the basis functions \(I_{\mathbf{PR}}\) and the operations \(Op_{\mathbf{PartRec}}\). Such an \(e\) is known as an index for \(f(\vec{x})\). As we will see in  Section 3,  the availability of such indices is one of the central features of the partial recursive functions which allows them to provide the basis for a general theory of computability and non-computability.