A Note on Recursively Enumerable Classes of Partial Recursive Functions

A class F of partial recursive functions is called recursively enumerable if there exists an r.e. set J ⊆ N such that F = {φi | i ∈ J}. We prove that every r.e. class F of partial recursive functions with infinite domains must have a recursive witness array, i.e. there is a computable array of finite sets X = [Xn]n∈ω such that (i) for every f ∈ F one has f(n) ∈ Xn for infinitely many n and (ii) Xn = ∅ for infinitely many n. The result gives a powerful diagonalisation tool for proving properties of r.e. classes. We show for example that no r.e. class of partial functions with infinite domains can contain all recursive involutions or all cyclefree recursive permutations. Keyword and phrases: Recursively enumerable classes, Rice-Shapiro theorem, recursive witness arrays, recursive permutations.