Invertible Classes

Abstract

This paper considers when one can invert general recursive operators which map a class of functions $\mathcal{F}$ to $\mathcal{F}$. In this regard, we study four different notions of inversion. We additionally consider enumeration of operators which cover all general recursive operators which map $\mathcal{F}$ to $\mathcal{F}$ in the sense that for every general recursive operator Ψ mapping $\mathcal{F}$ to $\mathcal{F}$, there is a general recursive operator in the enumerated sequence which behaves the same way as Ψ on $\mathcal{F}$. Three different possible types of enumeration are studied.