Total Recursion over Lexicographical Orderings: Elementary Recursive Operators Beyond PR

TL;DR

This paper introduces a hierarchy of total recursive operators called leveled primitive recursion, extending primitive recursion to higher orders and exploring their expressive power, including the ability to formalize the Ackermann function.

Contribution

It generalizes primitive recursion into a hierarchy of operators, showing that higher levels can formalize more complex functions like Ackermann's, and establishing initial relationships between hierarchy levels.

Findings

PR1 is equivalent to primitive recursion.

PR2 is a conservative extension of PR1.

Hierarchy beyond PR2 can formalize Ackermann function.

Abstract

In this work we generalize primitive recursion in order to construct a hierarchy of terminating total recursive operators which we refer to as {\em leveled primitive recursion of order $i$}($\mathbf{PR}_{i}$). Primitive recursion is equivalent to leveled primitive recursion of order $1$ ($\mathbf{PR}_{1}$). The functions constructable from the basic functions make up $\mathbf{PR}_{0}$. Interestingly, we show that $\mathbf{PR}_{2}$ is a conservative extension of $\mathbf{PR}_{1}$. However, members of the hierarchy beyond $\mathbf{PR}_{2}$, that is $\mathbf{PR}_{i}$ where $i\geq 3$, can formalize the Ackermann function, and thus are more expressive than primitive recursion. It remains an open question which members of the hierarchy are more expressive than the previous members and which are conservative extensions. We conjecture that for all $i\geq 1$ $\mathbf{PR}_{2i} \subset \mathbf{PR}_{2i+1}$. Investigation of further extensions is left for future work.