Total Recursion over Lexicographical Orderings: Elementary Recursive Operators Beyond PR

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.