A Characterization of Turing Machines that Compute Primitive Recursive Functions
Abstract
This paper provides a new and more direct proof of the assertion that a Turing computable function of the natural numbers is primitive recursive if and only if the time complexity of the corresponding Turing machine is bounded by a primitive recursive function of the function's arguments. In addition, it provides detailed proofs of two consequences of this fact, which, although well-known in some circles, do not seem to have ever been published. The first is that the Satisfiability Problem, properly construed as a function of natural numbers, is primitive recursive. The second is a generalization asserting that all the problems in NP are similarly primitive recursive. The purpose here is to present these theorems, fully detailed, in an archival journal, thereby giving them a status of permanence and general availability.
Cite
@article{arxiv.2510.18283,
title = {A Characterization of Turing Machines that Compute Primitive Recursive Functions},
author = {Daniel G. Schwartz},
journal= {arXiv preprint arXiv:2510.18283},
year = {2025}
}