A Characterization of Turing Machines that Compute Primitive Recursive Functions

TL;DR

The paper investigates when a Turing-computable function $f: \mathbb{N}^n \to \mathbb{N}$ is primitive recursive based on whether the machine's time complexity is bounded by a primitive recursive function. Using a direct approach grounded in Hermes' bounded $\mu$-operator and Kleene's normal form, it proves that time-boundedness by a PR function implies $f$ is primitive recursive and conversely that PR functions have PR-bounded time, with explicit constructions. It further derives that the satisfiability problem (SAT) is primitive recursive and that every problem in $NP$ corresponds to a primitive recursive function, thereby extending the characterization to these fundamental computational classes. The work consolidates the link between TM time bounds and primitive recursive computation, delivering thorough proofs intended for archival permanence and broad accessibility, and clarifies the role of bounded minimization in achieving primitive recursivity.

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.