A Characterization of Turing Machines that Compute Primitive Recursive Functions

TL;DR

This paper offers a new proof linking primitive recursive functions and Turing machine time complexity, and explores implications for problems like SAT and NP within this framework.

Contribution

It provides a direct proof of the equivalence between primitive recursive functions and bounded Turing machine time, and details consequences for SAT and NP problems.

Findings

Turing computable primitive recursive functions have bounded primitive recursive time complexity.

The Satisfiability Problem is primitive recursive as a function of natural numbers.

All problems in NP are primitive recursive.

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.