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arXiv · 2510.18283

A Characterization of Turing Machines that Compute Primitive Recursive Functions

Daniel G. Schwartz
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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.

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Daniel G. Schwartz. 2025-10-21. A Characterization of Turing Machines that Compute Primitive Recursive Functions. https://arxiv.org/abs/2510.18283

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