A Rice-like theorem for primitive recursive functions

TL;DR

This paper extends Rice's theorem to primitive recursive functions, providing a characterization of decidable properties based on primitive recursive indices, applicable to broader classes of total computable functions.

Contribution

It offers a generalized Rice-like theorem for primitive recursive functions and c.e. classes of total computable functions, expanding the theoretical understanding of decidability.

Findings

Characterization of decidable properties of primitive recursive functions

Extension of Rice's theorem to c.e. classes of total functions

Applicable to broader classes of total computable functions

Abstract

We provide an explicit characterization of the properties of primitive recursive functions that are decidable or semi-decidable, given a primitive recursive index for the function. The result is much more general as it applies to any c.e. class of total computable functions. This is an analog of Rice and Rice-Shapiro theorem, for restricted classes of total computable functions.