A note on the degree structure of primitive recursive m-reducibility

TL;DR

This paper proves that the first-order theory of the degree structure of primitive recursive m-reducibility is hereditarily undecidable, highlighting complex logical properties of this computational hierarchy.

Contribution

It establishes the hereditarily undecidable nature of the first-order theory of the degree structure under primitive recursive m-reducibility, a novel result in computability theory.

Findings

First-order theory of $C^{pr}_m$ is hereditarily undecidable.

Degree structure forms an upper semilattice.

Results contribute to understanding the logical complexity of computational degrees.

Abstract

Let $C^{pr}_m$ be the upper semilattice of degrees of computable sets with respect to primitive recursive $m$-reducibility. We prove that the first-order theory of $C^{pr}_m$ is hereditarily undecidable.