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

TL;DR

The paper proves that the first-order theory of the upper semilattice $\mathbf{C}^{pr}_m$ of degrees of computable sets under primitive recursive $m$-reducibility is hereditarily undecidable. It introduces a modified notion of super $\mathrm{pr}$-sparse sets and shows that, for such a set $A$, the associated Boolean algebra $\mathcal{B}(\mathbf{a})$ is effectively dense; it then constructs an interpretation of the lattice $\mathcal{I}(\mathcal{B}(\mathbf{a}))$ inside $\mathbf{C}^{pr}_m$ using a main lemma. The argument relies on embedding an undecidable structure into $\mathbf{C}^{pr}_m$ and applying Nies’ theorem on effectively dense $\Sigma^0_n$-Boolean algebras, yielding hereditary undecidability of $Th(\mathbf{C}^{pr}_m)$. The results bridge primitive recursive reductions with lattice theory to establish undecidability in a computability-theoretic degree framework.

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.

Theorems & Definitions (24)

• Lemma 1.1
• Theorem 1
• Lemma 1.2: folklore
• proof
• Definition 2.1: Nies-97
• Theorem 2: Nies Nies-97
• Definition 3.1
• Proposition 3.2
• Lemma 3.3
• proof