Nonexistence of Minimal Pairs for Generic Computability

TL;DR

This paper proves that there are no minimal pairs in the context of generic computability, resolving a question in recursion theory and revealing counterintuitive properties of partial computation methods.

Contribution

It establishes the nonexistence of minimal pairs for generic computability, a significant theoretical result in recursion theory and related fields.

Findings

No minimal pairs exist for generic computability.

The result answers a question posed by Jockusch and Schupp.

The analysis reveals counterintuitive aspects of partial computation.

Abstract

A generic computation of a subset A of the natural numbers consists of a a computation that correctly computes most of the bits of A, and which never incorrectly computes any bits of A, but which does not necessarily give an answer for every input. The motivation for this concept comes from group theory and complexity theory, but the purely recursion theoretic analysis proves to be interesting, and often counterintuitive. The primary result of this paper is that there are no minimal pairs for generic computability, answering a question of Jockusch and Schupp.