A minimal pair in the generic degrees
Denis R. Hirschfeldt
TL;DR
This paper proves the existence of a minimal pair in the nonuniform and uniform generic degrees, contrasting with previous results and addressing a fundamental structural question in the field.
Contribution
It establishes the existence of a minimal pair in the generic degrees, providing new insights into their structure and resolving an open question.
Findings
Existence of a minimal pair in nonuniform generic degrees
Existence of a minimal pair in uniform generic degrees
Contrasts with Igusa's result on relative generic computability
Abstract
We show that there is a minimal pair in the nonuniform generic degrees, and hence also in the uniform generic degrees. This fact contrasts with Igusa's result that there are no minimal pairs for relative generic computability, and answers a basic structural question mentioned in several papers in the area.
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