A weak-2 generic that bounds a minimal degree
Prof. Satyadev Nandakumar
31st January 2019
The downward density of a class of Turing degrees is the following
property: given two Turing degrees a and b from that class such that a is less than
b, there is a Turing degree c of that class such that a is less than c and c is less than b? If the
class of sets computes a minimal degree, then density does not hold.
In 1980, Carl Jokusch showed that the class of sufficiently generic sets
is dense by establishing that 2-generics are downward dense below a given
2-generic. In 1990 Kumabe, and independently, Chong and Downey established
that 1-generics are not dense, by constructing a minimal degree computable
from a 1-generic. For a well-studied intermediate notion, weakly
2-generics, the density question was open.
We settle the question by showing that there is a minimal degree
computable from a weakly 2-generic. The proof uses a technique from
recursion theory called full approximation.
(This is joint work with Rod Downey.)