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A WEAKLY 2-GENERIC WHICH BOUNDS A MINIMAL DEGREE

Published online by Cambridge University Press:  04 November 2019

RODNEY G. DOWNEY  and
SATYADEV NANDAKUMAR
RODNEY G. DOWNEY
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY WELLINGTON P. O. BOX 600, WELLINGTON, NEW ZEALAND E-mail: [email protected]
SATYADEV NANDAKUMAR
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY KANPUR KANPUR208016, UTTAR PRADESH, INDIA E-mail: [email protected]
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Abstract

Jockusch showed that 2-generic degrees are downward dense below a 2-generic degree. That is, if a is 2-generic, and $0 < {\bf{b}} < {\bf{a}}$, then there is a 2-generic g with $0 < {\bf{g}} < {\bf{b}}.$ In the case of 1-generic degrees Kumabe, and independently Chong and Downey, constructed a minimal degree computable from a 1-generic degree. We explore the tightness of these results.

We solve a question of Barmpalias and Lewis-Pye by constructing a minimal degree computable from a weakly 2-generic one. While there have been full approximation constructions of ${\rm{\Delta }}_3^0$ minimal degrees before, our proof is rather novel since it is a computable full approximation construction where both the generic and the minimal degrees are ${\rm{\Delta }}_3^0 - {\rm{\Delta }}_2^0$.

Keywords

Primary 03D28Secondary 03D3003D55Turing degreeminimal degreeweakly 2-generic
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 4 , December 2019 , pp. 1326 - 1347
Copyright
Copyright © The Association for Symbolic Logic 2019 

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