Abstract

We define and study a new notion called k-immunity that lies between immunity and hyperimmunity in strength. Our interest in k-immunity is justified by the result that # # does not k-tt reduce to a k-immune set, which improves a previous result by Kobzev [7, 13]. We apply the result to show that # # does not btt-reduce to MIN, the set of minimal programs. Other applications include the set of Kolmogorov random strings, and retraceable and regressive sets. We also give a new characterization of e#ectively simple sets and show that simple sets are not btt-cuppable. Keywords: Computability, Recursion Theory, bounded reducibilities, minimal programs, immunity, k-immune, regressive, retraceable, e#ectively simple, cuppable. 1 Introduction There seems to be a large gap between immunity and hyperimmunity (h-immunity) that is waiting to be filled. What happens, one wonders if the disjoint strong arrays that try to witness that a set is not h-immune are subjected to additional conditions...

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