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 θ does not k-tt reduce to a k-immune set, which improves a previous result by Kobzev [7]. We apply the result to show that Φ′ 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 effectively simple sets and show that simple sets are not btt-cuppable