AsetAof nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E =({We}e∈ω,⊆). Most previously known automorphisms Φ of the structure E of

A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E = (fW e g e2! ; `). We previously exhibited a first order E-definable property Q(X) such that Q(X) guarantees that X is not Turing complete (i.e., does not code complete information about c.e. sets). Here we show first that Q(X) implies that X has a certain "slowness " property whereby the elements must enter X slowly (under a certain precise complexity measure of speed of computation) even though X may have high information content. Second we prove that every X with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable A 2 E there exists B in the orbit of A such that X T B under relative Turing computability ( T ). We produce B using the \Delta 0 3 -automorphism method we introduced earli...

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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...

Unter den Linden 6

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. The following theorems on the structure inside nonrecursive truthtable degrees are established: Degtev's result that the number of bounded truth-table degrees inside a truth-table degree is at least two is improved by showing that this number is infinite. There are even infinite chains and antichains of bounded truth-table degrees inside the truth-table degrees which implies an affirmative answer to a question of Jockusch whether every truthtable degree contains an infinite antichain of many-one degrees. Some but not all truth-table degrees have a least bounded truth-table degree. The technique to construct such a degree is used to solve an open problem of Beigel, Gasarch and Owings: there are Turing degrees (constructed as hyperimmunefree truth-table degrees) which consist only of 2-subjective sets and do therefore not contain any objective set. Furthermore a truth-table degree consisting of three positive degrees is constructed where one positive degree consists of enum...

We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. Gerald Sacks has had a major influence on the development of logic, particularly recursion theory, over the past thirty years through his research, writing and teaching. Here, I would like to concentrate on just one instance of that influence that I feel has been of special significance to the study of the degrees of unsolvability in general and on my own work in particular--- the conjectures and questions posed at the end of the two editions of Sacks's first book, the classic monograph Degrees of Unsolvability (Annals

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Abstract. The goal of this paper is to show there is a single orbit of the c.e. sets with inclusion, E, such that the question of membership in this orbit is Σ1 1-complete. This result and proof have a number of nice corollaries: the Scott rank of E is ωCK 1 + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of E; for all finite α ≥ 9, there is a properly ∆0 α orbit (from the proof). 1.