The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that...

Given a computably enumerable set B; there is a Turing degree which is the least jump of any set in which B is computably enumerable, namely 0 : Remarkably, this is not a phenomenon of computably enumerable sets. We show that for every subset A of N; there is a Turing degree, c (A); which is the least degree of the jumps of all sets X for which A is \Sigma 1 (X): 1

The notion of a recursively enumerable (r.e.) set, i.e. a set of integers whose members can be e ectively listed, is a fundamental one. Another way of

We show that the intersection of the class of 2-REA degrees with that of the #-r.e. degrees consists precisely of the class of d.r.e. degrees. We also include some applications and show that there is no natural generalization of this result to higher levels of the REA hierarchy. 1 Introduction The # 0 2 degrees of unsolvability are basic objects of study in classical recursion theory, since they are the degrees of those sets whose characteristic functions are limits of recursive functions. A natural tool for understanding the Turing degrees is the introduction of hierarchies to classify various kinds of complexity. Because of its coarseness, the most common such hierarchy, the arithmetical hierarchy, is itself not of much use in the classification of the # 0 2 degrees. This fact leads naturally to the consideration of hierarchies based on finer distinctions than quantifier alternation. Two such hierarchies are by now well established. One, the REA hierarchy defined by Jockusch and S...

This paper is a contribution to the investigation of the relationship between the

.e. degree. Note that an isolated d.r.e. degree must be properly d.r.e., that is, it cannot be of r.e. degree. Recall that the d.r.e. degrees are 2-REA: if B = W \Gamma V with W and V both r.e. sets and h is any one-to-one onto recursive function from ! to W , then it is straightforward to show B is recursively enumerable in and above h \Gamma1 [V T W ]. While the degree of h \Gamma1 [V T W ] depends on the particular representation using W and V , it is independent of the enumerating function h, so by a slight abuse of notation we write ~ B for h \Gamma1 [V T W ] whenev

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