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Parameter Definability in the Recursively Enumerable Degrees
"... 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 defin ..."
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Cited by 36 (13 self)
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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...
Every Set Has a Least Jump Enumeration
 Journal of the London Mathematical Society
, 1998
"... 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 ..."
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Cited by 12 (0 self)
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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
Relative Enumerability in the Difference Hierarchy
 J. Symb. Logic
"... We show that the intersection of the class of 2REA 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 ..."
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Cited by 3 (1 self)
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We show that the intersection of the class of 2REA 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...
Iterated Relative Recursive Enumerability
, 2003
"... A result of Soare and Stob asserts that for any nonrecursive r.e. set C, there exists a r.e.[C] set A such that A⊕C is not of r.e. degree. A set Y is called [of] mREA (mREA[C]) [degree] iff it is [Turing equivalent to] the result of applying mmany iterated ‘hops ’ to the empty set (to C), where ..."
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Cited by 2 (0 self)
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A result of Soare and Stob asserts that for any nonrecursive r.e. set C, there exists a r.e.[C] set A such that A⊕C is not of r.e. degree. A set Y is called [of] mREA (mREA[C]) [degree] iff it is [Turing equivalent to] the result of applying mmany iterated ‘hops ’ to the empty set (to C), where a hop is any function of the form X ↦ → X ⊕ W X e. The cited result is the special case m = 0, n = 1 of our Theorem. For m = 0, 1, and any (m + 1)REA set C, if C is not of mREA degree, then for all n there exists a nr.e.[C] set A such that A ⊕ C is not of (m + n)REA degree.
Isolation in Higher Degree Types
, 1999
"... Examining various kinds of isolation phenomena in the Turing degrees, we show that there are, for every n ? 0, (n + 1)c.e. sets isolated in the nc.e. degrees by nc.e. sets below them. We show that for n 1 such phenomena arise below any computably enumerable degree, and conjecture that this r ..."
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Examining various kinds of isolation phenomena in the Turing degrees, we show that there are, for every n ? 0, (n + 1)c.e. sets isolated in the nc.e. degrees by nc.e. sets below them. We show that for n 1 such phenomena arise below any computably enumerable degree, and conjecture that this result holds densely in the c.e. degrees as well. We show that in fact the basic result can be strengthened to produce isolation (n + 1)CEA sets isolated in the nCEA degrees by nCEA degrees below them. We also point out that there are properly (n+1)c.e. degrees which are not isolated in the nc.e. degrees. 1 Introduction Our interest is in relative computable enumerability  the relation that obtains between sets A and B when A is \Sigma B 1 definable. It is often particularly interesting to study those sets which are computably enumerable in and above a given set, since these are the result of considering the lower set to actually define computability. Rather than study the en...
Interpolating dr. e. and REA degrees between r. e. degrees
, 1995
"... This paper is a contribution to the investigation of the relationship between the ..."
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This paper is a contribution to the investigation of the relationship between the
The Isolated D.r.e Degrees Are Dense in the R.e. Degrees
"... .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 2REA: if B = W \Gamma V with W and V both r.e. sets and h is any onetoone onto recursive function from ! to W , then it is straightforward to show B is ..."
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.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 2REA: if B = W \Gamma V with W and V both r.e. sets and h is any onetoone 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