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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 definabl ..."
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Cited by 34 (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...
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
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On the Distribution of Lachlan Nonsplitting Bases
 Arch. for Math. Logic
, 1998
"... We say that a computably enumerable (c.e.) degree b is a Lachlan nonsplitting base (LNB), if there is a computably enumerable degree a such that a > b, and for any c.e. degrees w, v _< a, if a _< w V v V b then either a _< w V b or a _< v V b. In this paper we investigate the relationship betwee ..."
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Cited by 2 (1 self)
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We say that a computably enumerable (c.e.) degree b is a Lachlan nonsplitting base (LNB), if there is a computably enumerable degree a such that a > b, and for any c.e. degrees w, v _< a, if a _< w V v V b then either a _< w V b or a _< v V b. In this paper we investigate the relationship between bounding and nonbounding of Lachlan nonsplitting bases and the high/low hierarchy. We prove that there is a noraLow2 c.e. degree which bounds no Lachlan nonsplitting base.
Splitting and Nonsplitting, II: A Low_2 C.E. Degree Above Which 0' Is Not Splittable
, 2001
"... It is shown that there exists a low 2 Harrington nonsplitting base  that is, a low 2 computably enumerable (c.e.) degree a such that for any c.e. degrees x, y, if 0 # = x # y, then either 0 # = x # a or 0 # = y # a. Contrary to prior expectations, the standard Harrington nonsplitting con ..."
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Cited by 2 (2 self)
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It is shown that there exists a low 2 Harrington nonsplitting base  that is, a low 2 computably enumerable (c.e.) degree a such that for any c.e. degrees x, y, if 0 # = x # y, then either 0 # = x # a or 0 # = y # a. Contrary to prior expectations, the standard Harrington nonsplitting construction is incompatible with the low 2 ness requirements to be satisfied, and the proof given involves new techniques with potentially wider application. 1
R.E. Degree with the Extension of Embeddings Properties of a Low Degree
"... We construct a nonlow 2 r.e. degree d such that every extension of embedding property that holds below every low 2 degree holds below d. Indeed, we can also guarantee the converse so that there is a low r.e. degree c such that that the extension of embedding properties true below c are exactly th ..."
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Cited by 1 (1 self)
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We construct a nonlow 2 r.e. degree d such that every extension of embedding property that holds below every low 2 degree holds below d. Indeed, we can also guarantee the converse so that there is a low r.e. degree c such that that the extension of embedding properties true below c are exactly the ones true below d. Moreover, we can also guarantee that no b # d is the base of a nonsplitting pair. 1 Introduction Our goal in this paper is to show how constructions establishing properties shared by all low 2 r.e. degrees (i.e. ones c such that c ## = 0 ## ) can be modified to construct a nonlow 2 degree with the same properties. This implies that none of the properties considered (nor indeed all of them together) can separate the low r.e. degrees from the nonlow 2 ones. In particular, none of them can define the low 2 r.e. degrees inside the structure R of all the r.e. degrees. As examples, we consider the properties established for all low 2 r.e. degrees in Shore and Slaman [1...