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15
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...
Array Nonrecursive Degrees and Genericity
 London Mathematical Society Lecture Notes Series 224
, 1996
"... A class of r.e. degrees, called the array nonrecursive degrees, previously studied by the authors in connection with multiple permitting arguments relative to r.e. sets, is extended to the degrees in general. This class contains all degrees which satisfy a (i.e. a 2 GL 2 ) but in addition ..."
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Cited by 24 (7 self)
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A class of r.e. degrees, called the array nonrecursive degrees, previously studied by the authors in connection with multiple permitting arguments relative to r.e. sets, is extended to the degrees in general. This class contains all degrees which satisfy a (i.e. a 2 GL 2 ) but in addition there exist low r.e. degrees which are array nonrecursive (a.n.r.).
The theory of the degrees below 0
 J. London Math. Soc
, 1981
"... Degree theory, that is the study of the structure of the Turing degrees (or degrees of unsolvability) has been divided by Simpson [24; §5] into two parts—global and local. By the global theory he means the study of general structural properties of 3d— the degrees as a partially ordered set or uppers ..."
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Cited by 18 (6 self)
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Degree theory, that is the study of the structure of the Turing degrees (or degrees of unsolvability) has been divided by Simpson [24; §5] into two parts—global and local. By the global theory he means the study of general structural properties of 3d— the degrees as a partially ordered set or uppersemilattice. The local theory concerns
1995], Degree theoretic definitions of the low 2 recursively enumerable sets
 J. Symbolic Logic
, 1995
"... 1. Introduction. The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤T B, where arbitrary (Turing) mac ..."
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Cited by 7 (5 self)
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1. Introduction. The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤T B, where arbitrary (Turing) machines, ϕe, can be used; access to
Local initial segments of the Turing degrees
 Bull. Symbolic Logic
, 2002
"... Abstract. Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of nontrivial automorphisms of the Turing degrees are indicated. §1. Introduction. This article concerns the algebraic study of the u ..."
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Cited by 6 (2 self)
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Abstract. Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of nontrivial automorphisms of the Turing degrees are indicated. §1. Introduction. This article concerns the algebraic study of the upper semilattice of Turing degrees. Upper semilattices of interest in this regard tend to have a least element, hence for convenience the following definition is made. Definition 1.1. A unital semilattice (usl) is a structure L = (L, ∗, e) satisfying the following equalities for all a, b, c ∈ L.
Embedding Lattices with Top Preserved Below NonGL2 Degrees
, 1997
"... this paper, we answer this question by showing that every recursively presented lattice can be embedded into D (0 ..."
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Cited by 5 (1 self)
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this paper, we answer this question by showing that every recursively presented lattice can be embedded into D (0
Generalized high degrees have the complementation property
 Journal of Symbolic Logic
"... Abstract. We show that if d ∈ GH1 then D( ≤ d) has the complementation property, i.e. for all a < d there is some b < d such that a ∧ b = 0 and a ∨ b = d. §1. Introduction. A major theme in the investigation of the structure of the Turing degrees, (D, ≤T), has been the relationship between the order ..."
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Cited by 3 (0 self)
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Abstract. We show that if d ∈ GH1 then D( ≤ d) has the complementation property, i.e. for all a < d there is some b < d such that a ∧ b = 0 and a ∨ b = d. §1. Introduction. A major theme in the investigation of the structure of the Turing degrees, (D, ≤T), has been the relationship between the order theoretic properties of a degree and its complexity of definition in arithmetic as expressed by the Turing jump operator which embodies a single step in the hierarchy of quantification. For example, there is a long history of results showing that 0 ′
Domination, forcing, array nonrecursiveness and relative recursive enumerability
, 2009
"... We present some abstract theorems showing how domination properties equivalent to not being GL2 or array recursive can be used to construct sets generic for different notions of forcing. These theorems are then applied to give simple proofs of several old results. We also give a direct uniform proof ..."
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Cited by 3 (3 self)
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We present some abstract theorems showing how domination properties equivalent to not being GL2 or array recursive can be used to construct sets generic for different notions of forcing. These theorems are then applied to give simple proofs of several old results. We also give a direct uniform proof of a recent result of AmbosSpies, Ding, Wang and Yu [2009] that every degree above any not in GL2 is recursively enumerable in a 1generic degree strictly below it. Our major new result is that every array nonrecursive degree is r.e. in some degree strictly below it. Our analysis of array nonrecursiveness and construction of generic sequences below ANR degrees also reveal a new level of uniformity in these types of results.
Jumps of minimal degrees below 0
 J. London Math. Soc
, 1996
"... Abstract. We show that there is a degree a REA in and low over 0 ′ such that no minimal degree below 0 ′ jumps to a degree above a. We also show that every nonlow r.e. degree bounds a nonlow minimal degree. Introduction. An important and longstanding area of investigation in recursion theory has be ..."
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Cited by 2 (0 self)
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Abstract. We show that there is a degree a REA in and low over 0 ′ such that no minimal degree below 0 ′ jumps to a degree above a. We also show that every nonlow r.e. degree bounds a nonlow minimal degree. Introduction. An important and longstanding area of investigation in recursion theory has been the relationship between quantifier complexity of the definitions of sets in arithmetic as expressed by the jump operator and the basic notion of relative computability as expressed by the ordering of the (Turing) degrees. In this paper we
Every 1generic computes a properly 1generic
 Journal of Symbolic Logic
"... Abstract. A real is called properly ngeneric if it is ngeneric but not n + 1generic. We show that every 1generic real computes a properly 1generic real. On the other hand, if m> n � 2 then an mgeneric real cannot compute a properly ngeneric real. ..."
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Cited by 1 (1 self)
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Abstract. A real is called properly ngeneric if it is ngeneric but not n + 1generic. We show that every 1generic real computes a properly 1generic real. On the other hand, if m> n � 2 then an mgeneric real cannot compute a properly ngeneric real.