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19
Enumeration Reducibility, Nondeterministic Computations and Relative . . .
 RECURSION THEORY WEEK, OBERWOLFACH 1989, VOLUME 1432 OF LECTURE NOTES IN MATHEMATICS
, 1990
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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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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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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
Degree structures: Local and global investigations
 Bulletin of Symbolic Logic
"... $1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead. ..."
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$1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.
Completing Pseudojump Operators
 ANN. PURE AND APPL. LOGIC
, 1999
"... We investigate operators which take a set X to a set relatively computably enumerable in and above X by studying which such sets X can be so mapped into the Turing degree of K. We introduce notions of nontriviality for such operators, and use these to study which additional properties can be req ..."
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Cited by 5 (4 self)
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We investigate operators which take a set X to a set relatively computably enumerable in and above X by studying which such sets X can be so mapped into the Turing degree of K. We introduce notions of nontriviality for such operators, and use these to study which additional properties can be required of sets which can be completed to the jump in this way by given operators.
Double Jump Inversions and Strong Minimal Covers in the Turing Degrees
, 2004
"... Decidability problems for (fragments of) the theory of the structure D of Turing degrees, form a wide and interesting class, much of which is yet unsolved. Lachlan showed in 1968 that the first order theory of D with the Turing reducibility relation is undecidable. Later results concerned the decida ..."
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Decidability problems for (fragments of) the theory of the structure D of Turing degrees, form a wide and interesting class, much of which is yet unsolved. Lachlan showed in 1968 that the first order theory of D with the Turing reducibility relation is undecidable. Later results concerned the decidability (or undecidability) of fragments of this theory, and of other theories obtained by extending the language (e.g. with 0 or with the Turing jump operator). Proofs of these results often hinge on the ability to embed certain classes of structures (lattices, jumphierarchies, etc.) in certain ways, into the structure of Turing degrees. The first part of the dissertation presents two results which concern embeddings onto initial segments of D with known double jumps, in other words a double jump inversion of certain degree structures onto initial segments. These results may prove to be useful tools in uncovering decidability results for (fragments of) the theory of the Turing degrees in languages containing the double jump operator. The second part of the dissertation relates to the problem of characterizing the Turing degrees which have a strong minimal cover, an issue first raised by Spector in 1956. Ishmukhametov solved the problem for the recursively enumerable degrees, by showing that those which have a strong minimal cover are exactly the r.e. weakly recursive degrees. Here we show that this characterization fails outside the r.e. degrees, and also construct a minimal degree below 0 ′ which is not weakly recursive, thereby answering a question from Ishmukhametov’s paper.
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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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 ′
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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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
Strong minimal covers and a question of Yates: the story so far
 the proceedings of the ASL meeting
, 2006
"... Abstract. An old question of Yates as to whether all minimal degrees have a strong minimal cover remains one of the longstanding problems of degree theory, apparently largely impervious to present techniques. We survey existing results in this area, focussing especially on some recent progress. 1. ..."
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Abstract. An old question of Yates as to whether all minimal degrees have a strong minimal cover remains one of the longstanding problems of degree theory, apparently largely impervious to present techniques. We survey existing results in this area, focussing especially on some recent progress. 1.
A single minimal complement for the c.e. degrees
"... Abstract. We show that there exists a minimal (Turing) degree b<0 ′ such that for all nonzero c.e. degrees a, 0 ′ = a ∨ b. Since b is minimal this means that b complements all c.e. degrees other than 0 and 0 ′. Since every nc.e. degree bounds a nonzero c.e. degree, b complements every nc.e. deg ..."
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Abstract. We show that there exists a minimal (Turing) degree b<0 ′ such that for all nonzero c.e. degrees a, 0 ′ = a ∨ b. Since b is minimal this means that b complements all c.e. degrees other than 0 and 0 ′. Since every nc.e. degree bounds a nonzero c.e. degree, b complements every nc.e. degree other than 0 and 0 ′. 1.