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Defining the Turing Jump
 MATHEMATICAL RESEARCH LETTERS
, 1999
"... The primary notion of effective computability is that provided by Turing machines (or equivalently any of the other common models of computation). We denote the partial function computed by the eth Turing machine in some standard list by # e . When these machines are equipped with an "oracle&q ..."
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The primary notion of effective computability is that provided by Turing machines (or equivalently any of the other common models of computation). We denote the partial function computed by the eth Turing machine in some standard list by # e . When these machines are equipped with an "oracle" for a subset A of the natural numbers #, i.e. an external procedure that answers questions of the form "is n in A", they define the basic notion of relative computability or Turing reducibility (from Turing (1939)). We say that A is computable from (or recursive in) B if there is a Turing machine which, when equipped with an oracle for B, computes (the characteristic function of) A, i.e. for some e, # B e = A. We denote this relation by A # T<F10
Decomposing Borel functions using the ShoreSlaman join theorem. submitted
, 2013
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Noncomputable Spectral Sets
, 2007
"... iii For my Mama, whose *minimal index is computable (because it’s finite). ..."
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iii For my Mama, whose *minimal index is computable (because it’s finite).
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
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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 ..."
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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 ′
A Splitting Theorem for nREA Degrees
"... We prove that, for any D, A and U with D > T A # U and r.e. in A# U , there are pairs X 0 , X 1 and Y 0 , Y 1 such that D # T X 0 #X 1 ; D # T Y 0 # Y 1 ; and, for any i and j from {0, 1} and any set B, if X i #A # T B and Y j # A # T B then A # T B. We then deduce that for any degrees d, a, and ..."
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We prove that, for any D, A and U with D > T A # U and r.e. in A# U , there are pairs X 0 , X 1 and Y 0 , Y 1 such that D # T X 0 #X 1 ; D # T Y 0 # Y 1 ; and, for any i and j from {0, 1} and any set B, if X i #A # T B and Y j # A # T B then A # T B. We then deduce that for any degrees d, a, and b such that a and b are recursive in d, a ## T b, and d is nREA in to a, d can be split over a avoiding b. This shows that the Main Theorem of Cooper [1990] and [1993] is false.
STRONG JUMPTRACEABILITY II: KTRIVIALITY
, 2010
"... We show that every strongly jumptraceable set is Ktrivial. Unlike other results, we do not assume that the sets in question are computably enumerable. ..."
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We show that every strongly jumptraceable set is Ktrivial. Unlike other results, we do not assume that the sets in question are computably enumerable.
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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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.
Conjectures and Questions from Gerald Sacks’s Degrees of Unsolvability
 Archive for Mathematical Logic
, 1993
"... We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. Gerald Sacks has had a major influence on the development of logic, parti ..."
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We describe the important role that the conjectures and questions posed at the end of the two editions of Gerald Sacks's Degrees of Unsolvability have had in the development of recursion theory over the past thirty years. Gerald Sacks has had a major influence on the development of logic, particularly recursion theory, over the past thirty years through his research, writing and teaching. Here, I would like to concentrate on just one instance of that influence that I feel has been of special significance to the study of the degrees of unsolvability in general and on my own work in particular the conjectures and questions posed at the end of the two editions of Sacks's first book, the classic monograph Degrees of Unsolvability (Annals
Global Properties of the Turing Degrees and the Turing Jump
"... We present a summary of the lectures delivered to the Institute for Mathematical Sciences, Singapore, during the 2005 Summer School in Mathematical Logic. The lectures covered topics on the global structure of the Turing degrees D, the countability of its automorphism group, and the definability of ..."
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We present a summary of the lectures delivered to the Institute for Mathematical Sciences, Singapore, during the 2005 Summer School in Mathematical Logic. The lectures covered topics on the global structure of the Turing degrees D, the countability of its automorphism group, and the definability of the Turing jump within D.