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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...
A splitting theorem for the Medvedev and Muchnik lattices
 Mathematical Logic Quarterly
, 2003
"... This is a contribution to the study of the Muchnik and Medvedev lattices of nonempty Π0 1 subsets of 2ω. In both these lattices, any nonminimum element can be split, i.e. it is the nontrivial join of two other elements. In fact, in the Medvedev case, if P>MQ, thenP can be split above Q. Both of t ..."
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Cited by 34 (1 self)
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This is a contribution to the study of the Muchnik and Medvedev lattices of nonempty Π0 1 subsets of 2ω. In both these lattices, any nonminimum element can be split, i.e. it is the nontrivial join of two other elements. In fact, in the Medvedev case, if P>MQ, thenP can be split above Q. Both of these facts are then generalised to the embedding of arbitrary finite distributive lattices. A consequence of this is that both lattices have decidible ∃theories. 1
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" for a ..."
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Cited by 10 (6 self)
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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
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
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Beyond strong jump traceability
"... Abstract. Strong jump traceability has been studied by various authors. In this paper we study a variant of strong jump traceability by looking at a partial relativization of traceability. We discover a new subclass H of the c.e. Ktrivials with some interesting properties. These sets are computatio ..."
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Cited by 4 (1 self)
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Abstract. Strong jump traceability has been studied by various authors. In this paper we study a variant of strong jump traceability by looking at a partial relativization of traceability. We discover a new subclass H of the c.e. Ktrivials with some interesting properties. These sets are computationally very weak, but yet contains a cuppable member. Surprisingly they cannot be constructed using cost functions, and is the first known example of a subclass of the Ktrivials which does not contain any promptly simple member. Furthermore there is a single c.e. set which caps every member of H, demonstrating that they are in fact very far away from being promptly simple. 1.
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, particular ..."
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Cited by 4 (1 self)
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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
Highness and Bounding Minimal Pairs
, 1993
"... We show the existence of a high r.e. degree bounding only joins of minimal pairs and of a high 2 nonbounding r.e. degree. 0 Introduction An important topic in the study of recursively enumerable sets and degrees has been the interaction between the jump operator and the order theoretic properties o ..."
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Cited by 3 (2 self)
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We show the existence of a high r.e. degree bounding only joins of minimal pairs and of a high 2 nonbounding r.e. degree. 0 Introduction An important topic in the study of recursively enumerable sets and degrees has been the interaction between the jump operator and the order theoretic properties of an r. e. set A (in the lattice E of all r. e. sets) and of its degree a in R, the upper semilattice of the r. e. degrees. An early theme in this area was the idea that sets with "low" jumps should behave like the recursive sets while those with "high" jumps should exhibit properties like the complete sets. For example, in the lattice E of r. e. sets modulo finite sets, we know from Soare[23] that if A is low, i. e. A 0 j T ; 0 , then L (A), the lattice of r. e. supersets of A, is isomorphic to E . In R there are many instances of the low Partially supported by IGC of Vicoria University, Wellington and ARO through MSI, Cornell University, DAAL03C0027. y Partailly suppo...
2004], The 89theory of R( ; _; ^) is undecidable
 Trans. Am. Math. Soc
"... Abstract The three quantifier theory of (R; ^T), the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman [1998]. The two quantifier theory includes the lattice embedding problem and its decidability is a long standing open question. A negative s ..."
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Abstract The three quantifier theory of (R; ^T), the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman [1998]. The two quantifier theory includes the lattice embedding problem and its decidability is a long standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of R that lies between the two and three quantifier theories with ^T but includes function symbols.
On Lachlan's major subdegree problem, to
 in: Set Theory and the Continuum, Proceedings of Workshop on Set Theory and the Continuum
, 1989
"... The Major Subdegree Problem of A. H. Lachlan (first posed in 1967) has become a longstanding open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the ..."
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Cited by 2 (2 self)
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The Major Subdegree Problem of A. H. Lachlan (first posed in 1967) has become a longstanding open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the theory of the c.e. Turing degrees. A c.e. degree a is a major subdegree of a c.e. degree b> a if for any c.e. degree x, 0 ′ = b ∨ x if and only if 0 ′ = a ∨ x. In this paper, we show that every c.e. degree b ̸ = 0 or 0 ′ has a major subdegree, answering Lachlan’s question affirmatively. 1