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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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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...
Enumeration Reducibility, Nondeterministic Computations and Relative . . .
 RECURSION THEORY WEEK, OBERWOLFACH 1989, VOLUME 1432 OF LECTURE NOTES IN MATHEMATICS
, 1990
"... ..."
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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Cited by 6 (2 self)
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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.
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
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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.
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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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
The ∀∃ theory of D(≤, ∨, ′ ) is undecidable
 In Proceedings of Logic Colloquium
, 2003
"... We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable. ..."
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We prove that the two quantifier theory of the Turing degrees with order, join and jump is undecidable.
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.
The ∀∃theory of R(≤, ∨, ∧) is undecidable
 Trans. Amer. Math. Soc
, 2004
"... 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 longstanding open question. A negative ..."
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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 longstanding 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. Theorem. The two quantifier theory of (R, ≤, ∨, ∧), the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on R) is undecidable. The same result holds for various lattices of ideals of R which are natural extensions of R preserving join and infimum when it exits. 1.
A ... Set With Barely ... Degree
"... We construct a \Delta 0 2 degree which fails to be computably enumerable in any computably enumerable set strictly below ; 0 . 1 Introduction The lion's share of effort in classical computability theory over the last fifty years has been directed toward the study of relative computability. Th ..."
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We construct a \Delta 0 2 degree which fails to be computably enumerable in any computably enumerable set strictly below ; 0 . 1 Introduction The lion's share of effort in classical computability theory over the last fifty years has been directed toward the study of relative computability. This paper is concerned with another, more neglected, yet still fundamental, notion of classical computability theory, namely that of relative enumerability. Specifically, we ask questions concerning the relationship between sets A and B when A is computably enumerable using B as an oracle. For example, given a set B, we might ask what properties the class of sets A which are c.e. relative to B has. Conversely, for fixed A, we might wonder which degrees contain sets relative to which A can be computably enumerated. In the present paper, our particular concern is with this latter type of question. We study \Sigma 0 2 sets and degrees, and their relations with the computably enumerable sets f...