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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...
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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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
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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Cited by 2 (2 self)
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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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Cited by 2 (0 self)
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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.
Undecidability and 1types in intervals of the computably enumerable degrees
 Ann. Pure Appl. Logic
, 2000
"... We show that the theory of the partial ordering of the computably enumerable degrees in any given nontrivial interval is undecidable and has uncountably many 1types. subject code classifications: 03D25 (03C65 03D35 06A06) ..."
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We show that the theory of the partial ordering of the computably enumerable degrees in any given nontrivial interval is undecidable and has uncountably many 1types. subject code classifications: 03D25 (03C65 03D35 06A06)
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"... We present an explicit measurement in the Fourier basis that solves an important case of the Hidden Subgroup Problem, including the case to which Graph Isomorphism reduces. This entangled measurement uses k = log 2 G  registers, and each of the 2 k subsets of the registers contributes some informa ..."
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We present an explicit measurement in the Fourier basis that solves an important case of the Hidden Subgroup Problem, including the case to which Graph Isomorphism reduces. This entangled measurement uses k = log 2 G  registers, and each of the 2 k subsets of the registers contributes some information. 1