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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 defin ..."
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Cited by 40 (15 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...
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
"... ..."
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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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
On Lachlan's major subdegree problem
 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
Extension of embeddings in the recursively enumerable degrees
"... The extension of embeddings problem for the recursively enumerable degrees R = (R;!; 0; 0 0) asks for given finite partially ordered sets P ` Q with least and greatest elements, whether every embedding of P into R can be extended to an embedding of Q into R. Many of the landmark theorems giving an a ..."
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Cited by 1 (0 self)
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The extension of embeddings problem for the recursively enumerable degrees R = (R;!; 0; 0 0) asks for given finite partially ordered sets P ` Q with least and greatest elements, whether every embedding of P into R can be extended to an embedding of Q into R. Many of the landmark theorems giving an algebraic insight into R assert either extension or nonextension of embeddings. We extend, strengthen, and unify these results and their proofs to produce complete and complementary criteria and techniques to analyze instances of extension and nonextension. We conclude that the full extension of embeddings problem is decidable.
Lattice Embeddings below a Nonlow Recursively Enumerable Degree
 Israel J. Math
, 1996
"... We introduce techniques that allow us to embed below an arbitary nonlow 2 recursively enumerable degree any lattice currently known to be embedable into the recursively enumerable degrees. 1 Introduction One of the most basic and important questions concerning the structure of the upper semilattice ..."
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We introduce techniques that allow us to embed below an arbitary nonlow 2 recursively enumerable degree any lattice currently known to be embedable into the recursively enumerable degrees. 1 Introduction One of the most basic and important questions concerning the structure of the upper semilattice R of recursively enumerable degrees is the embedding question: what (finite) lattices can be embedded as lattices into R? This question has a long and rich history. After the proof of the density theorem by Sacks [31], Shoenfield [32] made a conjecture, one consequence of which would be that no lattice embeddings into R were possible. Lachlan [21] and Yates [40] independently refuted Shoenfield's conjecture by proving that the 4 element boolean algebra could be embedded into R (even preserving 0). Using a little lattice representation theory, this result was subsequently extended by LachlanLermanThomason [38], [36] who proved that all countable distributive lattices could be embedded (pre...
1 Introduction Degrees of Unsolvability
, 2006
"... Modern computability theory began with Turing [Turing, 1936], where he introduced ..."
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Modern computability theory began with Turing [Turing, 1936], where he introduced
EMBEDDINGS INTO THE COMPUTABLY ENUMERABLE DEGREES
"... Abstract. We discuss the status of the problem of characterizing the finite (weak) lattices which can be embedded into the computably enumerable degrees. In particular, we summarize the current status of knowledge about the problem, provide an overview of how to prove these results, discuss directio ..."
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Abstract. We discuss the status of the problem of characterizing the finite (weak) lattices which can be embedded into the computably enumerable degrees. In particular, we summarize the current status of knowledge about the problem, provide an overview of how to prove these results, discuss directions which have been pursued to try to solve the problem, and present some related open questions. 1.
The Theories of the T, tt and wtt R. E. Degrees: Undecidability and Beyond
"... We discuss the structure of the recursively enumerable sets under three reducibilities: Turing, truthtable and weak truthtable. Weak truthtable reducibility requires that the questions asked of the oracle be effectively bounded. Truthtable reducibility also demands such a bound on the the length ..."
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We discuss the structure of the recursively enumerable sets under three reducibilities: Turing, truthtable and weak truthtable. Weak truthtable reducibility requires that the questions asked of the oracle be effectively bounded. Truthtable reducibility also demands such a bound on the the length of the computations. We survey what is known about the algebraic structure and the complexity of the decision procedure for each of the associated degree structures. Each of these structures is an upper semilattice with least and greatest element. Typical algebraic questions include the existence of infima, distributivity, embeddings of partial orderings or lattices and extension of embedding problems such as density. We explain how the algebraic information is used to decide fragments of the theories and then to prove their undecidability (and more). Finally, we discuss some results and open problems concerning automorphisms, definability and the complexity of the decision problems for the...
The nr.e. degrees: undecidability and Σ1 substructures
, 2012
"... We study the global properties of Dn, the Turing degrees of the nr.e. sets. In Theorem 1.5, we show that the first order theory of Dn is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, Dn is not a Σ1substructure of Dm. 1 ..."
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We study the global properties of Dn, the Turing degrees of the nr.e. sets. In Theorem 1.5, we show that the first order theory of Dn is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, Dn is not a Σ1substructure of Dm. 1