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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...
Totally ωcomputably enumerable degrees and bounding critical triples, preprint
"... Abstract. We characterize the class of c.e. degrees that bound a critical triple (equivalently, a weak critical triple) as those degrees that compute a function that has no ωc.e. approximation. 1. ..."
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Cited by 8 (4 self)
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Abstract. We characterize the class of c.e. degrees that bound a critical triple (equivalently, a weak critical triple) as those degrees that compute a function that has no ωc.e. approximation. 1.
1995], Degree theoretic definitions of the low 2 recursively enumerable sets
 J. Symbolic Logic
, 1995
"... 1. Introduction. The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤T B, where arbitrary (Turing) mac ..."
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Cited by 7 (5 self)
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1. Introduction. The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤T B, where arbitrary (Turing) machines, ϕe, can be used; access to
The recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
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Embeddings of N_5 and the Contiguous Degrees
"... Downey and Lempp [8] have shown that the contiguous computably enumerable (c.e.) degrees, i.e. the c.e. Turing degrees containing only one c.e. weak truthtable degree, can be characterized by a local distributivity property. Here we extend their result by showing that a c.e. degree a is noncont ..."
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Cited by 4 (1 self)
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Downey and Lempp [8] have shown that the contiguous computably enumerable (c.e.) degrees, i.e. the c.e. Turing degrees containing only one c.e. weak truthtable degree, can be characterized by a local distributivity property. Here we extend their result by showing that a c.e. degree a is noncontiguous if and only if there is an embedding of the nonmodular 5element lattice N5 into the c.e. degrees which maps the top to the degree a. In particular, this shows that local nondistributivity coincides with local nonmodularity in the computably enumerable degrees. 1 Introduction Lachlan [13] has shown that the two fiveelement nondistributive lattices, the modular lattice M 3 and the nonmodular lattice N 5 (see Figure 1 below) can be embedded into the upper semilattice (E; ) of the computably enumerable degrees. These lattices capture nondistributivity and nonmodularity in the sense that every nondistributive lattice contains one of these lattices as a sublattice and that every nonmo...
Maximal Contiguous Degrees
"... . A computably enumerable (c.e.) degree is a maximal contiguous degree if it is contiguous and no c.e. degree strictly above it is contiguous. We show that there are infinitely many maximal contiguous degrees. Since the contiguous degrees are definable, the class of maximal contiguous degrees pr ..."
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Cited by 2 (1 self)
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. A computably enumerable (c.e.) degree is a maximal contiguous degree if it is contiguous and no c.e. degree strictly above it is contiguous. We show that there are infinitely many maximal contiguous degrees. Since the contiguous degrees are definable, the class of maximal contiguous degrees provides the first example of a definable infinite antichain in the c.e. degrees. In addition, we show that the class of maximal contiguous degrees forms an automorphism base for the c.e. degrees and therefore for the Turing degrees in general. Finally we note that the construction of a maximal contiguous degree can be modified to answer a question of Walk about the array computable degrees and a question of Li about isolated formulas. 1. Introduction. We will work within the c.e. Turing degrees, R, ordered by Turing reducibility, # t . (We will suppress the # t for readability.) Our concern is that of definability and, specifically, the relationships between automorphisms of R ...
Embedding Distributive Lattices Preserving 1 Below a Nonzero Recursively Enumerable Turing Degree
"... this paper, we show that they can be, by proving that for every nonzero a 2 R, every countable distributive lattice can be embedded into R( a) preserving 1 (Theorem 18). The long gap between the appearance of [16] and our paper is perhaps explained by the fact that the techniques of Lachlan's paper ..."
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Cited by 1 (1 self)
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this paper, we show that they can be, by proving that for every nonzero a 2 R, every countable distributive lattice can be embedded into R( a) preserving 1 (Theorem 18). The long gap between the appearance of [16] and our paper is perhaps explained by the fact that the techniques of Lachlan's paper have not been wellunderstood. Our first step in proving our result was to reprove the Lachlan Splitting Theorem using techniques which have been developed since Lachlan's paper appeared. In Section 2 of this paper, we give this reproof of the Splitting Theorem. In Section 3, we extend the techniques of Section 2 to show our result, and in Section 4, we discuss related results and open questions. In particular, in this section, we show how our results, together with some other recent and older results, provide a complete answer to all questions of the form Which finite distributive lattices can be embedded into A preserving B? where A can be:
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...
Structural Properties of D.C.E. Degrees and Presentations of C.E. Reals
"... To my wife Caixia and my daughter Jiahui Abstract In this thesis, we are mainly concerned with the structural properties of the d.c.e. degrees and the distribution of the simple reals among the c.e. degrees. In chapters 2 and 3, we study the relationship between the isolation phenomenon and the jump ..."
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To my wife Caixia and my daughter Jiahui Abstract In this thesis, we are mainly concerned with the structural properties of the d.c.e. degrees and the distribution of the simple reals among the c.e. degrees. In chapters 2 and 3, we study the relationship between the isolation phenomenon and the jump operator. We prove in chapter 2 that there is a high d.c.e. degree d isolated by a low2 degree a. We improve this result in chapter 3 by showing that the isolating degree a can be low. Chapters 4 and 5 are devoted to the study of the pseudoisolation in the d.c.e. degrees. We prove that pseudoisolated d.c.e. degrees are dense in the c.e. degrees, and that there is a high d.c.e. degree pseudoisolated by a low d.c.e. degree.