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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 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.
Degree theoretic definitions of the low_2 recursively enumerable sets
 J. SYMBOLIC LOGIC
, 1995
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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...
Embedding finite lattices into the computably enumerable degrees  a status survey
 In Proceedings of Logic Colloquium
, 2002
"... Abstract. We survey the current status of an old open question in classical computability theory: Which finite lattices can be embedded into the degree structure of the computably enumerable degrees? Does the collection of embeddable finite lattices even form a computable set? Two recent papers by t ..."
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Cited by 4 (0 self)
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Abstract. We survey the current status of an old open question in classical computability theory: Which finite lattices can be embedded into the degree structure of the computably enumerable degrees? Does the collection of embeddable finite lattices even form a computable set? Two recent papers by the second author show that for a large subclass of the finite lattices, the socalled joinsemidistributive lattices (or lattices without socalled “critical triple”), the collection of embeddable lattices forms a Π0 2set. This paper surveys recent joint work by the authors, concentrating on restricting the number of meets by considering “quasilattices”, i.e., finite upper semilattices in which only some meets of incomparable elements are specified. In particular, we note that all finite quasilattices with one meet specified are embeddable; and that the class of embeddable finite quasilattices with two meets specified, while nontrivial, forms a computable set. On the other hand, more sophisticated techniques may be necessary for finite quasilattices with three meets specified. 1.
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 p ..."
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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:
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