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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 36 (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...
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...
Intervals without critical triples
 In: Logic Colloquium '95 (J.A. Makowsky and E.V. Ravve, eds.), Lect. Notes in Logic 11
, 1998
"... Abstract. This paper is concerned with the construction of intervals of computably enumerable degrees in which the lattice M5 (see Figure 1) cannot be embedded. Actually, we construct intervals I of computably enumerable degrees without any weak critical triples (this implies that M5 cannot be embed ..."
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Cited by 4 (3 self)
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Abstract. This paper is concerned with the construction of intervals of computably enumerable degrees in which the lattice M5 (see Figure 1) cannot be embedded. Actually, we construct intervals I of computably enumerable degrees without any weak critical triples (this implies that M5 cannot be embedded in I, see Section 2). Our strongest result is that there is a low2 computably enumerable degree e such that there are no weak critical triples in either of the intervals [0,e]or[e,0 ′]. 1.
Contiguity and Distributivity in the Enumerable Turing Degrees \Lambda
, 1996
"... Abstract We prove that a (recursively) enumerable degree is contiguous iff it is locally distributive. This settles a twentyyear old question going back to Ladner and Sasso. We also prove that strong contiguity and contiguity coincide, settling a question of the first author, and prove that no mtop ..."
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Abstract We prove that a (recursively) enumerable degree is contiguous iff it is locally distributive. This settles a twentyyear old question going back to Ladner and Sasso. We also prove that strong contiguity and contiguity coincide, settling a question of the first author, and prove that no mtopped degree is contiguous, settling a question of the first author and Carl Jockusch [11]. Finally, we prove some results concerning local distributivity and relativized weak truth table reducibility.
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...
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...