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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...
Embedding Lattices with Top Preserved Below NonGL2 Degrees
, 1997
"... this paper, we answer this question by showing that every recursively presented lattice can be embedded into D (0 ..."
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Cited by 5 (1 self)
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this paper, we answer this question by showing that every recursively presented lattice can be embedded into D (0
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 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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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:
COMPUTABILITY, TRACEABILITY AND BEYOND
"... This thesis is concerned with the interaction between computability and randomness. In the first part, we study the notion of traceability. This combinatorial notion has an increasing influence in the study of algorithmic randomness. We prove a separation result about the bounds on jump traceability ..."
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This thesis is concerned with the interaction between computability and randomness. In the first part, we study the notion of traceability. This combinatorial notion has an increasing influence in the study of algorithmic randomness. We prove a separation result about the bounds on jump traceability, and show that the index set of the strongly jump traceable computably enumerable (c.e.) sets is Π0 4complete. This shows that the problem of deciding if a c.e. set is strongly jump traceable, is as hard as it can be. We define a strengthening of strong jump traceability, called hyper jump traceability, and prove some interesting results about this new class. Despite the fact that the hyper jump traceable sets have their origins in algorithmic randomness, we are able to show that they are natural examples of several Turing degree theoretic properties. For instance, we show that the hyper jump traceable sets are the first example of a lowness class with no promptly simple members. We also study the dual highness notions obtained from strong jump traceability, and explore their degree theoretic properties.
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...
Resolute sequences in . . .
, 2012
"... We study infinite sequences whose initial segment complexity is invariant under effective insertions of blocks of zeros inbetween their digits. Surprisingly, such resolute sequences may have nontrivial initial segment complexity. In fact, we show that they occur in many well known classes from comp ..."
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We study infinite sequences whose initial segment complexity is invariant under effective insertions of blocks of zeros inbetween their digits. Surprisingly, such resolute sequences may have nontrivial initial segment complexity. In fact, we show that they occur in many well known classes from computability theory, e.g. in every jump class and every high degree. Moreover there are degrees which consist entirely of resolute sequences, while there are degrees which do not contain any. Finally we establish connections with the contiguous c.e. degrees, the ultracompressible sequences, the anticomplex sequences thus demonstrating that this class is an interesting superclass of the sequences with trivial initial segment complexity.
RESOLUTE SEQUENCES IN INITIAL SEGMENT COMPLEXITY
"... Abstract. We study infinite sequences whose initial segment complexity is invariant under effective insertions of blocks of zeros inbetween their digits. Surprisingly, such resolute sequences may have nontrivial initial segment complexity. In fact, we show that they occur in many well known classes ..."
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Abstract. We study infinite sequences whose initial segment complexity is invariant under effective insertions of blocks of zeros inbetween their digits. Surprisingly, such resolute sequences may have nontrivial initial segment complexity. In fact, we show that they occur in many well known classes from computability theory, e.g. in every jump class and every high degree. Moreover there are degrees which consist entirely of resolute sequences, while there are degrees which do not contain any. Finally we establish connections with the contiguous c.e. degrees, the ultracompressible sequences, the anticomplex sequences thus demonstrating that this class is an interesting superclass of the sequences with trivial initial segment complexity. 1.