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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...
Randomness, lowness and degrees
 J. of Symbolic Logic
, 2006
"... Abstract. We say that A ≤LR B if every Brandom number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally a ..."
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Cited by 9 (4 self)
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Abstract. We say that A ≤LR B if every Brandom number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e. restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α is not GL2 the LR degree of α bounds 2 ℵ0 degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees. 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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Highness and Bounding Minimal Pairs
, 1993
"... We show the existence of a high r.e. degree bounding only joins of minimal pairs and of a high 2 nonbounding r.e. degree. 0 Introduction An important topic in the study of recursively enumerable sets and degrees has been the interaction between the jump operator and the order theoretic properties o ..."
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Cited by 3 (2 self)
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We show the existence of a high r.e. degree bounding only joins of minimal pairs and of a high 2 nonbounding r.e. degree. 0 Introduction An important topic in the study of recursively enumerable sets and degrees has been the interaction between the jump operator and the order theoretic properties of an r. e. set A (in the lattice E of all r. e. sets) and of its degree a in R, the upper semilattice of the r. e. degrees. An early theme in this area was the idea that sets with "low" jumps should behave like the recursive sets while those with "high" jumps should exhibit properties like the complete sets. For example, in the lattice E of r. e. sets modulo finite sets, we know from Soare[23] that if A is low, i. e. A 0 j T ; 0 , then L (A), the lattice of r. e. supersets of A, is isomorphic to E . In R there are many instances of the low Partially supported by IGC of Vicoria University, Wellington and ARO through MSI, Cornell University, DAAL03C0027. y Partailly suppo...
Definability and Global Degree Theory
 Logic Colloquium '90, Association of Symbolic Logic Summer Meeting in Helsinki, Berlin 1993 [Lecture Notes in Logic 2
"... Gödel's work [Gö34] on undecidable theories and the subsequent formalisations of the notion of a recursive function ([Tu36], [K136] etc.) have led to an ever deepening understanding of the nature of the noncomputable universe (which as Gödel himself showed, includes sets and functions of everyday s ..."
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Cited by 2 (1 self)
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Gödel's work [Gö34] on undecidable theories and the subsequent formalisations of the notion of a recursive function ([Tu36], [K136] etc.) have led to an ever deepening understanding of the nature of the noncomputable universe (which as Gödel himself showed, includes sets and functions of everyday significance). The nontrivial aspect of Church's Thesis (any function not contained within one of the equivalent definitions of recursive/Turing computable, cannot be considered to be effectively computable) still provides a basis not only for classical and generalised recursion theory, but also for contemporary theoretical computer science. Recent years, in parallel with the massive increase in interest in the computable universe and the development of much subtler concepts of 'practically computable', have seen remarkable progress with some of the most basic and challenging questions concerning the noncomputable universe, results both of philosophical significance and of potentially wider technical importance. Relativising Church's Thesis, Kleene and Post [KP54] proposed the now
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.