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Some ComputabilityTheoretical Aspects of Reals and Randomness
 the Lect. Notes Log. 18, Assoc. for Symbol. Logic
, 2001
"... We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. ..."
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Cited by 24 (7 self)
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We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.
DegreeTheoretic Aspects of Computably Enumerable Reals
 in Models and Computability
, 1998
"... A real # is computable if its left cut, L###; is computable. If #q i # i is a computable sequence of rationals computably converging to #; then fq i g; the corresponding set, is always computable. A computably enumerable #c.e.# real # is a real which is the limit of an increasing computable sequ ..."
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Cited by 13 (0 self)
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A real # is computable if its left cut, L###; is computable. If #q i # i is a computable sequence of rationals computably converging to #; then fq i g; the corresponding set, is always computable. A computably enumerable #c.e.# real # is a real which is the limit of an increasing computable sequence of rationals, and has a left cut which is c.e. We study the Turing degrees of representations of c.e. reals, that is the degrees of increasing computable sequences converging to #: For example, every representation A of # is Turing reducible to L###: Every noncomputable c.e. real has both a computable and noncomputable representation. In fact, the representations of noncomputable c.e. reals are dense in the c.e. Turing degrees, and yet not every c.e. Turing degree below deg T L### necessarily contains a representation of #: 1 Introduction Computability theory essentially studies the relative computability of sets of natural numbers. Since G#odel introduced a method for coding s...
Isomorphisms Of Splits Of Computably Enumerable Sets
 J. OF SYMBOLIC LOGIC
, 2002
"... We show that if A and A are automorphic via # then the structures SR (A) and SR ( 3 isomorphic via an isomorphism # induced by #. Then we use this result to classify completely the orbits of hhsimple sets. ..."
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Cited by 4 (4 self)
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We show that if A and A are automorphic via # then the structures SR (A) and SR ( 3 isomorphic via an isomorphism # induced by #. Then we use this result to classify completely the orbits of hhsimple sets.
On the Universal Splitting Property
 Mathematical Logic Quarterly
, 1996
"... We prove that if an incomplete computably enumerable set has the the universal splitting property then it is low 2 . This solves a question from AmbosSpies and Fejer [1] and Downey and Stob [7]. Some technical improvements are discussed. 1 Introduction Two computably enumerable sets A 1 and A 2 ar ..."
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Cited by 3 (2 self)
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We prove that if an incomplete computably enumerable set has the the universal splitting property then it is low 2 . This solves a question from AmbosSpies and Fejer [1] and Downey and Stob [7]. Some technical improvements are discussed. 1 Introduction Two computably enumerable sets A 1 and A 2 are said to split A if A = A 1 [ A 2 and A 1 " A 2 = ;. We write A 1 t A 2 = A in the case that A 1 and A 2 split A. Splitting theorems for computably enumerable sets have played a central role in the history of classical computability theory. For instance, Sack's splitting theorm [14], demonstrated that every nonzero computably enumerable degree could be Downey's research supported by Cornell University, an IGC grant from Victoria University and the New Zealand Marsden Fund via grant 95VICMIS0698 under contract VIC509. Some of these results were obtained whilst Downey was a Visiting Professor at Cornell University in fall 1995. decomposed into a pair of incomparible nonzero computa...
Kolmogorov complexity and computably enumerable sets
, 2011
"... Abstract. We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments ..."
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Abstract. We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefixfree complexity. A computably enumerable set A is Ktrivial if and only if the family of sets with complexity bounded by the complexity of A is uniformly computable from the halting problem. 1.
Computability, Definability and Algebraic Structures
, 1999
"... In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set ..."
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Cited by 2 (1 self)
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In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set
NONCUPPING, MEASURE AND COMPUTABLY ENUMERABLE SPLITTINGS
"... Abstract. We show that there is a computably enumerable function f (i.e. computably approximable from below) which dominates almost all functions and f ⊕ W is incomplete, for all incomplete computably enumerable sets W. Our main methodology is the LR equivalence relation on reals: A ≡LR B iff the no ..."
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Abstract. We show that there is a computably enumerable function f (i.e. computably approximable from below) which dominates almost all functions and f ⊕ W is incomplete, for all incomplete computably enumerable sets W. Our main methodology is the LR equivalence relation on reals: A ≡LR B iff the notions of Arandomness and Brandomness coincide. We also show that there are c.e. sets which cannot be split into two c.e. sets of the same LR degree. Moreover a c.e. set is low for random iff it computes no c.e. set with this property. 1.
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...