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Lowness Properties and Randomness
 ADVANCES IN MATHEMATICS
"... The set A is low for MartinLof random if each random set is already random relative to A. A is Ktrivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of AmbosSpies and Kucera [2 ..."
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Cited by 78 (21 self)
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The set A is low for MartinLof random if each random set is already random relative to A. A is Ktrivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of AmbosSpies and Kucera [2], showing that each low for MartinLof random set is # 2 . Our class induces a natural intermediate # 3 ideal in the r.e. Turing degrees (which generates the whole class under downward closure). Answering
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...
Lowness properties and approximations of the jump
 Proceedings of the Twelfth Workshop of Logic, Language, Information and Computation (WoLLIC 2005). Electronic Lecture Notes in Theoretical Computer Science 143
, 2006
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Codable Sets and Orbits of Computably Enumerable Sets
 J. Symbolic Logic
, 1995
"... A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E = (fW e g e2! ; `). We previously exhibited a first order ..."
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Cited by 10 (5 self)
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A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E = (fW e g e2! ; `). We previously exhibited a first order Edefinable property Q(X) such that Q(X) guarantees that X is not Turing complete (i.e., does not code complete information about c.e. sets). Here we show first that Q(X) implies that X has a certain "slowness " property whereby the elements must enter X slowly (under a certain precise complexity measure of speed of computation) even though X may have high information content. Second we prove that every X with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable A 2 E there exists B in the orbit of A such that X T B under relative Turing computability ( T ). We produce B using the \Delta 0 3 automorphism method we introduced earli...
Presentations of computably enumerable reals
 Theoretical Computer Science
, 2002
"... Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefixfree set of binary strings. ..."
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Cited by 10 (5 self)
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Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefixfree set of binary strings.
Benign cost functions and lowness properties
"... Abstract. We show that the class of strongly jumptraceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of wellbehaved cost function, called benign. This characterisation implies the containment of the class of strongly jumptraceable c.e. ..."
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Cited by 9 (5 self)
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Abstract. We show that the class of strongly jumptraceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of wellbehaved cost function, called benign. This characterisation implies the containment of the class of strongly jumptraceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LRhard random degrees, and all ωc.e. random degrees. The last result implies recent results of Diamondstone’s and Ng’s regarding cupping with supwerlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees. 1.
Lowness Properties of Reals and Randomness
 Advances in Mathematics
, 2002
"... We investigate three properties of the set of natural numbers which have been discovered independently by different... ..."
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Cited by 8 (3 self)
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We investigate three properties of the set of natural numbers which have been discovered independently by different...
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 structures: Local and global investigations
 Bulletin of Symbolic Logic
"... $1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead. ..."
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Cited by 6 (2 self)
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$1. Introduction. The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.
Natural Definability in Degree Structures
"... . A major focus of research in computability theory in recent years has involved denability issues in degree structures. There has been much success in getting general results by coding methods that translate rst or second order arithmetic into the structures. In this paper we concentrate on the ..."
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Cited by 5 (1 self)
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. A major focus of research in computability theory in recent years has involved denability issues in degree structures. There has been much success in getting general results by coding methods that translate rst or second order arithmetic into the structures. In this paper we concentrate on the issues of getting denitions of interesting, apparently external, relations on degrees that are ordertheoretically natural in the structures D and R of all the Turing degrees and of the r.e. Turing degrees, respectively. Of course, we have no formal denition of natural but we oer some guidelines, examples and suggestions for further research. 1. Introduction A major focus of research in computability theory in recent years has involved denability issues in degree structures. The basic question is, which interesting apparently external relations on degrees can actually be dened in the structures themselves, that is, in the rst order language with the single fundamental relation...