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Π 0 1 CLASSES WITH COMPLEX ELEMENTS.
"... Abstract. An infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a Π 0 1 class P contains a complex element if and only if it contains a wttcover for the Cantor set. That is, if and only if for every real ..."
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Abstract. An infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a Π 0 1 class P contains a complex element if and only if it contains a wttcover for the Cantor set. That is, if and only if for every real Y there is an X in the P such that X �wtt Y. We show that this is also equivalent to the Π 0 1 class’s being large in some sense. We give an example of how this result can be used in the study of scattered linear orders. §1. Introduction. There has been interest in the literature over many years in studying various notions of the size of subclasses of 2 ω. In this paper we have tried to generalise and consolidate some of these ideas. We investigate a notion of size that has appeared independently in [1] and [5], namely the notion of a computable perfect class (computably growing in [5] and nonuphi in [1]). It is
Universal recursively enumerable sets of strings
 Developments in Language Theory (DLT’08), Lectures Notes in Comput. Sci. 5257, SpringerVerlag, Berlin, 2008, 170–182. S. Calude 21
"... Abstract. The present work clarifies the relation between domains of universal machines and r.e. prefixfree supersets of such sets. One such characterisation can be obtained in terms of the spectrum function sW (n) mapping n to the number of all strings of length n in the set W. An r.e. prefixfree ..."
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Abstract. The present work clarifies the relation between domains of universal machines and r.e. prefixfree supersets of such sets. One such characterisation can be obtained in terms of the spectrum function sW (n) mapping n to the number of all strings of length n in the set W. An r.e. prefixfree set W is the superset of the domain of a universal machine iff there are two constants c, d such that sW (n) +sW (n + 1) +... + sW (n + c) is between 2 n−H(n)−d and 2 n−H(n)+d for all n; W is the domain of a universal machine iff there is a constant c such that for each n the pair of n and sW (n)+sW (n + 1) +...+ sW (n + c) has at least the prefixfree description complexity n. There exists a prefixfree r.e. superset W of a domain of a universal machine which is the not a domain of a universal machine; still, the halting probability ΩW of such a set W is MartinLöf random. Furthermore, it is investigated to which extend this results can be transferred to plain universal machines. 1
ON THE NUMBER OF INFINITE SEQUENCES WITH TRIVIAL INITIAL SEGMENT COMPLEXITY
"... Abstract. The sequences which have trivial prefixfree initial segment complexity are known as Ktrivial sets, and form a cumulative hierarchy of length ω. We show that the problem of finding the number of Ktrivial sets in the various levels of the hierarchy is ∆0 3. This answers a question of Down ..."
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Abstract. The sequences which have trivial prefixfree initial segment complexity are known as Ktrivial sets, and form a cumulative hierarchy of length ω. We show that the problem of finding the number of Ktrivial sets in the various levels of the hierarchy is ∆0 3. This answers a question of Downey/Miller/Yu (see [DH10, Section 10.1.4]) which also appears in [Nie09, Problem 5.2.16]. We also show the same for the hierarchy of the low for K sequences, which are the ones that (when used as oracles) do not give shorter initial segment complexity compared to the computable oracles. In both cases the classification ∆0 3 is sharp. 1.
WORKING WITH STRONG REDUCIBILITIES ABOVE TOTALLY ωC.E. DEGREES
"... Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ω ..."
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Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ωc.e. iff every set in it is weak truthtable reducible to a hypersimple, or ranked, set. We also show that a c.e. degree is array computable iff every leftc.e. real of that degree is reducible in a computable Lipschitz way to a random leftc.e. real (an Ωnumber). 1.
Algorithmic Randomness and Computability
"... Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability. ..."
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Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability.
classes and strong degree spectra of relations
 Journal of Symbolic Logic 72 (2007), 1003
"... Abstract. We study the weak truthtable and truthtable degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truthtable reducible to any initial segment of any scattered compu ..."
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Abstract. We study the weak truthtable and truthtable degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truthtable reducible to any initial segment of any scattered computable linear order. Countable Π0 1 subsets of 2ω and Kolmogorov complexity play a major role in the proof.
Π 0 1 CLASSES AND STRONG DEGREE SPECTRA OF RELATIONS
"... Abstract. We study the weak truthtable and truthtable degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truthtable reducible to any initial segment of any scattered compu ..."
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Abstract. We study the weak truthtable and truthtable degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truthtable reducible to any initial segment of any scattered computable linear order. Countable Π 0 1 subsets of 2 ω and Kolmogorov complexity play a major role in the proof.
HIGHER KURTZ RANDOMNESS
"... Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it is in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1Kurtz randomness as being ∆ 1 1dominated and ∆ 1 1semitraceable. 1. ..."
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Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it is in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1Kurtz randomness as being ∆ 1 1dominated and ∆ 1 1semitraceable. 1.
CLASSES, STRONG MINIMAL COVERS AND HYPERIMMUNEFREE DEGREES
"... Abstract. We investigate issues surrounding an old question of Yates’ as to the existence of a minimal degree with no strong minimal cover, specifically with respect to the hyperimmunefree degrees. 1. ..."
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Abstract. We investigate issues surrounding an old question of Yates’ as to the existence of a minimal degree with no strong minimal cover, specifically with respect to the hyperimmunefree degrees. 1.