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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
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.
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.
Demuth’s path to randomness
 In Proceedings of the 2012 international conference on Theoretical Computer Science: computation, physics and beyond, WTCS’12
, 2012
"... Abstract. Osvald Demuth (1936–1988) studied constructive analysis from the viewpoint of the Russian school of constructive mathematics. In the course of his work he introduced various notions of effective null set which, when phrased in classical language, yield a number of major algorithmic random ..."
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Abstract. Osvald Demuth (1936–1988) studied constructive analysis from the viewpoint of the Russian school of constructive mathematics. In the course of his work he introduced various notions of effective null set which, when phrased in classical language, yield a number of major algorithmic randomness notions. In addition, he proved several results connecting constructive analysis and randomness that were rediscovered only much later. In this paper, we trace the path that took Demuth from his constructivist roots to his deep and innovative work on the interactions between constructive analysis, algorithmic randomness, and computability theory. We will focus specifically on (i) Demuth’s work on the differentiability of Markov computable functions and his study of constructive versions of the Denjoy alternative, (ii) Demuth’s independent discovery of the main notions of algorithmic randomness, as well as the development of Demuth randomness, and (iii) the interactions of truthtable reducibility, algorithmic randomness, and semigenericity in Demuth’s work. §1. Introducing Demuth. The mathematician Osvald Demuth worked primarily on constructive analysis in the Russian style, which was initiated by Markov, Šanin, Cĕıtin, and others in the 1950s. Born in 1936 in Prague, Demuth
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.
Traceable sets
"... Abstract. We investigate systematically into the various possible notions of traceable sets and the relations they bear to each other and to other notions such as diagonally noncomputable sets or complex and autocomplex sets. We review known notions and results that appear in the literature in diffe ..."
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Abstract. We investigate systematically into the various possible notions of traceable sets and the relations they bear to each other and to other notions such as diagonally noncomputable sets or complex and autocomplex sets. We review known notions and results that appear in the literature in different contexts, put them into perspective and provide simplified or at least more direct proofs. In addition, we introduce notions of traceability and complexity such as infinitely often versions of jump traceability and of complexity, and derive results about these notions that partially can be viewed as a natural completion of the results known before. Finally, we give a result about polynomialtime bounded notions of traceability and complexity that shows that in principle the equivalences derived so far can be transferred to the timebounded setting. 1 Introduction and
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.