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10
Randomness, relativization, and Turing degrees
 J. Symbolic Logic
, 2005
"... We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is nrandom if it is MartinLof random relative to . We show that a set is 2random if and only if there is a constant c such that infinitely many initial segments x of the set are cincompre ..."
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We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is nrandom if it is MartinLof random relative to . We show that a set is 2random if and only if there is a constant c such that infinitely many initial segments x of the set are cincompressible: C(x) c. The `only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of timebounded Ccomplexity.
Effective Hausdorff dimension
 In Logic Colloquium ’01
, 2005
"... ABSTRACT. We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective sdimensional Hausdorff measures, similar to the effectivization ..."
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ABSTRACT. We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective sdimensional Hausdorff measures, similar to the effectivization of Lebesgue measure by MARTINLÖF. It turns out that effective Hausdorff dimension allows to classify sequences according to their ‘degree ’ of algorithmic randomness, i.e., their algorithmic density of information. Earlier the works of STAIGER and RYABKO showed a deep connection between Kolmogorov complexity and Hausdorff dimension. We further develop this relationship and use it to give effective versions of some important properties of (classical) Hausdorff dimension. Finally, we determine the effective dimension of some objects arising in the context of computability theory, such as degrees and spans. 1.
LOWNESS FOR COMPUTABLE MACHINES
"... Abstract. Two lowness notions in the setting of Schnorr randomness have been studied (lowness for Schnorr randomness and tests, by Terwijn and Zambella [19], and by KjosHanssen, Stephan, and Nies [7]; and Schnorr triviality, by Downey, Griffiths and LaForte [3, 4] and Franklin [6]). We introduce lo ..."
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Cited by 1 (1 self)
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Abstract. Two lowness notions in the setting of Schnorr randomness have been studied (lowness for Schnorr randomness and tests, by Terwijn and Zambella [19], and by KjosHanssen, Stephan, and Nies [7]; and Schnorr triviality, by Downey, Griffiths and LaForte [3, 4] and Franklin [6]). We introduce lowness for computable machines, which by results of Downey and Griffiths [3] is an analog of lowness for K. We show that the reals that are low for computable machines are exactly the computably traceable ones, and so this notion coincides with that of lowness for Schnorr randomness and for Schnorr tests. 1.
STRONG JUMPTRACEABILITY II: KTRIVIALITY
, 2010
"... Abstract. We show that every strongly jumptraceable set is Ktrivial. Unlike other results, we do not assume that the sets in question are computably enumerable. 1. ..."
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Abstract. We show that every strongly jumptraceable set is Ktrivial. Unlike other results, we do not assume that the sets in question are computably enumerable. 1.
PSEUDOJUMP INVERSION AND SJTHARD SETS
, 2011
"... Abstract. There are noncomputable c.e. sets, computable from every SJThard c.e. set. This yields a natural pseudojump operator, increasing on all sets, which cannot be inverted back to a minimal pair or even avoiding an upper cone. 1. ..."
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Abstract. There are noncomputable c.e. sets, computable from every SJThard c.e. set. This yields a natural pseudojump operator, increasing on all sets, which cannot be inverted back to a minimal pair or even avoiding an upper cone. 1.
LIMIT COMPUTABILITY AND CONSTRUCTIVE MEASURE
"... Abstract. In this paper we study constructive measure and dimension in the class ∆0 2 of limit computable sets. We prove that the lower cone of any Turingincomplete set in ∆0 2 has ∆0 2dimension 0, and in contrast, that although the upper cone of a noncomputable set in ∆0 2 always has ∆0 2measure ..."
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Abstract. In this paper we study constructive measure and dimension in the class ∆0 2 of limit computable sets. We prove that the lower cone of any Turingincomplete set in ∆0 2 has ∆0 2dimension 0, and in contrast, that although the upper cone of a noncomputable set in ∆0 2 always has ∆0 2measure 0, upper cones in ∆0 2 have nonzero ∆0 2dimension. In particular the ∆0 2dimension of the Turing degree of ∅ ′ (the Halting Problem) is 1. Finally, it is proved that the low sets do not have ∆0 2measure 0, which means that the low sets do not form a small subset of ∆0 2. This result has consequences for the existence of biimmune sets. 1.
On the Quantitative Structure of ...
, 2000
"... We analyze the quantitative structure of 0 2 . Among other things, we prove that a set is Turing complete if and only if its lower cone is nonnegligible, and that the sets of r.e.degree form a small subset of 0 2 . Mathematical Subject Classification: 03D15, 03D30, 28E15 Keywords: Comput ..."
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We analyze the quantitative structure of 0 2 . Among other things, we prove that a set is Turing complete if and only if its lower cone is nonnegligible, and that the sets of r.e.degree form a small subset of 0 2 . Mathematical Subject Classification: 03D15, 03D30, 28E15 Keywords: Computable measure theory, Turing degrees, completeness. 1 Introduction We study an eective measure theory suited for the study of 0 2 , the second level of the arithmetical hierarchy (alternatively, the sets computable relative to the halting problem K). This work may be seen as part of the constructivist tradition in mathematics as documented in [6]. The framework for eectivizing measure theory that we employ uses martingales. Martingales were rst applied to the study of random sequences by J. Ville [22]. Recursive martingales were studied in Schnorr [19], and became popular in complexity theory in more recent years through the work of Lutz [14, 15]. Lutz Research supported by a Ma...
The Computational Complexity Column
, 1998
"... Introduction Investigation of the measuretheoretic structure of complexity classes began with the development of resourcebounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resourcebounded measure to be a powerful too ..."
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Introduction Investigation of the measuretheoretic structure of complexity classes began with the development of resourcebounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resourcebounded measure to be a powerful tool that sheds new light on many aspects of computational complexity. Recent survey papers by Lutz [60], AmbosSpies and Mayordomo [3], and Buhrman and Torenvliet [22] describe many of the achievements of this line of inquiry. In this column, we give a more recent snapshot of resourcebounded measure, focusing not so much on what has been achieved to date as on what we hope will be achieved in the near future. Section 2 below gives a brief, nontechnical overview of resourcebounded measure in terms of its motivation and principal ideas. Sections 3, 4, and 5 describe twelve specific open problems in the area. We have used the following three criteria in choosing these problems. 1. Their
EFFECTIVE PACKING DIMENSION AND TRACEABILITY
"... The concern of this paper is with effective packing dimension. This concept can be traced back to the work of Borel and Lebesgue who studied measure as a way of specifying the size of sets. Carathéodory later generalized Lebesgue measure to the ndimensional Euclidean space, and this was taken furth ..."
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The concern of this paper is with effective packing dimension. This concept can be traced back to the work of Borel and Lebesgue who studied measure as a way of specifying the size of sets. Carathéodory later generalized Lebesgue measure to the ndimensional Euclidean space, and this was taken further by Hausdorff [Hau19]