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34
Mass problems and hyperarithmeticity
, 2006
"... A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of we ..."
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Cited by 24 (16 self)
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A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of the Cantor space. The lattice Pw has been studied in previous publications. The purpose of this paper is to show that Pw partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in Pw which are indexed by the ordinal numbers less than ω CK 1 and which correspond to the hyperarithmetical hierarchy. Namely, for each α < ω CK 1 let hα be the weak degree of 0 (α) , the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p ∗ be the weak degree
A lower cone in the wtt degrees of nonintegral effective dimension
 In Proceedings of IMS workshop on Computational Prospects of Infinity
, 2006
"... ABSTRACT. For any rational number r, we show that there exists a set A (weak truthtable reducible to the halting problem) such that any set B weak truthtable reducible to it has effective Hausdorff dimension at most r, where A itself has dimension at least r. This implies, for any rational r, the e ..."
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Cited by 23 (2 self)
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ABSTRACT. For any rational number r, we show that there exists a set A (weak truthtable reducible to the halting problem) such that any set B weak truthtable reducible to it has effective Hausdorff dimension at most r, where A itself has dimension at least r. This implies, for any rational r, the existence of a wttlower cone of effective dimension r. 1.
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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Lowness for the class of Schnorr random reals
 SIAM Journal on Computing
, 2005
"... We answer a question of AmbosSpies and Kučera in the affirmative. They asked whether, when a real is low for Schnorr randomness, it is already low for Schnorr tests. ..."
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Cited by 12 (5 self)
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We answer a question of AmbosSpies and Kučera in the affirmative. They asked whether, when a real is low for Schnorr randomness, it is already low for Schnorr tests.
EXTRACTING INFORMATION IS HARD: A TURING DEGREE OF NONINTEGRAL EFFECTIVE HAUSDORFF DIMENSION
"... Abstract. We construct a ∆0 2 infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimension can be viewed as a measure of the information density of a sequence. In particular, the dimension of A ..."
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Cited by 8 (0 self)
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Abstract. We construct a ∆0 2 infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimension can be viewed as a measure of the information density of a sequence. In particular, the dimension of A ∈ 2ω is the lim inf of the ratio between the information content and length of initial segments of A. Thus the main result demonstrates that it is not always possible to extract information from a partially random source to produce a sequence that has higher information density. 1.
Lowness for weakly 1generic and Kurtzrandom
 in Theory and Applications of Models of Computation: Third Internationa l Conference, TAMC 2006
, 2006
"... Abstract. We prove that a set is low for weakly 1generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, we refute a recent conjecture on the characterization of these sets. Furthermore, we show that every set which is low for w ..."
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Cited by 7 (2 self)
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Abstract. We prove that a set is low for weakly 1generic iff it has neither dnr nor hyperimmune Turing degree. As this notion is more general than being recursively traceable, we refute a recent conjecture on the characterization of these sets. Furthermore, we show that every set which is low for weakly 1generic is also low for Kurtzrandom. 1
Effectively closed sets of measures and randomness
 Ann. Pure Appl. Logic
"... We show that if a real x ∈ 2ω is strongly Hausdorff Hhrandom, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure µ such that the µmeasure of the basic open cylinders shrinks according to h. The proof uses a new method to con ..."
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Cited by 7 (1 self)
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We show that if a real x ∈ 2ω is strongly Hausdorff Hhrandom, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure µ such that the µmeasure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for Π0 1classes applied to closed sets of probability measures. We use the main result to give a new proof of Frostman’s Lemma, to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman’s Theorem. 1
CALCULUS OF COST FUNCTIONS
"... Abstract. We study algebraic properties of cost functions. We give an application: building sets close to being Turing complete. 1. ..."
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Cited by 6 (4 self)
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Abstract. We study algebraic properties of cost functions. We give an application: building sets close to being Turing complete. 1.
KOLMOGOROV COMPLEXITY OF INITIAL SEGMENTS OF SEQUENCES AND ARITHMETICAL DEFINABILITY
"... Abstract. The structure of the Kdegrees provides a way to classify sets of natural numbers or infinite binary sequences with respect to the level of randomness of their initial segments. In the Kdegrees of infinite binary sequences, X is below Y if the prefixfree Kolmogorov complexity of the firs ..."
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Cited by 5 (4 self)
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Abstract. The structure of the Kdegrees provides a way to classify sets of natural numbers or infinite binary sequences with respect to the level of randomness of their initial segments. In the Kdegrees of infinite binary sequences, X is below Y if the prefixfree Kolmogorov complexity of the first n bits of X is less than the complexity of the first n bits of Y, for each n. Identifying infinite binary sequences with subsets of N, we study the Kdegrees of arithmetical sets and explore the interactions between arithmetical definability and prefix free Kolmogorov complexity. We show that in the Kdegrees, for each n> 1 there exists a Σ0 n nonzero degree which does not bound any ∆0 n nonzero degree. An application of this result is that in the Kdegrees there exists a Σ0 2 degree which forms a minimal pair with all Σ0 1 degrees. This extends work of Csima/Montalbán [CM06] and Merkle/Stephan [MS07]. Our main result is that, given any ∆0 2 family C of sequences, there is a ∆0 2 sequence of nontrivial initial segment complexity which is not larger than the initial segment complexity of any nontrivial member of C. This general theorem has the following surprising consequence. There is a 0 ′computable sequence of nontrivial initial segment complexity which is not larger than the initial segment complexity of any nontrivial computably enumerable set. Our analysis and results demonstrate that, examining the extend to which arithmetical definability interacts with the K reducibility (and in general any ‘weak reducibility’) is a fruitful way of studying the induced structure. 1.
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