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Mass problems and almost everywhere domination
 Mathematical Logic Quarterly
, 2007
"... We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and MartinLöf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED ..."
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Cited by 10 (7 self)
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We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and MartinLöf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED, MLR×AED, MLR∩AED respectively. Let Pw be the lattice of degrees of unsolvability of mass problems associated with nonempty Π 0 1 subsets of 2 ω. Let 1 and 0 be the top and bottom elements of Pw. We show that inf(b1,1) and inf(b2,1) and inf(b3,1) belong to Pw and that 0 < inf(b1,1) < inf(b2,1) < inf(b3,1) < 1. Under the natural embedding of the recursively enumerable Turing degrees into Pw, we show that inf(b1,1) and inf(b3,1) but not inf(b2,1) are comparable with some recursively enumerable Turing degrees other than 0 and 0 ′. In order to make this paper more selfcontained, we exposit the proofs of some recent theorems due to Hirschfeldt, Miller, Nies, and Stephan.
Some fundamental issues concerning degrees of unsolvability
 In [6], 2005. Preprint
, 2007
"... Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a ..."
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Cited by 9 (8 self)
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Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a “smallness property ” of an infinite, corecursively enumerable set A ⊆ ω which ensures that the Turing degree deg T (A) = a ∈ RT is> 0 and < 0 ′. In order to address these issues, we embed RT into a slightly larger degree structure, Pw, which is much better behaved. Namely, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. We define a specific, natural embedding of RT into Pw, and we present some recent and new research results.
A basis theorem for Π0 1 classes of positive measure and jump inversion for random reals
 Proceedings of the American Mathematical Society
, 2006
"... We extend the Shoenfield jump inversion theorem to the members of any Π0 1 class P⊆2ω with nonzero measure; i.e., for every Σ0 2 set S ≥T ∅ ′, there is a ∆0 2 real A ∈Psuch that A ′ ≡T S. In particular, we get jump inversion for ∆0 2 1random reals. This paper is part of an ongoing program to stud ..."
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Cited by 6 (1 self)
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We extend the Shoenfield jump inversion theorem to the members of any Π0 1 class P⊆2ω with nonzero measure; i.e., for every Σ0 2 set S ≥T ∅ ′, there is a ∆0 2 real A ∈Psuch that A ′ ≡T S. In particular, we get jump inversion for ∆0 2 1random reals. This paper is part of an ongoing program to study the relationship between two fundamental notions of complexity for real numbers. The first is the computational complexity of a real as captured, for example, by its Turing degree. The second is the intrinsic randomness of a real. In particular, we are interested in the 1random reals, which were introduced by MartinLöf [13] and represent the most widely studied randomness class. For the purposes of this introduction, we will assume that the reader is somewhat familiar with basic algorithmic randomness, as per LiVitányi [12], and with computability theory [18]. A review of notation and terminology will be given in Section 1. Intuitively, a 1random real is very complex. This complexity can be captured formally in terms of unpredictability or incompressibility, but is it reflected in the
Lowness and Π0 2 nullsets
 J. Symbolic Logic
, 2006
"... Abstract. We prove that there exists a noncomputable c.e. real which is low for weak 2randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2randomness are low for MartinLöf randomness. 1. ..."
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Abstract. We prove that there exists a noncomputable c.e. real which is low for weak 2randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2randomness are low for MartinLöf randomness. 1.
The Kolmogorov Complexity of Random Reals
 Ann. Pure Appl. Logic
, 2003
"... We investigate the initial segment complexity of random reals. Let K(... ..."
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Cited by 5 (1 self)
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We investigate the initial segment complexity of random reals. Let K(...
Mass problems and measuretheoretic regularity
, 2009
"... Research supported by NSF grants DMS0600823 and DMS0652637. ..."
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Cited by 4 (3 self)
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Research supported by NSF grants DMS0600823 and DMS0652637.
Random Continuous Functions
, 2006
"... We investigate notions of randomness in the space C(2 N) of continuous functions on 2 N. A probability measure is given and a version of the MartinLöf Test for randomness is defined. Random ∆ 0 2 continuous functions exist, but no computable function can be random and no random function can map a c ..."
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We investigate notions of randomness in the space C(2 N) of continuous functions on 2 N. A probability measure is given and a version of the MartinLöf Test for randomness is defined. Random ∆ 0 2 continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any y ∈ 2 N, there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set.
Relative Randomness via RKReducibility
, 2006
"... Its focus is relative randomness as measured by rKreducibility, a refinement of Turing reducibility defined as follows. An infinite binary sequence A is rKreducible to an infinite binary sequence B, written A ≤rK B, if ∃d ∀n. K(A ↾ nB ↾ n) < d, where K(στ) is the conditional prefixfree descript ..."
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Cited by 2 (0 self)
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Its focus is relative randomness as measured by rKreducibility, a refinement of Turing reducibility defined as follows. An infinite binary sequence A is rKreducible to an infinite binary sequence B, written A ≤rK B, if ∃d ∀n. K(A ↾ nB ↾ n) < d, where K(στ) is the conditional prefixfree descriptional complexity of σ given τ. Herein i study the relationship between relative randomness and (standard) absolute randomness and that between relative randomness and computable analysis. i Acknowledgements Foremost, i would like to thank my advisor, Steffen Lempp, for all his words of wisdom and encouragement throughout the long years of the Ph.D. Also, thanks to Frank Stephan who worked with me on some of the questions herein at the Computational Prospects of
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.