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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.
Mass problems and measuretheoretic regularity
, 2009
"... Research supported by NSF grants DMS0600823 and DMS0652637. ..."
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Research supported by NSF grants DMS0600823 and DMS0652637.
Jump inversions inside effectively closed sets and applications to randomness
 J. Symbolic Logic
"... Abstract. We study inversions of the jump operator on Π0 1 classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2random ..."
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Abstract. We study inversions of the jump operator on Π0 1 classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2random sets which are not 2random, and the jumps of the weakly 1random relative to 0 ′ sets which are not 2random. Both of the classes coincide with the degrees above 0 ′ which are not 0 ′dominated. A further application is the complete solution of [Nie09, Problem 3.6.9]: one direction of van Lambalgen’s theorem holds for weak 2randomness, while the other fails. Finally we discuss various techniques for coding information into incomplete randoms. Using these techniques we give a negative answer to [Nie09, Problem 8.2.14]: not all weakly 2random sets are array computable. In fact, given any oracle X, there is a weakly 2random which is not array computable relative to X. This contrasts with the fact that all 2random sets are array computable. 1.
Complexity of Randomness Notions
"... Abstract. Schnorr famously proved that MartinLöfrandomness of a sequence A can be characterised via the complexity of A’s initial segments. Nies, Stephan and Terwijn as well as independently Miller showed that Kolmogorov randomness coincides with MartinLöf randomness relative to the halting pro ..."
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Abstract. Schnorr famously proved that MartinLöfrandomness of a sequence A can be characterised via the complexity of A’s initial segments. Nies, Stephan and Terwijn as well as independently Miller showed that Kolmogorov randomness coincides with MartinLöf randomness relative to the halting problem K; that is, a set A is MartinLöf random relative to K iff there is no function f such that for all m and all n> f(m) it holds that C(A(0)A(1)... A(n)) ≤ n−m. In the present work it is shown that characterisations of this style can also be given for other randomness criteria like strongly random, Kurtz random relative to K, PAincomplete MartinLöf random and strongly Kurtz random; here one does not just quantify over all functions f but over functions f of a specific form. For example, A is MartinLöf random and PAincomplete iff there is no Arecursive function f such that for all m and all n> f(m) it holds that C(A(0)A(1)... A(n)) ≤ n −m. The characterisation for strong randomness relates to functions which are the concatenation of an Arecursive function executed after a Krecursive function; this solves an open problem of Nies. In addition to this, characterisations of a similar style are also given for Demuth randomness and Schnorr randomness relative to K. Although the unrelativised versions of Kurtz randomness and Schnorr randomness do not admit such a characterisation in terms of plain Kolmogorov complexity, Bienvenu and Merkle gave one in terms of Kolmogorov complexity defined by computable machines. 1
CHARACTERIZING LOWNESS FOR DEMUTH RANDOMNESS
"... Abstract. We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [35]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial rel ..."
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Abstract. We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [35]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable. 1.