Results 1  10
of
16
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
Lowness notions, measure and domination
, 2008
"... Abstract. We show that positive measure domination implies uniform almost everywhere domination and that this proof translates into a proof in the subsystem WWKL0 (but not in RCA0) of the equivalence of various Lebesgue measure regularity statements introduced by Dobrinen and Simpson. This work also ..."
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Cited by 15 (1 self)
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Abstract. We show that positive measure domination implies uniform almost everywhere domination and that this proof translates into a proof in the subsystem WWKL0 (but not in RCA0) of the equivalence of various Lebesgue measure regularity statements introduced by Dobrinen and Simpson. This work also allows us to prove that low for weak 2randomness is the same as low for MartinLöf randomness (a result independently obtained by Nies). Using the same technique, we show that ≤LR implies ≤LK, generalizing the fact that low for MartinLöf randomness implies low for K. 1.
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.
Benign cost functions and lowness properties
"... Abstract. We show that the class of strongly jumptraceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of wellbehaved cost function, called benign. This characterisation implies the containment of the class of strongly jumptraceable c.e. ..."
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Cited by 9 (5 self)
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Abstract. We show that the class of strongly jumptraceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of wellbehaved cost function, called benign. This characterisation implies the containment of the class of strongly jumptraceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LRhard random degrees, and all ωc.e. random degrees. The last result implies recent results of Diamondstone’s and Ng’s regarding cupping with supwerlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees. 1.
Randomness, lowness and degrees
 J. of Symbolic Logic
, 2006
"... Abstract. We say that A ≤LR B if every Brandom number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally a ..."
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Cited by 9 (4 self)
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Abstract. We say that A ≤LR B if every Brandom number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e. restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α is not GL2 the LR degree of α bounds 2 ℵ0 degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees. 1.
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.
NONCUPPING, MEASURE AND COMPUTABLY ENUMERABLE SPLITTINGS
"... Abstract. We show that there is a computably enumerable function f (i.e. computably approximable from below) which dominates almost all functions and f ⊕ W is incomplete, for all incomplete computably enumerable sets W. Our main methodology is the LR equivalence relation on reals: A ≡LR B iff the no ..."
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Cited by 1 (1 self)
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Abstract. We show that there is a computably enumerable function f (i.e. computably approximable from below) which dominates almost all functions and f ⊕ W is incomplete, for all incomplete computably enumerable sets W. Our main methodology is the LR equivalence relation on reals: A ≡LR B iff the notions of Arandomness and Brandomness coincide. We also show that there are c.e. sets which cannot be split into two c.e. sets of the same LR degree. Moreover a c.e. set is low for random iff it computes no c.e. set with this property. 1.
BJØRN KJOSHANSSEN AND ANDRÉ NIES
"... Abstract. We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2random. We also study the class superhigh ✸ , and show that it contains some, but not all, of the noncomputable Ktrivial sets. 1. ..."
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Abstract. We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2random. We also study the class superhigh ✸ , and show that it contains some, but not all, of the noncomputable Ktrivial sets. 1.