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26
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 35 (17 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
Uniform almost everywhere domination
 Journal of Symbolic Logic
, 2006
"... ABSTRACT. We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed t ..."
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Cited by 34 (1 self)
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ABSTRACT. We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the prooftheoretic strength of the regularity of Lebesgue measure for G δ sets. Our constructions essentially settle the reverse mathematical classification of this principle. 1.
Low for random reals and positivemeasure domination
 Proceedings of the American Mathematical Society, 2007. Preprint
, 2005
"... Abstract. The low for random reals are characterized topologically, as well as in terms of domination of Turing functionals on a set of positive measure. 1. ..."
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Cited by 30 (1 self)
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Abstract. The low for random reals are characterized topologically, as well as in terms of domination of Turing functionals on a set of positive measure. 1.
Almost everywhere domination and superhighness
 Mathematical Logic Quarterly
"... Let ω denote the set of natural numbers. For functions f, g: ω → ω, we say that f is dominated by g if f(n) < g(n) for all but finitely many n ∈ ω. We consider the standard “fair coin ” probability measure on the space 2 ω of infinite sequences of 0’s and 1’s. A Turing oracle B is said to be almo ..."
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Cited by 28 (8 self)
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Let ω denote the set of natural numbers. For functions f, g: ω → ω, we say that f is dominated by g if f(n) < g(n) for all but finitely many n ∈ ω. We consider the standard “fair coin ” probability measure on the space 2 ω of infinite sequences of 0’s and 1’s. A Turing oracle B is said to be almost everywhere dominating if, for measure one many X ∈ 2 ω, each function which is Turing computable from X is dominated by some function which is Turing computable from B. Dobrinen and Simpson have shown that the almost everywhere domination property and some of its variant properties are closely related to the reverse mathematics of measure theory. In this paper we exposit some recent results of KjosHanssen, KjosHanssen/Miller/Solomon, and others concerning LRreducibility and almost everywhere domination. We also prove the following new result: If B is almost everywhere dominating, then B is superhigh, i.e., 0 ′′ is
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 27 (5 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 ASSOCIATED WITH EFFECTIVELY CLOSED SETS
, 2011
"... earlier draft of this paper. The study of mass problems and Muchnik degrees was originally motivated by Kolmogorov’s nonrigorous 1932 interpretation of intuitionism as a calculus of problems. The purpose of this paper is to summarize recent investigations into the lattice of Muchnik degrees of none ..."
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Cited by 14 (3 self)
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earlier draft of this paper. The study of mass problems and Muchnik degrees was originally motivated by Kolmogorov’s nonrigorous 1932 interpretation of intuitionism as a calculus of problems. The purpose of this paper is to summarize recent investigations into the lattice of Muchnik degrees of nonempty effectively closed sets in Euclidean space. Let Ew be this lattice. We show that Ew provides an elegant and useful framework for the classification of certain foundationally interesting problems which are algorithmically unsolvable. We exhibit some specific degrees in Ew which are associated with such problems. In addition, we present some structural results concerning the lattice Ew. One of these results answers a question which arises naturally from the Kolmogorov interpretation. Finally, we show how Ew can be applied in symbolic dynamics, toward the classification of tiling problems
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 f ..."
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Cited by 10 (7 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.
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 (6 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.
THE STRENGTH OF SOME COMBINATORIAL PRINCIPLES RELATED TO RAMSEY’S THEOREM FOR PAIRS
"... We study the reverse mathematics and computabilitytheoretic strength of (stable) Ramsey’s Theorem for pairs and the related principles COH and DNR. We show that SRT 2 2 implies DNR over RCA0 but COH does not, and answer a question of Mileti by showing that every computable stable 2coloring of pa ..."
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Cited by 8 (5 self)
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We study the reverse mathematics and computabilitytheoretic strength of (stable) Ramsey’s Theorem for pairs and the related principles COH and DNR. We show that SRT 2 2 implies DNR over RCA0 but COH does not, and answer a question of Mileti by showing that every computable stable 2coloring of pairs has an incomplete ∆ 0 2 infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT 2 2 does not imply RT 2 2.
Mass problems and measuretheoretic regularity
, 2009
"... Research supported by NSF grants DMS0600823 and DMS0652637. ..."
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Cited by 6 (3 self)
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Research supported by NSF grants DMS0600823 and DMS0652637.