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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
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 almost ..."
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Cited by 17 (9 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
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.
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.
Special Session
, 2009
"... Let A be a finite set of symbols. The 2dimensional shift space on A is A Z×Z with shift operators S1 and S2 given by S1(x)(m, n) = x(m + 1, n) and S2(x)(m, n) = x(m, n + 1). A 2dimensional subshift is a nonempty, closed subset of A Z×Z which is invariant under S1 and S2. A 2dimensional subshift ..."
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Let A be a finite set of symbols. The 2dimensional shift space on A is A Z×Z with shift operators S1 and S2 given by S1(x)(m, n) = x(m + 1, n) and S2(x)(m, n) = x(m, n + 1). A 2dimensional subshift is a nonempty, closed subset of A Z×Z which is invariant under S1 and S2. A 2dimensional subshift is said to be of finite type if it is defined by a finite set of excluded finite configurations of symbols. We regard real numbers and points of A Z×Z as Turing oracles. If X and Y are sets of Turing oracles, we say that X is Muchnik reducible to Y if each y ∈ Y can be used to compute some x ∈ X. The Muchnik degree of X is the equivalence class of X under mutual Muchnik reducibility. We prove that the Muchnik degrees of 2dimensional subshifts of finite type are the same as the Muchnik degrees of nonempty, effectively closed sets of real numbers. We then apply known results about such Muchnik degrees to obtain an infinite family of 2dimensional subshifts of finite type which are, in a certain strong sense, mutually independent. Our application is stated purely in terms of symbolic dynamics, with no mention of Muchnik reducibility.
Recent Aspects of Mass Problems: Symbolic Dynamics and Intuitionism
, 2008
"... This note consists of an abstract and references for a talk given on February 21, ..."
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This note consists of an abstract and references for a talk given on February 21,
EMBEDDING FD(ω) INTO Ps DENSELY
, 708
"... Abstract. Let Ps be the lattice of degrees of nonempty Π 0 1 subsets of 2ω under Medvedev reducibility. Binns and Simpson proved that FD(ω), the free distributive lattice on countably many generators, is latticeembeddable below any nonzero element in Ps. Cenzer and Hinman proved that Ps is dense, ..."
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Abstract. Let Ps be the lattice of degrees of nonempty Π 0 1 subsets of 2ω under Medvedev reducibility. Binns and Simpson proved that FD(ω), the free distributive lattice on countably many generators, is latticeembeddable below any nonzero element in Ps. Cenzer and Hinman proved that Ps is dense, by adapting the Sacks Preservation and Sacks Coding Strategies used in the proof of the density of the c.e. Turing degrees. With a construction that is a modification of the one by Cenzer and Hinman, we improve on the result of Binns and Simpson by showing that for any U <s V, we can lattice embed FD(ω) into Ps strictly between degs(U) and degs(V). We also note that, in contrast to the infinite injury in the proof of the Sacks Density Theorem, in our proof all injury is finite, and that this is also true for the proof of Cenzer and Hinman, if a straightforward simplification is made. 1.