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An extension of the recursively enumerable Turing degrees
 Journal of the London Mathematical Society
, 2006
"... Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overco ..."
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Cited by 26 (15 self)
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Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overcome this difficulty, we embed RT into a larger degree structure which is better behaved. Namely, consider the countable distributive lattice Pw consisting of the weak degrees (also known as Muchnik degrees) of mass problems associated with nonempty Π 0 1 subsets of 2ω. It is known that Pw contains a bottom degree 0 and a top degree 1 and is structurally rich. Moreover, Pw contains many specific, natural degrees other than 0 and 1. In particular, we show that in Pw one has 0 < d < r1 < inf(r2, 1) < 1. Here, d is the weak degree of the diagonally nonrecursive functions, and rn is the weak degree of the nrandom reals. It is known that r1 can be characterized as the maximum weak degree ofaΠ 0 1 subset of 2ω of positive measure. We now show that inf(r2, 1) can be characterized as the maximum weak degree of a Π 0 1 subset of 2ω, the Turing upward closure of which is of positive measure. We exhibit a natural embedding of RT into Pw which is onetoone, preserves the semilattice structure of RT, carries 0 to 0, and carries 0 ′ to 1. Identifying RT with its image in Pw, we show that all of the degrees in RT except 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2, 1) inPw. 1.
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 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.