Abstract
A result of Soare and Stob asserts that for any non-recursive r.e. setC, there exists a r.e.[C] setA such thatA⊕C is not of r.e. degree. A setY is called [of]m-REA (m-REA[C] [degree] iff it is [Turing equivalent to] the result of applyingm-many iterated ‘hops’ to the empty set (toC), where a hop is any function of the formX→X ⊕W Xe . The cited result is the special casem=0,n=1 of our Theorem. Form=0,1, and any (m+1)-REA setC, ifC is not ofm-REA degree, then for alln there exists an-r.e.[C] setA such thatA ⊕C is not of (m+n)-REA degree. We conjecture that this holds also form≥2.
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German speakers should not be unduely influenced by the acronym for this title
Partially supported by an NSF Postdoctoral Fellowship and the US Army Research Office through the Mathematical Sciences Institute of Cornell University
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Cholak, P.A., Hinman, P.G. Iterated relative recursive enumerability. Arch Math Logic 33, 321–346 (1994). https://doi.org/10.1007/BF01278463
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DOI: https://doi.org/10.1007/BF01278463