Abstract
We study a strong enumeration reducibility, called bounded enumeration reducibility and denoted by ≤be, which is a natural extension of s-reducibility ≤s. We show that ≤s, ≤be, and enumeration reducibility do not coincide on the \({\Pi^0_1}\) –sets, and the structure \({\boldsymbol{\mathcal{D}_{\rm be}}}\) of the be-degrees is not elementarily equivalent to the structure of the s-degrees. We show also that the first order theory of \({\boldsymbol{\mathcal{D}_{\rm be}}}\) is computably isomorphic to true second order arithmetic: this answers an open question raised by Cooper (Z Math Logik Grundlag Math 33:537–560, 1987).
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The authors were partially supported by the project Computability with partial information, sponsored by BNSF, Contract No: D002-258/18.12.08.
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Marsibilio, D., Sorbi, A. Bounded enumeration reducibility and its degree structure. Arch. Math. Logic 51, 163–186 (2012). https://doi.org/10.1007/s00153-011-0259-2
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DOI: https://doi.org/10.1007/s00153-011-0259-2