Medvedev Degrees, Muchnik Degrees, Subsystems of Z2 and Reverse Mathematics
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چکیده
منابع مشابه
An extension of the recursively enumerable Turing degrees
We embed the upper semilattice of r.e. Turing degrees into a slightly larger structure which is better behaved and more foundationally relevant. For P,Q ⊆ 2, we say P is Muchnik reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y . We let Pw denote the lattice of Muchnik degrees of nonempty Π1 subsets of 2. Pw is a countable distributive lattice with 0 and 1....
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We prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order theory of true arithmetic are pairwise recursively isomorphic (obtained independently by Lewis, Nies, and Sorbi [7]). We then restrict our attention to the degrees of closed sets and prove that the following theories are pairwise recursively isomorphic: the first-orde...
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This is a report for my presentation at the upcoming meeting on Berechenbarkeitstheorie (“Computability Theory”), Oberwolfach, January 21–27, 2001. We use 2 to denote the space of infinite sequences of 0’s and 1’s. For X, Y ∈ 2, X ≤T Y means that X is Turing reducible to Y . For P,Q ⊆ 2 we say that P is Muchnik reducible to Q, abbreviated P ≤w Q, if for all Y ∈ Q there exists X ∈ P such that X ...
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