Medvedev and Muchnik Degrees of Nonempty Π 01 Subsets of 2 ω

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 ≤T Y . We say that P is Medvedev reducible to Q, abbreviated P ≤M Q, if there exists a recursive functional Φ : Q→ P . Note that P ≤M Q implies P ≤w Q, but not conversely. Sorbi [13] gives a useful survey of Medvedev and Muchnik degrees of arbitrary subsets of 2. Here we initiate a study of Medvedev and Muchnik degrees of nonempty Π1 subsets of 2 . Theorems 1 and 2 below have been proved in collaboration with my Ph. D. student Stephen Binns. Let P be the set of nonempty Π1 subsets of 2. Let PM (respectively Pw) be the set of Medvedev (respectively Muchnik) degrees of members of P, ordered by Medvedev (respectively Muchnik) reducibility. We say that P ∈ P is Medvedev complete (respectively Muchnik complete) if P ≥M Q (respectively P ≥w Q) for all Q ∈ P. For example, the set of complete extensions of Peano Arithmetic is Medvedev complete. Clearly every Medvedev