Small Π 01 Classes . Stephen Binns

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

Embeddings into the Medvedev and Muchnik lattices of Π1 classes

Let Pw and PM be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty Π1 subsets of 2 , under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of Pw. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of PM .

Muchnik and Medvedev Degrees of Π 01 Subsets of 2 ω

Effectively Closed Sets Π 01 Classes DRAFT August 2014

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 ...