Muchnik and Medvedev Degrees of Π 01 Subsets of 2 ω

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

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 ...

Inside the Muchnik Degrees II: The Degree Structures induced by the Arithmetical Hierarchy of Countably Continuous Functions

It is known that infinitely many Medvedev degrees exist inside the Muchnik degree of any nontrivial Π1 subset of Cantor space. We shed light on the fine structures inside these Muchnik degrees related to learnability and piecewise computability. As for nonempty Π1 subsets of Cantor space, we show the existence of a finite-∆2-piecewise degree containing infinitely many finite-(Π 0 1)2-piecewise ...

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 .

Coding true arithmetic in the Medvedev and Muchnik degrees

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...