Non-branching degrees in the Medvedev lattice of Π01 classes

Special thanks to my advisor Steffen Lempp for his guidance and conversation. Abstract Always Sometimes 2 In this talk I will present the necessary background and then describe and characterize the non-branching degrees. Significantly, I will show three distinct classes of non-branching degrees. The Medvedev lattice of Π 0 1 classes is a lattice of certain subsets of 2 ω under a natural reduction. It is known that this lattice always splits. It is also known that branching does not always hold. In particular, the separating classes of c.e. sets induce non-branching degrees. 3 In this context we can think of a computable function φ : T → S. If this function respects the tree structure then it will induce a functional ϕ : [T] → [S] which is called computably continuous and is continuous in the topology of ω ω. Specifically, φ must satisfy σ τ ⇒ φ (σ) φ (τ); ∀X ∈ [T] lim n→∞ φ (X n) ∈ [S]. A tree T is a subset of ω <ω which is closed under initial prefixes. The prefix relation is denoted by ; the set of infinite paths through T by [T]; and the set of extendible nodes, those that are prefixes of members of [T], by Ext(T). Note that Ext(T) is itself a tree and [T] = [Ext(T)]. In 2 ω the clopen sets are finite unions of cones. Formally, for σ ∈ 2 <ω define I(σ) = {X : σ X}.