Classifying the Branching Degrees in the Medvedev Lattice of Π 1 0 Classes

Abstract

A ${\Pi }_1^0$ class can be defined as the set of infinite paths through a computable tree. For classes $P$ and $Q$, say that $P$ is Medvedev reducible to $Q$, $P{\le }_{M}Q$, if there is a computably continuous functional mapping $Q$ into $P$. Let ${ℒ}_M$ be the lattice of degrees formed by ${\Pi }_1^0$ subclasses of $2^{\omega }$ under the Medvedev reducibility. In "Non-branching degrees in the Medvedev lattice of ${\Pi }_1^0$ classes," I provided a characterization of nonbranching/branching and a classification of the nonbranching degrees. In this paper, I present a similar classification of the branching degrees. In particular, $P$ is separable if there is a clopen set $C$ such that $P\cap C\ne \varnothing \ne P\cap C^c$ and $P\cap C{\perp }_{M}P\cap C^c$. By the results in the first paper, separability is an invariant of a Medvedev degree and a degree is branching if and only if it contains a separable member. Further define $P$ to be hyperseparable if, for all such $C$, $P\cap C{\perp }_{M}P\cap C^c$ and totally separable if, for all $X,Y\in P$, $X{\perp }_{T}Y$. I will show that totally separable implies hyperseparable implies separable and that the reverse implications do not hold, that is, that these are three distinct types of branching degrees. Along the way I will show some related results and present a combinatorial framework for constructing ${\Pi }_1^0$ classes with priority arguments.