Inside the Muchnik degrees II: The degree structures induced by the arithmetical hierarchy of countably continuous functions

Abstract

It is known that infinitely many Medvedev degrees exist inside the Muchnik degree of any nontrivial Π10 subset of Cantor space. We shed light on the fine structures inside these Muchnik degrees related to learnability and piecewise computability. As for nonempty Π10 subsets of Cantor space, we show the existence of a finite-Δ20-piecewise degree containing infinitely many finite-(Π10)2-piecewise degrees, and a finite-(Π20)2-piecewise degree containing infinitely many finite-Δ20-piecewise degrees (where (Πn0)2 denotes the difference of two Πn0 sets), whereas the greatest degrees in these three “finite-Γ-piecewise” degree structures coincide. Moreover, as for nonempty Π10 subsets of Cantor space, we also show that every nonzero finite-(Π10)2-piecewise degree includes infinitely many Medvedev (i.e., one-piecewise) degrees, every nonzero countable-Δ20-piecewise degree includes infinitely many finite-piecewise degrees, every nonzero finite-(Π20)2-countable-Δ20-piecewise degree includes infinitely many countable-Δ20-piecewise degrees, and every nonzero Muchnik (i.e., countable-Π20-piecewise) degree includes infinitely many finite-(Π20)2-countable-Δ20-piecewise degrees. Indeed, we show that any nonzero Medvedev degree and nonzero countable-Δ20-piecewise degree of a nonempty Π10 subset of Cantor space have the strong anticupping properties. Finally, we obtain an elementary difference between the Medvedev (Muchnik) degree structure and the finite-Γ-piecewise degree structure of all subsets of Baire space by showing that none of the finite-Γ-piecewise structures is Brouwerian, where Γ is any of the Wadge classes mentioned above.