Inside the Muchnik degrees II: The degree structures induced by the arithmetical hierarchy of countably continuous functions
Annals of Pure and Applied Logic
It is known that infinitely many Medvedev degrees exist inside the Muchnik degree of any nontrivial Π 0 1 subset of Cantor space. We shed light on the fine structures inside these Muchnik degrees related to learnability and piecewise computability. As for nonempty Π 0 1 subsets of Cantor space, we show the existence of a finite-∆ 0 2 -piecewise degree containing infinitely many finite-(Π 0 1 ) 2 -piecewise degrees, and a finite-(Π 0 2 ) 2piecewise degree containing infinitely many finite-∆ 0 2
... piecewise degrees (where (Π 0 n ) 2 denotes the difference of two Π 0 n sets), whereas the greatest degrees in these three "finite-Γ-piecewise" degree structures coincide. Moreover, as for nonempty Π 0 1 subsets of Cantor space, we also show that every nonzero finite-(Π 0 1 ) 2 -piecewise degree includes infinitely many Medvedev (i.e., one-piecewise) degrees, every nonzero countable-∆ 0 2 -piecewise degree includes infinitely many finite-piecewise degrees, every nonzero finite-(Π 0 2 ) 2 -countable-∆ 0 2 -piecewise degree includes infinitely many countable-∆ 0 2 -piecewise degrees, and every nonzero Muchnik (i.e., countable-Π 0 2 -piecewise) degree includes infinitely many finite-(Π 0 2 ) 2 -countable-∆ 0 2 -piecewise degrees. Indeed, we show that any nonzero Medvedev degree and nonzero countable-∆ 0 2 -piecewise degree of a nonempty Π 0 1 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 are Brouwerian, where Γ is any of the Wadge classes mentioned above.