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 $\Pi^0_1$ subset of Cantor space. We shed light on the fine structures inside these Muchnik degrees related to learnability and piecewise computability. As for nonempty $\Pi^0_1$ subsets of Cantor space, we show the existence of a finite-$\Delta^0_2$-piecewise degree containing infinitely many finite-$(\Pi^0_1)_2$-piecewise degrees, and a finite-$(\Pi^0_2)_2$-piecewise degree containing infinitely many finite-$\Delta^0_2$-piecewise degrees (where $(\Pi^0_n)_2$ denotes the difference of two $\Pi^0_n$ sets), whereas the greatest degrees in these three "finite-$\Gamma$-piecewise" degree structures coincide. Moreover, as for nonempty $\Pi^0_1$ subsets of Cantor space, we also show that every nonzero finite-$(\Pi^0_1)_2$-piecewise degree includes infinitely many Medvedev (i.e., one-piecewise) degrees, every nonzero countable-$\Delta^0_2$-piecewise degree includes infinitely many finite-piecewise degrees, every nonzero finite-$(\Pi^0_2)_2$-countable-$\Delta^0_2$-piecewise degree includes infinitely many countable-$\Delta^0_2$-piecewise degrees, and every nonzero Muchnik (i.e., countable-$\Pi^0_2$-piecewise) degree includes infinitely many finite-$(\Pi^0_2)_2$-countable-$\Delta^0_2$-piecewise degrees. Indeed, we show that any nonzero Medvedev degree and nonzero countable-$\Delta^0_2$-piecewise degree of a nonempty $\Pi^0_1$ subset of Cantor space have the strong anticupping properties. Finally, we obtain an elementary difference between the Medvedev (Muchnik) degree structure and the finite-$\Gamma$-piecewise degree structure of all subsets of Baire space by showing that none of the finite-$\Gamma$-piecewise structures are Brouwerian, where $\Gamma$ is any of the Wadge classes mentioned above.