Inside the Muchnik Degrees II: The Degree Structures induced by the Arithmetical Hierarchy of Countably Continuous Functions

TL;DR

This paper explores the intricate internal structures of Muchnik degrees within the arithmetical hierarchy, revealing new relationships and properties related to learnability, piecewise computability, and degree structures in computability theory.

Contribution

It introduces novel structural insights into Muchnik degrees induced by arithmetical hierarchies, including the existence of specific piecewise degrees and their properties, and distinguishes these structures from Medvedev degrees.

Findings

Existence of finite-Δ^0_2-piecewise degrees with infinitely many finite-(Π^0_1)_2- and (Π^0_2)_2-piecewise degrees

Nonzero degrees in these structures include infinitely many Medvedev or finite-piecewise degrees

Finite-Γ-piecewise structures are not Brouwerian, unlike Medvedev and Muchnik degrees

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.