  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.
