1 research outputs found
Inside the Muchnik Degrees II: The Degree Structures induced by the Arithmetical Hierarchy of Countably Continuous Functions
It is known that infinitely many Medvedev degrees exist inside the Muchnik
degree of any nontrivial subset of Cantor space. We shed light on the
fine structures inside these Muchnik degrees related to learnability and
piecewise computability. As for nonempty subsets of Cantor space, we
show the existence of a finite--piecewise degree containing
infinitely many finite--piecewise degrees, and a
finite--piecewise degree containing infinitely many
finite--piecewise degrees (where denotes the
difference of two sets), whereas the greatest degrees in these three
"finite--piecewise" degree structures coincide. Moreover, as for
nonempty subsets of Cantor space, we also show that every nonzero
finite--piecewise degree includes infinitely many Medvedev (i.e.,
one-piecewise) degrees, every nonzero countable--piecewise degree
includes infinitely many finite-piecewise degrees, every nonzero
finite--countable--piecewise degree includes
infinitely many countable--piecewise degrees, and every nonzero
Muchnik (i.e., countable--piecewise) degree includes infinitely many
finite--countable--piecewise degrees. Indeed, we show
that any nonzero Medvedev degree and nonzero countable--piecewise
degree of a nonempty subset of Cantor space have the strong
anticupping properties. Finally, we obtain an elementary difference between the
Medvedev (Muchnik) degree structure and the finite--piecewise degree
structure of all subsets of Baire space by showing that none of the
finite--piecewise structures are Brouwerian, where is any of
the Wadge classes mentioned above