On bounded \(m\)-reducibilities

Abstract

Conditions for classes \({\mathfrak F}^1,{\mathfrak F}^0\) of non-decreasing total one-place arithmetic functions to define reducibility   \(\leq_m[^{{\mathfrak R}^1}_{{\mathfrak R}^0}]\leftrightharpoons\{(A,B)|A,B\subseteq\mathbb N\ \&\ (\exists \mbox{ r.f. }\ h) (\exists f_1\in{\mathfrak F}^1)(\exists f_0\in{\mathfrak F}^0) \) \([A\le_m^h\,B\ \&\ f_0\unlhd h\unlhd f_1]\}\) where \(k\unlhd l\) means that function \(l\) majors function \(k\) almost everywhere are studied. It is proved that the system of these reducibilities is highly ramified, and examples are constructed which differ drastically \(\leq_m[^{{\mathfrak R}^1}_{{\mathfrak R}^0}]\) from the standard m-reducibility  with respect to systems of degrees. Indecomposable and recursive degrees are considered.