Minimal Degrees For Honest Polynomial Reducibilities

The existence of minimal degrees is investigated for several polynomial reducibilities. It is shown that no set has minimal degree with respect to polynomial many-one or Turing reducibility. This extends a result of Ladner [L] whew reciirsive sets are considered. An "honest" polynomial reducibility, ⩽is defined which is a strengthening of polynomial Turing reducibility. We prove that no recursive set, (or igeeand P-immune set) has minimal < ;-degree. However, proving this same fact for all Δ_s sets (or even all 3 sets) would imply P 2 .y/l. Finally, a partial converse of this result is obtained, proving that if a certain class of one-way functions exists then no set has minimal (h/t)-degree.