Q-degrees of n-C.E. sets

Abstract

In this paper we study Q-degrees of n-computably enumerable (n-c.e.) sets. It is proved that n-c.e. sets form a true hierarchy in terms of Q-degrees, and that for any n ≥ 1 there exists a 2n-c.e. Q- degree which bounds no noncomputable c.e. Q-degree, but any (2n + l)- c.e. non 2n-c.e. Q-degree bounds a c.e. noncomputable Q-degree. Studying weak density properties of n-c.e. Q-degrees, we prove that for any n ≥ 1, properly n-c.e. Q-degrees are dense in the ordering of c.e. Q-degrees, but there exist c.e. sets A and B such that A - B <Q A ≡Q φ′, and there are no c.e. sets for which the Q-degrees are strongly between A - B and A. ©2007 University of Illinois.