A Comparison of Weak Completeness Notions 1 Extended

We compare the weak completeness notions for E in the sense of Lutz's resource-bounded measure theory 11] with respect to the standard polynomial time reducibilities. Our results parallel results for classical completeness by Watanabe 17] and others. We show that the weak completeness notions for 1-query reductions coincide: A set is weakly complete for E under 1-truth-table reducibility ii it is weakly complete for length-increasing one-one reducibility. For most of the other polynomial reducibilities, however, we obtain separations of the weak completeness notions where these reducibilities diier on E (Ladner et al. 10]). In fact our separations simultaneously hold for the corresponding weak completeness notions for E and E2, for the classical completeness notions, and for the weak completeness notions in the sense of the resource-bounded Baire category concepts of Ambos-Spies et al. 2] and Ambos-Spies 1].