Helping by unambiguous computation and probabilistic computation

Schöning S] introduced a notion of helping and suggested the study of the class P_help(C) of languages that can be helped by oracles in a given classC. He showed that P_help(BPP) is included in ZPP. Later, Ko K] introduced a weaker notion of helping, called one-sided helping, and proved that P_1-help(BPP) is included in R and that UP is included in P_1-help(UP). The three inverse inclusions are open problems (see Hem]). Caiet al. CHV] constructed a relativized world in which the third open inclusion fails. We show relativized worlds in which all three open inclusions fail in a strong way. In particular, we strengthen the result of Caiet al., showing that P_help(UP) is not included in Few. This is obtained as a corollary of a general result that gives sufficient conditions, on a relativizable complexity classC, to ensure the relativized separation of P_help(UP) fromC. Further separations are also derived. The other two open inclusions are showed to fail strongly by the relativized separation of ZPP from P_1-help(AM ∩ co-AM).