On the probabilistic closure of the loose unambiguous hierarchy

Unambiguous hierarchies [NR93, LR94, NR98] are defined similarly to the polynomial hierarchy; however, all witnesses must be unique. These hierarchies have subtle differences in the mode of using oracles. We consider a “loose” unambiguous hierarchy prUH• with relaxed definition of oracle access to promise problems. Namely, we allow to make queries that miss the promise set; however, the oracle answer in this case can be arbitrary (a similar definition of oracle access has been used in [CR08]). We prove that the first part of Toda’s theorem PH ⊆ BP · ⊕P ⊆ PPP can be rectified to PH = BP · prUH•, that is, the closure of our hierarchy under Schöning’s BP operator equals the polynomial hierarchy. It is easily seen that BP · prUH• ⊆ BP · ⊕P. The proof follows the same lines as Toda’s proof, so the main contribution of the present note is a new definition.