On the probabilistic closure of the loose unambiguous hierarchy

•We introduce a new version prUH•prUH• of the unambiguous hierarchy of promise problems.•We prove that PH=BP⋅prUH•PH=BP⋅prUH•.•Our result strengthens the first part of Toda's theorem PH⊆BP⋅⊕P⊆PPPPH⊆BP⋅⊕P⊆PPP, as BP⋅prUH•⊆BP⋅⊕PBP⋅prUH•⊆BP⋅⊕P.

Unambiguous hierarchies [1], [2] and [3] 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•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 [4]).In this short note we prove that the first part of Toda's theorem PH⊆BP⋅⊕P⊆PPPPH⊆BP⋅⊕P⊆PPP can be strengthened to PH=BP⋅prUH•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⋅⊕PBP⋅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 that allows to characterize PH as a probabilistic closure of unambiguous computations.