Class Description

P^PP: P With PP Oracle

A level of the counting hierarchy CH.

It is not known whether there exists an oracle relative to which P^PP does not equal PSPACE.

Contains PP^PH [Tod89].

Equals P^#P (exercise for the visitor).

Since the permanent of a matrix is #P-complete [Val79], Toda's theorem implies that any problem in the polynomial hierarchy can be solved by computing a sequence of permanents.

Linked From

P^#P: P With #P Oracle

I decided this class is so important that it deserves an entry of its own, apart from #P.

Contains PH [Tod89], and is contained in CH and in PSPACE.

Equals P^PP (exercise for the visitor).

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PP: Probabilistic Polynomial-Time

The class of decision problems solvable by an NP machine such that

1. If the answer is 'yes' then at least 1/2 of computation paths accept.
2. If the answer is 'no' then less than 1/2 of computation paths accept.

Defined in [Gil77].

PP is closed under union and intersection [BRS91] (this was an open problem for 14 years). More generally, P^PP[log] = PP. Even more generally, PP is closed under polynomial-time truth-table reductions, or even k-round reductions for any constant k [FR96].

Contains P^NP[log] [BHW89] and QMA (see [MW05]). However, there exists an oracle relative to which PP does not contain Δ₂P [Bei94].

PH is in P^PP [Tod89].

BPP [KST+89b] and even BQP [FR98] and YQP* [Yir24] are low for PP: i.e., PP^BQP = PP.
APP is PP-low [Li93].

Equals PostBQP [Aar05b].

For a random oracle A, PP^A is strictly contained in PSPACE^A with probability 1 [ABF+94].

For any fixed k, there exists a language in PP that does not have circuits of size n^k [Vin04b].
Indeed, there exists a language in PP that does not even have quantum circuits of size n^k with quantum advice [Aar06] and [Yir24].

By contrast, there exists an oracle relative to which PP has linear-size circuits [Aar06].

PP can be generalized to the counting hierarchy CH.

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