Class Description

PPP: P With PP Oracle

A level of the counting hierarchy CH.

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

Contains PPPH [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 PPP (exercise for the visitor).

More about...


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

PH is in PPP [Tod89].

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

Equals PostBQP [Aar05b].

For a random oracle A, PPA is strictly contained in PSPACEA with probability 1 [ABF+94].

For any fixed k, there exists a language in PP that does not have circuits of size nk [Vin04b].
Indeed, there exists a language in PP that does not even have quantum circuits of size nk 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.

More about...