Class Description

APP: Amplified PP

Roughly, the class of decision problems for which the following holds. For all polynomials p(n), there exist GapP functions f and g such that for all inputs x with n=|x|,

1. If the answer is "yes" then 1 > f(x)/g(1ⁿ) > 1-2^-p(n).
2. If the answer is "no" then 0 < f(x)/g(1ⁿ) < 2^-p(n).

Defined in [Li93], where the following was also shown:

• APP is contained in PP, and indeed is low for PP.
• APP is closed under intersection, union, and complement.

APP contains AWPP [Fen02].

The abbreviation APP is also used for Approximable in Probabilistic Polynomial Time, see AxPP.

Contains FewP [Li93] and contains YQP*, YMA*, and YP* [Yir24].

Linked From

AWPP: Almost WPP

The class of decision problems solvable by an NP machine such that for some polynomial-time computable (i.e. FP) function f,

1. If the answer is "no," then the difference between the number of accepting and rejecting paths is non-negative and at most 2^-poly(n)f(x).
2. If the answer is "yes," then the difference is between (1-2^-poly(n))f(x) and f(x).

Defined in [FFK94].

Contains BQP [FR98], WAPP [BGM02], LWPP, and WPP.

Contained in APP [Fen02].

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AxPP: Approximable in Probabilistic Polynomial Time

Usually called APP. I've renamed it AxPP to distinguish it from the "other" APP.

The class of real-valued functions from {0,1}ⁿ to [0,1] that can be approximated within any ε>0 by a probabilistic Turing machine in time polynomial in n and 1/ε.

Defined by [KRC00], who also show the following:

• Approximating the acceptance probability of a Boolean circuit is AxPP-complete. The authors argue that this makes AxPP a more natural class than BPP, since the latter is not believed to have complete problems.
• If AxPP = AxP, then BPP = P.
• On the other hand, there exists an oracle relative to which BPP = P but AxPP does not equal AxP.

AxPP is recursively enumerable [Jeř07].

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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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YQP*: Yaroslav BQP star

Is to YQP as YP* is to YP [AD14].

Is contained in APP, and so is low for PP [Yir24].

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