Class Description

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:

AxPP is recursively enumerable [Jeř07].

Linked From

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 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].

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