Class Description

MAE: Exponential-Time MA With Linear Exponent

Same as MA, except now Arthur is E instead of polynomial-time.

If MAE = NEE then MA = NEXPcoNEXP [IKW01].

Linked From

MA: Merlin-Arthur

The class of decision problems solvable by a Merlin-Arthur protocol, which goes as follows. Merlin, who has unbounded computational resources, sends Arthur a polynomial-size purported proof that the answer to the problem is "yes." Arthur must verify the proof in BPP (i.e. probabilistic polynomial-time), so that

  1. If the answer is "yes," then there exists a proof such that Arthur accepts with probability at least 2/3.
  2. If the answer is "no," then for all proofs Arthur accepts with probability at most 1/3.

Defined in [Bab85].

An alternative definition requires that if the answer is "yes," then there exists a proof such that Arthur accepts with certainty. However, the definitions with one-sided and two-sided error can be shown to be equivalent (see [FGM+89]).

Contains NP and BPP (in fact also ∃BPP), and is contained in AM and in QMA.

Also contained in Σ2PΠ2P.

There exists an oracle relative to which BQP is not in MA [Wat00].

Equals NP under a derandomization assumption: if E requires exponentially-sized circuits, then PromiseBPP = PromiseP, implying that MA = NP [IW97].

Shown in [San07] that MA/1 (that is, MA with 1 bit of advice) does not have circuits of size for any . In the same paper, the result was used to show that MA/1 cannot be solved on more than a fraction of inputs having length by any circuit of size . Finally, it was shown that MA does not have arithmetic circuits of size .

See also: MAE, MAEXP, OMA.

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NEE: Nondeterministic EE

Nondeterministic double-exponential time with linear exponent (i.e. NTIME(22O(n))).

If MAE = NEE then MA = NEXPcoNEXP [IKW01].

Contained in NEEXP.

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