BPP ⊆ AM

BPP is contained in AM.

Statement

Migrated verbatim from bounded-error-probabilistic-polynomial-time § Known relationships:

• $\mathbf{BPP}\subseteq \mathbf{AM}\cap \mathbf{coAM}$: BPP problems have trivial one-message Arthur-Merlin protocols (Arthur decides without Merlin) — GS86.

Migrated verbatim from co-arthur-merlin § Known relationships:

• $\mathbf{BPP}\subseteq \mathbf{coAM}$: BPP problems have a trivial one-message coAM protocol where Merlin’s message is ignored (Arthur decides alone). Symmetrically, $\mathbf{BPP}\subseteq \mathbf{AM}$.

Notes

class: unstated: no citing page says which notion of reduction is meant. Recording a class the wiki does not state would add a claim.

class: free because a containment between complexity classes is proved by any argument at all; the reduction-class axis does not discriminate here, and unstated would wrongly suggest the information is missing.

Recorded during migration and not fixed — these are claims about the source text, not changes to it:

• The GS86 attribution is doubtful (recorded, not fixed): BPP inside AM is immediate from the definition, not a result of the private-versus-public-coin paper.
• Duplicates co-arthur-merlin.md:18 sub-edge 1, which states the same inclusion with NO citation - the two pages disagree on whether a citation is needed.
• Uncited and unlabelled folklore; same disagreement with bounded-error-probabilistic-polynomial-time.md:24.
• The ‘Symmetrically’ clause carries no justification of its own, so the split leaves this link unsupported.