BPP ⊆ coAM

BPP is contained in coAM.

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:

• Same doubtful GS86 attribution.
• Duplicates co-arthur-merlin.md:18 sub-edge 0, which is uncited.
• Uncited and unlabelled folklore, while bounded-error-probabilistic-polynomial-time.md:24 attributes the identical inclusion to GS86 - the two pages disagree on whether a citation is needed.