BPP ⊆ PP

BPP is contained in PP.

Statement

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

• $\mathbf{BPP}\subseteq \mathbf{PP}\subseteq \mathbf{PSPACE}$: BPP has a constant gap (and thus sits inside PP), and PP’s computation can be simulated in polynomial space.

Notes

source: folklore: the claim carried no citation on the page it was migrated from, and none was invented.

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:

• Uncited and not labelled ’- standard’ or ’- folklore’ as the house style requires.