RP ⊆ BPP

RP is contained in BPP.

Statement

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

• $\mathbf{P}\subseteq \mathbf{ZPP}\subseteq \mathbf{RP}\subseteq \mathbf{BPP}$: deterministic algorithms are a special case of Las Vegas, which are a special case of one-sided error, which are a special case of two-sided error.

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

• $\mathbf{P}\subseteq \mathbf{RP}\subseteq \mathbf{BPP}$: RP is a “one-sided” restriction of BPP (which allows two-sided error).

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

• $\mathbf{P}\subseteq \mathbf{ZPP}\subseteq \mathbf{RP}\subseteq \mathbf{BPP}$.

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 unlabelled folklore.
• The single prose justification on the bullet covers all three links at once, so no per-link justification survives the split.
• As literally stated the containment needs amplification: RP’s 1/2 acceptance must be boosted past BPP’s 2/3 threshold, which the one-line gloss elides.
• Duplicates the RP-to-BPP sub-edge of record Complexity/randomized-polynomial-time:23; migration must dedupe.
• Uncited; the amplification step is elided here as it is there.