TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-12

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

RP admits one-sided amplification to exponentially small error

Statement

If $L \in R P$ and $k \in N$ , then $L$ has a probabilistic polynomial-time decider with no false positives and false-negative probability at most $2^{- k}$ .

Facts & Assumptions

Given: a language $L \in R P$ and an integer $k \geq 1$ .

[L1]

An RP machine rejects every no-instance with probability $1$ and accepts every yes-instance with probability at least $1 / 2$ (The classes RP, coRP, ZPP, BPP, and PP).

Proof

technique · direct

1.1L1givenconstruct

Let $M$ be an RP machine for $L$ . Run $M$ independently $k$ times on the same input, and accept iff at least one run accepts. This remains polynomial time because $k$ independent repetitions of a polynomial-time machine still use polynomial time for fixed or polynomially bounded $k$ .

2.1L1step 1.1algebra

If $x \notin L$ , then by [L1] every run rejects, so the repeated machine also rejects with probability $1$ . If $x \in L$ , then each run fails to accept with probability at most $1 / 2$ , and independence makes the probability that all $k$ runs fail at most $(1 / 2)^{k} = 2^{- k}$ .

3.1step 2.1∎

Thus the repeated machine keeps one-sided error and reduces the false negative probability to at most $2^{- k}$ .

Depends on

The classes RP, coRP, ZPP, BPP, and PP

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)
Eric Blais, Models of Computation, 14. Randomized Computation (standard reference, not scraped)