DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

PromiseBPP

Definition

Let $Π = (Y, N)$ be a promise problem, so $Y, N \subseteq {0, 1}^{*}$ are disjoint (Promise problem). We say $Π$ belongs to $P r o m i s e B P P$ if there are one fixed probabilistic Turing machine $M$ and one positive integer polynomial $p$ such that every random branch halts within $p (∣ x ∣)$ transitions and

$\Pr [M (x) accepts] \geq \frac{2}{3} (x \in Y), \Pr [M (x) accepts] \leq \frac{1}{3} (x \in N) .$

The probability is over the uniform finite tape ${0, 1}^{p (∣ x ∣)}$ of independent fair bits; after the machine halts, unused suffix bits are ignored, as in Probabilistic polynomial time machine interface. The pair has no correctness requirement outside $Y \cup N$ . For total promise pairs this is the ordinary BPP acceptance convention of Rp corp zpp bpp and pp classes.

Depends on

Used by

PromiseP is contained in PromiseBPP and PromiseBPP in BQP Theorem

Dependency tree · two levels

7 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

John Watrous, Quantum Computational Complexity, Section IV.1 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, Chapter 13, Section 13.2 (standard reference, not scraped)