DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Promise preserving reduction

Definition

For promise problems $(Y, N)$ and $(Y^{'}, N^{'})$ in Promise problem, a polynomial-time many-one promise reduction is a total polynomial-time function $f$ satisfying $f (Y) \subseteq Y^{'}$ and $f (N) \subseteq N^{'}$ . There is no condition on $f$ outside $Y \cup N$ .

A polynomial-time oracle promise reduction is a polynomially clocked deterministic oracle machine $M$ which solves $(Y, N)$ for every total language $B$ satisfying $Y^{'} \subseteq B$ and $B \cap N^{'} = \emptyset$ . This uses binary, consistent membership completions: an off-promise word may have either bit, but repeated queries to that word receive the same bit. The clock is uniform over all completions.

Depends on

Promise problem

Used by

Promise oracle off promise answers Example
Oracle and promise conventions are distinct Remark

Dependency tree · one level

1 result within one dependency step 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

Goldreich, On Promise Problems; §1.2 Definition1.3 and oracle reduction paragraph, pp4–5. (standard reference, not scraped)