DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-02

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.

The search problem attached to an NP verifier

Definition

Let $L \subseteq {0, 1}^{*}$ be a language in $N P$ , witnessed by a verifier relation $R$ and a polynomial certificate bound $p$ as in Polynomial-time verifiers with polynomially bounded certificates. The search problem attached to $R$ is:

input: a word $x \in L$ ,
output: any certificate $u \in {0, 1}^{*}$ such that $∣ u ∣ \leq p (∣ x ∣)$ and $(x, u) \in R$ .

Equivalently, it asks not merely whether a witness exists, but to produce one.

Remarks

On inputs $x \notin L$ , no valid certificate exists, so the search problem has no required output there.
The output need not be unique; any witnessing certificate is acceptable.

Depends on

Used by

Dependency tree · two levels

4 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)