DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Completeness and soundness of an interactive protocol

Definition

For a language $L$ , an interactive protocol has $c$ -completeness when for every $x \in L$ an honest prover $P_{h}$ makes $V$ accept with probability at least $c$ . It has $s$ -soundness when for every $x \notin L$ and every prover strategy $P$ , the acceptance probability is at most $s$ . Probabilities are over $V$ 's coins. Thus completeness has an existential honest strategy, while soundness has a universal cheating-strategy quantifier.

Depends on

Interactive-proof transcripts, rounds, and prover strategies

Used by

The class IP Definition
False: soundness quantifies only over honest provers False statement
Sequential repetition amplifies completeness and soundness gaps Lemma

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

Arora and Barak, Definition 8.5 (standard reference, not scraped)