DefinitionDefinition: Literature-sourcedProof: 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.

NumberSAT

Definition

An input to $N u m b e r S A T$ is well formed when it encodes a Boolean formula $φ$ together with an ordered list $(x_{1}, \dots, x_{n})$ of distinct variables and every variable occurring in $φ$ appears in that list. Its value is the exact number of assignments $a : {x_{1}, \dots, x_{n}} \to {0, 1}$ that satisfy $φ$ . Declared variables count even when unused, so the count ranges over all $2^{n}$ assignments to the list. Every input not having this form is malformed and has value zero.

Used by

NumberSAT is not a Boolean membership predicate Counterexample
MajoritySAT Definition
Deciding satisfiability by a NumberSAT oracle query Example
Every decision many-one reduction is parsimonious False statement
Sharp-P is a language class False statement
The Cook--Levin construction can be made parsimonious Lemma
NumberSAT belongs to Sharp-P Proposition
NumberSAT is Sharp-P-complete under parsimonious reductions Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Arora and Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)
Lance Fortnow, Counting Complexity (standard reference, not scraped)