Alphabeta Math
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.

Arithmetization of Boolean formulas

Definition

Let F be a field and let φ be a Boolean formula on variables x1,…,xn, with n≥0. Use the syntax of Boolean formulas, conjunctive normal form, and the satisfiability language SAT, with binary AND and OR. Its arithmetization is the formal polynomial Pφ∈F[X1,…,Xn] defined recursively by P⊥=0,P⊤=1,Pxi=Xi,P¬ψ=1−Pψ, Pψ∧θ=PψPθ,Pψ∨θ=Pψ+Pθ−PψPθ. Here field and polynomial ring mean Field and Polynomial rings in finitely many commuting indeterminates by iteration. Retain the original formula tree with these gate operations; a gate can use its two already computed child values more than once. Expanding into monomials is unnecessary. Boolean false and true are identified with the distinct field elements 0 and 1.

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