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A polynomial-size 3-SAT reduction to arbitrarily small projection-game value

Lax253009.SmallValueSatisfiability · concepts/Lax253009/SmallValueSatisfiability.lean · lax-253009

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    Natural Language Statement

    Theorem

    For every fixed positive soundness δ, exact 3-CNF formulas reduce to centered projection games over fixed answer alphabets. Satisfiable formulas give perfectly complete games, and unsatisfiable formulas give value at most δ. The sum of all three question-space sizes is bounded by one fixed polynomial in the number of clauses.

    This combines the regular Dinur gap, projection symmetrization, tuple fortification, repeated squaring, and Cauchy–Schwarz. It is a finite mathematical reduction; the uniform machine running-time proof is separate.

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    Proven claimThis conceptRelated conceptA → B: B builds on A
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    Lean source view on GitHub

    1import Lax253009.Amplification
    2import Lax253009.GapSatisfiability
    3
    4/-!
    5---
    6title: A polynomial-size 3-SAT reduction to arbitrarily small projection-game value
    7type: theorem
    8---
    9For every fixed positive soundness δ, exact 3-CNF formulas reduce to centered
    10projection games over fixed answer alphabets. Satisfiable formulas give
    11perfectly complete games, and unsatisfiable formulas give value at most δ.
    12The sum of all three question-space sizes is bounded by one fixed polynomial
    13in the number of clauses.
    14
    15This combines the regular Dinur gap, projection symmetrization, tuple
    16fortification, repeated squaring, and Cauchy–Schwarz. It is a finite
    17mathematical reduction; the uniform machine running-time proof is separate.
    18-/
    19
    20namespace Lax253009.SmallValueSatisfiability
    21
    22open CenteredProjection GapSatisfiability
    23
    24axiom reduction (δ : ℝ) (hδ : 0 < δ) :
    25 ∃ x y K e : ℕ, 0 < x ∧ 0 < y ∧
    26 ∀ φ : Formula, Is3CNF φ →
    27 ∃ u o w : ℕ, 0 < u ∧ 0 < o ∧ 0 < w ∧ u + o + w ≤ K * (φ.length + 1) ^ e ∧
    28 ∃ G : System (Fin u) (Fin o) (Fin w) (Fin x) (Fin y),
    29 (Satisfiable φ → G.Complete) ∧ (¬ Satisfiable φ → G.Sound δ)
    30
    31end Lax253009.SmallValueSatisfiability
    32
    Show Proof
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