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A fixed-alphabet regular gap reduction for 3-SAT

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

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    Natural Language Statement

    Theorem

    Dinur's gap construction followed by degree reduction gives regular binary constraint systems with a fixed alphabet, fixed degree, and a positive constant unsatisfiability gap. Their number of vertices is bounded by a fixed polynomial in the number of clauses. Satisfiable formulas give satisfiable systems.

    This is the finite mathematical reduction. Its proof uses the ported Dinur development from complexitylib (Apache-2.0). A polynomial bound on the output size is stated here; a machine running-time claim is separate.

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    Proven claimThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax253009.ProjectionGames
    2
    3/-!
    4---
    5title: A fixed-alphabet regular gap reduction for 3-SAT
    6type: theorem
    7---
    8Dinur's gap construction followed by degree reduction gives regular binary
    9constraint systems with a fixed alphabet, fixed degree, and a positive
    10constant unsatisfiability gap. Their number of vertices is bounded by a
    11fixed polynomial in the number of clauses. Satisfiable formulas give
    12satisfiable systems.
    13
    14This is the finite mathematical reduction. Its proof uses the ported Dinur
    15development from complexitylib (Apache-2.0). A polynomial bound on the
    16output size is stated here; a machine running-time claim is separate.
    17-/
    18
    19namespace Lax253009.GapSatisfiability
    20
    21abbrev Literal := Bool × ℕ
    22abbrev Formula := List (List Literal)
    23
    24def Is3CNF (φ : Formula) : Prop := ∀ c ∈ φ, c.length = 3
    25
    26def Satisfiable (φ : Formula) : Prop :=
    27 ∃ a : List Bool, ∀ c ∈ φ, ∃ l ∈ c, (a[l.2]?).getD false = l.1
    28
    29axiom regular_gap :
    30 ∃ a d K e : ℕ, 0 < a ∧ 0 < d ∧ 0 < K ∧
    31 ∃ γ : ℝ, 0 < γ ∧
    32 ∀ φ : Formula, Is3CNF φ →
    33 ∃ n : ℕ, 0 < n ∧ n ≤ K * (φ.length + 1) ^ e ∧
    34 ∃ C : ProjectionGames.System (Fin n) (Fin d) (Fin a),
    35 (Satisfiable φ → C.Satisfiable) ∧ (¬ Satisfiable φ → C.Sound γ)
    36
    37end Lax253009.GapSatisfiability
    38
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