File size: 1,200 Bytes
92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | import OAI.Computability.UniqueGames.Decoding.MatrixGapFromInverseLemmas
import OAI.Computability.UniqueGames.Reduction.FromMatrixGap
namespace OAI
section
namespace UniqueGamesTheorem.Theorem
theorem exists_binaryGapReduction (ε δ : ℝ)
(hε : 0 < ε) (_hεhalf : ε < 1 / 2)
(hδ : 0 < δ) (_hδhalf : δ < 1 / 2) :
Nonempty (Explicit.MachineOutputContract.BinaryGapReduction ε δ) := by
obtain ⟨P⟩ := ParameterSelection.exists_outerParameters ε δ hε hδ
obtain ⟨M⟩ := Decoder.MatrixGap.exists_parameters P.pStar P.pStar_pos
exact ⟨FromMatrixGap.reduction P M⟩
end UniqueGamesTheorem.Theorem
namespace UniqueGamesTheorem
/-- For every fixed pair of errors in `(0, 1/2)`, binary 3SAT reduces in
polynomial time to translation Unique Games with completeness at least `1 - ε`
and soundness at most `δ`. The alphabet and machine depend only on the errors. -/
theorem theorem11 (ε δ : ℝ)
(hε : 0 < ε) (hεhalf : ε < 1 / 2)
(hδ : 0 < δ) (hδhalf : δ < 1 / 2) :
Nonempty (Explicit.MachineOutputContract.BinaryGapReduction ε δ) :=
Theorem.exists_binaryGapReduction ε δ hε hεhalf hδ hδhalf
end UniqueGamesTheorem
end
end OAI
|