| import Mathlib |
| import OAI.Computability.MinUncut.PCP.Walsh |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
|
|
| namespace MinUncutGames.Foundations.Hastad |
|
|
| open scoped BigOperators |
| open Finset |
|
|
| variable {I J : Type*} [Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J] |
|
|
| def support (s : Cube I) : Finset I := Finset.univ.filter fun i => s i = true |
|
|
| def noiseWeight (ε : ℝ) (μ : Cube I) : ℝ := |
| ∏ i, if μ i then ε else 1 - ε |
|
|
| omit [DecidableEq I] in |
| theorem noiseWeight_nonneg {ε : ℝ} (hε : 0 ≤ ε) (hε' : ε ≤ 1) (μ : Cube I) : |
| 0 ≤ noiseWeight ε μ := by |
| apply Finset.prod_nonneg |
| intro i _ |
| split <;> linarith |
|
|
| theorem noiseWeight_sum (ε : ℝ) : (∑ μ : Cube I, noiseWeight ε μ) = 1 := by |
| unfold noiseWeight |
| rw [← Fintype.prod_sum (fun (_ : I) (b : Bool) => if b then ε else 1 - ε)] |
| simp |
|
|
| theorem noise_walsh (ε : ℝ) (s : Cube I) : |
| (∑ μ, noiseWeight ε μ * walsh s μ) = (1 - 2 * ε) ^ (support s).card := by |
| unfold noiseWeight walsh |
| simp only [← Finset.prod_mul_distrib] |
| rw [← Fintype.prod_sum (fun i b => |
| (if b then ε else 1 - ε) * bitSign (s i && b))] |
| have hc (i : I) : (∑ b : Bool, |
| (if b then ε else 1 - ε) * bitSign (s i && b)) = |
| if s i then 1 - 2 * ε else 1 := by |
| cases s i <;> simp [bitSign] |
| ring |
| simp_rw [hc] |
| simp [support, Finset.prod_ite] |
|
|
| def thirdQuery (π : J → I) (f : Cube I) (g μ : Cube J) : Cube J := |
| cubeXor g (cubeXor (fun y => f (π y)) μ) |
|
|
| theorem walsh_autocorrelation (B : Cube J → ℝ) (h : Cube J) : |
| (𝔼 g, B g * B (cubeXor g h)) = |
| ∑ s, coefficient B s ^ 2 * walsh s h := by |
| conv_lhs => |
| enter [2, g, 2] |
| rw [← walsh_inversion B (cubeXor g h)] |
| simp_rw [Finset.mul_sum, walsh_xor] |
| rw [Finset.expect_sum_comm] |
| apply Finset.sum_congr rfl |
| intro s _ |
| have hterm : (fun g => B g * (coefficient B s * (walsh s g * walsh s h))) = |
| (fun g => (coefficient B s * walsh s h) * (B g * walsh s g)) := by |
| funext g |
| ring |
| rw [hterm, ← Finset.mul_expect] |
| change (coefficient B s * walsh s h) * coefficient B s = _ |
| ring |
|
|
| def projectedCoefficient (π : J → I) (A : Cube I → ℝ) (s : Cube J) : ℝ := |
| 𝔼 f, A f * walsh s (fun y => f (π y)) |
|
|
| def testBias (ε : ℝ) (π : J → I) (A : Cube I → ℝ) (B : Cube J → ℝ) : ℝ := |
| 𝔼 f, ∑ μ, noiseWeight ε μ * A f * |
| (𝔼 g, B g * B (thirdQuery π f g μ)) |
|
|
| theorem testBias_fourier (ε : ℝ) (π : J → I) |
| (A : Cube I → ℝ) (B : Cube J → ℝ) : |
| testBias ε π A B = ∑ s, |
| projectedCoefficient π A s * coefficient B s ^ 2 * |
| (1 - 2 * ε) ^ (support s).card := by |
| unfold testBias thirdQuery |
| simp_rw [walsh_autocorrelation, Finset.mul_sum] |
| conv_lhs => |
| enter [2, f] |
| rw [Finset.sum_comm] |
| rw [Finset.expect_sum_comm] |
| apply Finset.sum_congr rfl |
| intro s _ |
| have hterm (f : Cube I) : |
| (∑ μ, noiseWeight ε μ * A f * |
| (coefficient B s ^ 2 * walsh s (cubeXor (fun y => f (π y)) μ))) = |
| (A f * walsh s (fun y => f (π y))) * coefficient B s ^ 2 * |
| (1 - 2 * ε) ^ (support s).card := by |
| simp_rw [walsh_xor] |
| have heq (μ : Cube J) : noiseWeight ε μ * A f * |
| (coefficient B s ^ 2 * (walsh s (fun y => f (π y)) * walsh s μ)) = |
| ((A f * walsh s (fun y => f (π y))) * coefficient B s ^ 2) * |
| (noiseWeight ε μ * walsh s μ) := by ring |
| simp_rw [heq] |
| rw [← Finset.mul_sum, noise_walsh] |
| simp_rw [hterm, ← Finset.expect_mul] |
| rfl |
|
|
| def testAcceptance (ε : ℝ) (π : J → I) |
| (A : Cube I → Bool) (B : Cube J → Bool) : ℝ := |
| 𝔼 f, ∑ μ, noiseWeight ε μ * |
| (𝔼 g, if A f ^^ B g ^^ B (thirdQuery π f g μ) then 0 else 1) |
|
|
| theorem threeBit_indicator (a b c : Bool) : |
| (if a ^^ b ^^ c then (0 : ℝ) else 1) = |
| (1 + bitSign a * bitSign b * bitSign c) / 2 := by |
| cases a <;> cases b <;> cases c <;> norm_num [bitSign] |
|
|
| theorem testAcceptance_eq (ε : ℝ) (π : J → I) |
| (A : Cube I → Bool) (B : Cube J → Bool) : |
| testAcceptance ε π A B = |
| (1 + testBias ε π (fun f => bitSign (A f)) (fun g => bitSign (B g))) / 2 := by |
| unfold testAcceptance testBias |
| simp_rw [threeBit_indicator, div_eq_mul_inv, |
| ← Finset.expect_mul, Finset.expect_add_distrib] |
| simp only [Fintype.expect_const] |
| have hterm (f : Cube I) (μ : Cube J) : |
| noiseWeight ε μ * ((1 + 𝔼 g, |
| bitSign (A f) * bitSign (B g) * bitSign (B (thirdQuery π f g μ))) * 2⁻¹) = |
| (noiseWeight ε μ + noiseWeight ε μ * bitSign (A f) * |
| (𝔼 g, bitSign (B g) * bitSign (B (thirdQuery π f g μ)))) * 2⁻¹ := by |
| simp_rw [mul_assoc (bitSign (A f)), ← Finset.mul_expect] |
| ring |
| simp_rw [hterm, ← Finset.sum_mul, Finset.sum_add_distrib, noiseWeight_sum, |
| ← Finset.expect_mul, Finset.expect_add_distrib] |
| simp |
|
|
| end MinUncutGames.Foundations.Hastad |
|
|
| end |
| end |
|
|
| end OAI |
|
|
