File size: 4,425 Bytes
9dbc64e | 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 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.Computability.DepthThree.HardSliceProbability
import OAI.Computability.DepthThree.HardSliceSelection
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace DepthThreeLowerBound
theorem actual_hash_coefficient_window_failure_le {d r h b : ℕ}
(p : Fin r → Bool) [Fact (Irreducible (BinaryAlgebra.inputPolynomialBits p))]
(hr : 0 < r) (hb : 1 ≤ b) (hh : 0 < h) (q : ℝ)
(σ : Restriction (Fin d)) (hσ : restrictionWindow q σ)
(hbound :
(2 : ℝ) ^ Fintype.card (Live σ) / (2 : ℝ) ^ r +
sparseMomentBound (Fintype.card (Live σ)) b
(hardSliceSparseConstant b hb * Fintype.card (Live σ)) h
((2 : ℝ) ^ (-(Fintype.card (Live σ) : ℝ) / 4)) ≤
2 * (2 : ℝ) ^ (-(q * d) / 16)) :
finiteProb (fun ω : (Fin (d + r - 1) → Bool) ×
(Fin (2 * h) → BinaryAlgebra.BitQuotient p) =>
¬ corr b (fun z => sign (hashAcceptance p (2 * h) ω.1 ω.2 (fill σ z))) ≤
(2 : ℝ) ^ (-(q * d) / 16)) ≤
2 * (2 : ℝ) ^ (-(q * d) / 16) := by
have hlow : (q * d) / 2 ≤ (Fintype.card (Live σ) : ℝ) := by
simpa only [restrictionWindow, Fintype.card_fin] using hσ.1
have hthreshold : (2 : ℝ) ^ (-(Fintype.card (Live σ) : ℝ) / 8) ≤
(2 : ℝ) ^ (-(q * d) / 16) :=
Real.rpow_le_rpow_of_exponent_le (by norm_num) (by linarith)
apply le_trans _ ((actual_hash_coefficient_failure_le p hr σ b hb h hh).trans hbound)
apply finiteProb_mono
intro ω hω hsmall
exact hω (hsmall.trans hthreshold)
theorem exists_good_hash_coefficients {d r h b : ℕ}
(p : Fin r → Bool) [Fact (Irreducible (BinaryAlgebra.inputPolynomialBits p))]
(hr : 0 < r) (hb : 1 ≤ b) (hh : 0 < h) (q : ℝ)
(hq : 0 ≤ q) (hq1 : q ≤ 1)
(hbound : ∀ m : ℕ, (q * d) / 2 ≤ (m : ℝ) → (m : ℝ) ≤ 2 * (q * d) →
(2 : ℝ) ^ m / (2 : ℝ) ^ r +
sparseMomentBound m b (hardSliceSparseConstant b hb * m) h
((2 : ℝ) ^ (-(m : ℝ) / 4)) ≤ 2 * (2 : ℝ) ^ (-(q * d) / 16)) :
∃ (u : Fin (d + r - 1) → Bool) (β : Fin (2 * h) → BinaryAlgebra.BitQuotient p),
restrictionAvg q (fun σ => if ¬ goodRestriction q
((2 : ℝ) ^ (-(q * d) / 16)) b (hashAcceptance p (2 * h) u β) σ
then 1 else 0) ≤ 4 * (2 : ℝ) ^ (-(q * d) / 16) := by
have hfixed (σ : Restriction (Fin d)) (hσ : restrictionWindow q σ) :
finiteProb (fun ω : (Fin (d + r - 1) → Bool) ×
(Fin (2 * h) → BinaryAlgebra.BitQuotient p) =>
¬ corr b (fun z => sign (hashAcceptance p (2 * h) ω.1 ω.2 (fill σ z))) ≤
(2 : ℝ) ^ (-(q * d) / 16)) ≤
2 * (2 : ℝ) ^ (-(q * d) / 16) := by
apply actual_hash_coefficient_window_failure_le p hr hb hh q σ hσ
apply hbound
· simpa only [restrictionWindow, Fintype.card_fin] using hσ.1
· simpa only [restrictionWindow, Fintype.card_fin] using hσ.2
obtain ⟨ω, hω⟩ := exists_good_parameter hq hq1 b
(fun ω : (Fin (d + r - 1) → Bool) × (Fin (2 * h) → BinaryAlgebra.BitQuotient p) =>
hashAcceptance p (2 * h) ω.1 ω.2)
(by simpa only [Fintype.card_fin] using hfixed)
exact ⟨ω.1, ω.2, by simpa only [Fintype.card_fin] using hω⟩
def coefficientInputBits {r t : ℕ} (p : Fin r → Bool)
(β : Fin t → BinaryAlgebra.BitQuotient p) : Fin t → Fin r → Bool :=
fun j i => BinaryAlgebra.fieldBit
(BinaryAlgebra.coordinates (fun i => BinaryAlgebra.bitValue (p i)) (β j) i)
@[simp] theorem encode_coefficientInputBits {r t : ℕ} (p : Fin r → Bool)
(β : Fin t → BinaryAlgebra.BitQuotient p) (j : Fin t) :
BinaryAlgebra.encode (fun i => BinaryAlgebra.bitValue (p i))
(fun i => BinaryAlgebra.bitValue (coefficientInputBits p β j i)) = β j := by
simp only [coefficientInputBits, BinaryAlgebra.bitValue_fieldBit]
exact BinaryAlgebra.encode_coordinates _ _
theorem hashAcceptance_coefficientInputBits {d r t : ℕ} (p : Fin r → Bool)
[Fact (Irreducible (BinaryAlgebra.inputPolynomialBits p))]
(u : Fin (d + r - 1) → Bool) (β : Fin t → BinaryAlgebra.BitQuotient p) :
hashAcceptance p t u
(fun j => BinaryAlgebra.encode (fun i => BinaryAlgebra.bitValue (p i))
(fun i => BinaryAlgebra.bitValue (coefficientInputBits p β j i))) =
hashAcceptance p t u β := by
simp only [encode_coefficientInputBits]
end DepthThreeLowerBound
end
end OAI
|