| import OAI.Computability.DepthThree.HardSliceProbability |
| import OAI.Computability.DepthThree.HardSliceSelection |
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| namespace OAI |
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| noncomputable section |
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| open scoped BigOperators Classical |
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| namespace DepthThreeLowerBound |
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| 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) |
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| 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⟩ |
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| 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) |
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| @[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 _ _ |
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| 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] |
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| end DepthThreeLowerBound |
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| end |
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| end OAI |
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