| import Mathlib |
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| namespace OAI |
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| section |
| noncomputable section |
| open scoped BigOperators |
| namespace SoftChannel204 |
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|
| abbrev Cube (n : ℕ) := Fin n → Bool |
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| def sign (b : Bool) : ℝ := if b then 1 else -1 |
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| def avg {n : ℕ} (g : Cube n → ℝ) : ℝ := (∑ x, g x) / (2 : ℝ) ^ n |
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| def L : ℝ := Real.log 2 |
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| def psi (r : ℝ) : ℝ := |
| (1 + r) / 2 * Real.log (1 + r) + (1 - r) / 2 * Real.log (1 - r) |
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| def H (r : ℝ) : ℝ := L - psi r |
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| def info {n : ℕ} (u : Cube n → ℝ) : ℝ := H (avg u) - avg (H ∘ u) |
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| def psiInv (s : ℝ) : ℝ := Function.invFunOn psi (Set.Icc 0 1) s |
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| def kernel {n : ℕ} (ρ : ℝ) (x y : Cube n) : ℝ := |
| ∏ i, if x i = y i then (1 + ρ) / 2 else (1 - ρ) / 2 |
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| def noise {n : ℕ} (ρ : ℝ) (u : Cube n → ℝ) (y : Cube n) : ℝ := |
| ∑ x, kernel ρ y x * u x |
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| def Closed {n : ℕ} (u : Cube n → ℝ) : Prop := ∀ x, u x ∈ Set.Icc (-1) 1 |
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| def Interior {n : ℕ} (g : Cube n → ℝ) : Prop := ∀ x, g x ∈ Set.Ioo (-1) 1 |
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| def softCoordinate {n : ℕ} (i : Fin n) (a : ℝ) (x : Cube n) : ℝ := a * sign (x i) |
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| def flip {n : ℕ} (i : Fin n) (x : Cube n) : Cube n := Function.update x i (!(x i)) |
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| def numberOperator {n : ℕ} (g : Cube n → ℝ) (x : Cube n) : ℝ := |
| (1 / 2 : ℝ) * ∑ i, (g x - g (flip i x)) |
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| def production {n : ℕ} (g : Cube n → ℝ) : ℝ := |
| avg (fun x => Real.artanh (g x) * numberOperator g x) |
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| / |
| def F (s : ℝ) : ℝ := psiInv s * Real.artanh (psiInv s) |
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| / |
| def S (s : ℝ) : ℝ := if s < 0 then 0 else F s - 2 * s |
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| / |
| def reserve (m I : ℝ) : ℝ := |
| 2 * I + (1 - m ^ 2) * S (L - (H m - I) / (1 - m ^ 2)) |
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| def hybrid (m I : ℝ) : ℝ := max (reserve m I) (F I) |
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| def softJoint {n : ℕ} (ρ : ℝ) (u : Cube n → ℝ) (b : Bool) (y : Cube n) : ℝ := |
| avg (fun x => (1 + sign b * u x) / 2 * kernel ρ x y) |
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| def boolJoint {n : ℕ} (ρ : ℝ) (f : Cube n → Bool) (b : Bool) (y : Cube n) : ℝ := |
| avg (fun x => if f x = b then kernel ρ x y else 0) |
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| def leftMarginal {n : ℕ} (p : Bool → Cube n → ℝ) (b : Bool) : ℝ := ∑ y, p b y |
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| def rightMarginal {n : ℕ} (p : Bool → Cube n → ℝ) (y : Cube n) : ℝ := ∑ b, p b y |
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| def mutualInformation {n : ℕ} (p : Bool → Cube n → ℝ) : ℝ := |
| ∑ b, ∑ y, if 0 < p b y then |
| p b y * Real.log (p b y / (leftMarginal p b * rightMarginal p y)) else 0 |
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| def binaryEntropyBits (ε : ℝ) : ℝ := |
| -(ε * Real.log ε + (1 - ε) * Real.log (1 - ε)) / L |
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| def SoftContraction : Prop := ∀ (n : ℕ) (u : Cube n → ℝ), Closed u → |
| ∀ ρ ∈ Set.Icc (-1 : ℝ) 1, |
| info (noise ρ u) ≤ psi (|ρ| * psiInv (info u)) |
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| def SoftAttainment : Prop := ∀ (n : ℕ) (i : Fin n) (a : ℝ), |a| ≤ 1 → |
| ∀ ρ ∈ Set.Icc (-1 : ℝ) 1, |
| info (noise ρ (softCoordinate i a)) = |
| psi (|ρ| * psiInv (info (softCoordinate i a))) |
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| def SoftInformationIdentity : Prop := ∀ (n : ℕ) (u : Cube n → ℝ), Closed u → |
| ∀ ρ ∈ Set.Icc (-1 : ℝ) 1, |
| mutualInformation (softJoint ρ u) = info (noise ρ u) |
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| def RefinedBoolean : Prop := ∀ (n : ℕ) (f : Cube n → Bool) |
| (ρ : ℝ), ρ ∈ Set.Icc (-1 : ℝ) 1 → |
| let m := avg (sign ∘ f) |
| let bound := psi (|ρ| * psiInv (H m)) |
| mutualInformation (boolJoint ρ f) = info (noise ρ (sign ∘ f)) ∧ |
| mutualInformation (boolJoint ρ f) ≤ bound ∧ bound ≤ psi |ρ| ∧ |
| (0 < |m| → |m| < 1 → 0 < |ρ| → bound < psi |ρ|) |
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| def BooleanAttainment : Prop := ∀ (n : ℕ) (i : Fin n) |
| (ρ : ℝ), ρ ∈ Set.Icc (-1 : ℝ) 1 → |
| mutualInformation (boolJoint ρ (fun x => x i)) = psi |ρ| ∧ |
| mutualInformation (boolJoint ρ (fun x => !(x i))) = psi |ρ| |
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| def BitsConversion : Prop := ∀ (n : ℕ) (f : Cube n → Bool) |
| (ε : ℝ), ε ∈ Set.Icc (0 : ℝ) (1/2) → |
| psi (1 - 2 * ε) / L = 1 - binaryEntropyBits ε ∧ |
| mutualInformation (boolJoint (1 - 2 * ε) f) / L ≤ 1 - binaryEntropyBits ε |
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| def SharpProduction : Prop := ∀ (n : ℕ) (g : Cube n → ℝ), Interior g → |
| hybrid (avg g) (info g) ≤ production g ∧ F (info g) ≤ hybrid (avg g) (info g) |
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| def FullMain : Prop := SoftContraction ∧ SoftAttainment ∧ SoftInformationIdentity ∧ |
| RefinedBoolean ∧ BooleanAttainment ∧ BitsConversion ∧ SharpProduction |
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| end SoftChannel204 |
| end |
| end |
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| end OAI |
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|
