| import OAI.Combinatorics.ContingencyTables.Dense.DenseBooleanTrial |
| import OAI.Combinatorics.ContingencyTables.Dense.DenseBoundedRejection |
| import OAI.Combinatorics.ContingencyTables.Machines.FiniteWordRejection |
| import OAI.Combinatorics.ContingencyTables.Dense.DenseBitBudget |
|
|
| namespace OAI |
|
|
| /-! # A bounded unbiased Boolean-word dense table draw -/ |
|
|
| namespace ContingencyTables.GridBoundary |
|
|
| open scoped BigOperators Classical |
| open ResidualMixture FirstSuccess FairBits |
|
|
| noncomputable section |
|
|
| variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] |
| variable {q : ℕ} |
|
|
| def denseDrawBits (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ) |
| (e : I × J ≃ Fin q) (h : ℕ) : ℕ := |
| (d ^ 4 * (h + 1)) * denseTrialBits R P d e h |
|
|
| def denseBooleanDraw (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ) |
| (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) |
| (e : I × J ≃ Fin q) (htotal : ∑ i, R i = ∑ j, P j) (h : ℕ) |
| (a : Grid (I := Fin q) (4 * d ^ 8)) (fallback : Table R P) |
| (bits : Fin (denseDrawBits R P d e h) → Bool) : Table R P := |
| wordRetry (fun word => trialSuccess R P d e htotal (denseBooleanTrial R P d hd hR hP e h a word)) |
| fallback (d ^ 4 * (h + 1)) (gridWordEquiv (d ^ 4 * (h + 1)) (denseTrialBits R P d e h) bits) |
|
|
| def denseBooleanDrawLaw (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ) |
| (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) |
| (e : I × J ≃ Fin q) (htotal : ∑ i, R i = ∑ j, P j) (h : ℕ) |
| (a : Grid (I := Fin q) (4 * d ^ 8)) (fallback : Table R P) : RationalLaw (Table R P) := |
| mapLaw (uniformLaw (α := Fin (denseDrawBits R P d e h) → Bool)) |
| (denseBooleanDraw R P d hd hR hP e htotal h a fallback) |
|
|
| theorem denseBooleanDrawLaw_eq (R : Option I → ℕ) (P : Option J → ℕ) |
| (d : ℕ) (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) |
| (hRmax : ∀ i, R (some i) ≤ R none) (hPmax : ∀ j, P (some j) ≤ P none) |
| (hdimR : Fintype.card (Option I) ≤ d) (hdimP : Fintype.card (Option J) ≤ d) |
| (hfree : Fintype.card (I × J) ≤ d) (htotal : ∑ i, R i = ∑ j, P j) |
| (e : I × J ≃ Fin q) (h : ℕ) (a : Grid (I := Fin q) (4 * d ^ 8)) (fallback : Table R P) : |
| denseBooleanDrawLaw R P d hd hR hP e htotal h a fallback = |
| denseBoundedLaw R P d hd hR hP hfree e htotal h a fallback := by |
| unfold denseBooleanDrawLaw denseBooleanDraw |
| change mapLaw (uniformLaw (α := Fin ((d^4*(h+1))*denseTrialBits R P d e h) → Bool)) |
| (fun bits => wordRetry |
| (fun word => trialSuccess R P d e htotal (denseBooleanTrial R P d hd hR hP e h a word)) |
| fallback (d^4*(h+1)) (gridWordEquiv (d^4*(h+1)) (denseTrialBits R P d e h) bits)) = _ |
| rw [← map_equiv (gridWordEquiv (d ^ 4 * (h + 1)) (denseTrialBits R P d e h)), uniform_equiv] |
| change wordRetryLaw |
| (fun word => trialSuccess R P d e htotal (denseBooleanTrial R P d hd hR hP e h a word)) |
| fallback (d ^ 4 * (h + 1)) = _ |
| rw [wordRetryLaw_eq, ← retryLaw_map, |
| denseBooleanTrialLaw_eq R P d hd hR hP hRmax hPmax hdimR hdimP hfree htotal e h a] |
| rfl |
|
|
| / |
| and its law satisfies the desired dense approximation guarantee. -/ |
| theorem denseBooleanDraw_variation (R : Option I → ℕ) (P : Option J → ℕ) |
| (d : ℕ) (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) |
| (hRmax : ∀ i, R (some i) ≤ R none) (hPmax : ∀ j, P (some j) ≤ P none) |
| (hdimR : Fintype.card (Option I) ≤ d) (hdimP : Fintype.card (Option J) ≤ d) |
| (hfree : Fintype.card (I × J) ≤ d) (htotal : ∑ i, R i = ∑ j, P j) |
| (e : I × J ≃ Fin q) (hPL : PublishedInputs.FiniteBoxPrekopaLeindler q) |
| (hCheeger : PublishedInputs.FiniteCheeger (Grid (I := Fin q) (4 * d ^ 8))) |
| (h : ℕ) (a : Grid (I := Fin q) (4 * d ^ 8)) (fallback : Table R P) : |
| let _ : Nonempty (Table R P) := ⟨fallback⟩ |
| (variation (denseBooleanDrawLaw R P d hd hR hP e htotal h a fallback) |
| (uniformLaw (α := Table R P)) : ℝ) ≤ (2 : ℝ) ^ (-(h : ℝ)) := by |
| let : Nonempty (Table R P) := ⟨fallback⟩ |
| rw [denseBooleanDrawLaw_eq R P d hd hR hP hRmax hPmax hdimR hdimP hfree htotal e h a fallback] |
| exact denseBoundedLaw_variation R P d hd hR hP hRmax hPmax hdimR hdimP hfree htotal |
| e hPL hCheeger h a fallback |
|
|
| omit [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] in |
| / |
| parameters. -/ |
| theorem denseDrawBits_polynomial (R : Option I → ℕ) (P : Option J → ℕ) |
| (d C b h : ℕ) (e : I × J ≃ Fin q) (hq : q ≤ d) |
| (hwidth : ∀ p, integerWidth R P d p ≤ C) (hb : Nat.clog 2 (C + 2) ≤ b) : |
| denseDrawBits R P d e h ≤ 25 * (d + b + h + 1) ^ 62 := by |
| have hoff : (∑ j : Fin q, offsetBitLength (integerWidth R P d (e.symm j)) h d) ≤ |
| d * (b + 1 + 4 * (h + d)) := by |
| calc |
| _ ≤ ∑ _j : Fin q, (b + 1 + 4 * (h + d)) := |
| Finset.sum_le_sum (fun j _ => denseOffsetBits_upper _ C b h d (hwidth _) hb) |
| _ = q * (b + 1 + 4 * (h + d)) := by simp |
| _ ≤ _ := Nat.mul_le_mul_right _ hq |
| apply (show denseDrawBits R P d e h ≤ denseRandomBitBudget d b h from ?_).trans |
| (denseRandomBitBudget_polynomial d b h) |
| unfold denseDrawBits denseTrialBits denseRandomBitBudget |
| exact Nat.mul_le_mul_left _ (Nat.add_le_add_left hoff _) |
|
|
| end |
|
|
| end ContingencyTables.GridBoundary |
|
|
| end OAI |
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|
