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97c3405 | 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 92 93 94 95 96 97 98 99 100 101 102 | 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
/-- The actual finite Boolean draw always has codomain the feasible tables
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
/-- The full Boolean source has polynomial length in the declared dense
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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