File size: 4,940 Bytes
5d5048e | 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 99 100 101 102 103 104 105 106 107 108 | import OAI.Combinatorics.LowSpaceSubsetSum.Separators.SeparatorCheck
namespace OAI
/-! # Coverage probability of the actual four-block checker -/
noncomputable section
namespace LowSpaceSubsetSum.SeparatorCheck
open SeparatorExpectation
variable {g : ℕ}
/-- The one-draw success probability for the prescribed left count. -/
def theta (e c : ℕ) : ℝ :=
(Nat.choose (g / 2) (e - c) : ℝ) / Nat.choose g e
abbrev Draws (g : ℕ) (e c : Fin 4 → ℕ) :=
∀ i : Fin 4, Fin (⌈10 / (theta (g := g) (e i) (c i))⌉₊) →
SeparatorSample (Finset.univ : Finset (Fin g)) (e i)
/-- Sample occurrences keep their indices, including repeated subsets. -/
def sampledFamilies {e c : Fin 4 → ℕ} (draws : Draws g e c) :
Fin 4 → Array (Finset (Fin g)) :=
fun i => Array.ofFn (fun j => (draws i j).val)
theorem sampled_covered_iff {e c : Fin 4 → ℕ} (draws : Draws g e c)
(x y : Record g) :
Covered (sampledFamilies draws) x y ↔
∀ i, ∃ j, localMask i x.mask ⊆ (draws i j).val ∧
Disjoint (draws i j).val (localMask i y.mask) := by
constructor
· intro h i
obtain ⟨j, hl, hr⟩ := h i
let k : Fin (⌈10 / (theta (g := g) (e i) (c i))⌉₊) :=
⟨j.val, by simpa [sampledFamilies] using j.isLt⟩
exact ⟨k, by simpa [sampledFamilies, k] using hl,
(by simpa [sampledFamilies, k, disjoint_comm] using hr)⟩
· intro h i
obtain ⟨j, hl, hr⟩ := h i
let k : Fin (sampledFamilies draws i).size :=
⟨j.val, by simp only [sampledFamilies, Array.size_ofFn]; exact j.isLt⟩
exact ⟨k, by simpa [sampledFamilies, k] using hl,
(by simpa [sampledFamilies, k, disjoint_comm] using hr)⟩
/-- All four families cover any fixed admissible pair with the paper's
`1-4 exp(-10)` probability under the actual independent sampling law. -/
theorem coverage_probability (e c : Fin 4 → ℕ) (x y : Record g)
(heven : 2 ∣ g) (hd : Disjoint x.mask y.mask)
(hc : ∀ i, (localMask i x.mask).card = c i)
(hhalf : ∀ i, c i + (localMask i y.mask).card = g / 2)
(hce : ∀ i, c i ≤ e i) (heg : ∀ i, e i ≤ g)
(hfeas : ∀ i, e i - c i ≤ g / 2) :
1 - 4 * Real.exp (-10) ≤ finiteUniformProbability
(fun draws : Draws g e c => Covered (sampledFamilies draws) x y) := by
classical
let Ψ := fun i : Fin 4 => Fin (⌈10 / (theta (g := g) (e i) (c i))⌉₊) →
SeparatorSample (Finset.univ : Finset (Fin g)) (e i)
have hsample (i : Fin 4) :
Nonempty (SeparatorSample (Finset.univ : Finset (Fin g)) (e i)) :=
(Finset.powersetCard_nonempty.mpr (by simpa using heg i)).to_subtype
let (i : Fin 4) : Nonempty (SeparatorSample (Finset.univ : Finset (Fin g)) (e i)) :=
hsample i
let Q (i : Fin 4) (family : Ψ i) : Prop :=
∃ j, localMask i x.mask ⊆ (family j).val ∧
Disjoint (family j).val (localMask i y.mask)
have hmarginal (i : Fin 4) :
1 - Real.exp (-10) ≤ finiteUniformProbability (Q i) := by
have hdi : Disjoint (localMask i x.mask) (localMask i y.mask) := by
apply Finset.disjoint_left.mpr
intro j hj hk
exact Finset.disjoint_left.mp hd ((localMask_mem i x.mask j).mp hj)
((localMask_mem i y.mask j).mp hk)
have hcov := separator_family_coverage (Finset.univ : Finset (Fin g))
(localMask i x.mask) (localMask i y.mask) (e i)
(Finset.subset_univ _) (Finset.subset_univ _) hdi
(by simpa only [hc i] using hce i) (by simpa using heg i)
(by simpa using heven) (by simpa only [Finset.card_univ, Fintype.card_fin, hc i] using (hhalf i))
(by simpa only [Finset.card_univ, Fintype.card_fin, hc i] using hfeas i)
dsimp only at hcov
rw [Finset.card_univ, Fintype.card_fin, hc i] at hcov
exact hcov
have hfour := four_family_union_bound (fun i (draws : ∀ i, Ψ i) => Q i (draws i))
(fun i => by rw [probability_pi_coordinate i (Q i)]; exact hmarginal i)
simpa only [sampled_covered_iff, Q] using hfour
/-- Applying the executable checker preserves this fixed-pair success bound. -/
theorem acceptance_probability (n : ℕ) (e c : Fin 4 → ℕ) (target : ℤ)
(xs ys : List (Record g)) (hsize : xs.length + ys.length ≤ 2 ^ n)
(x y : Record g) (hx : x ∈ xs) (hy : y ∈ ys)
(hw : x.value + y.value = target)
(heven : 2 ∣ g) (hd : Disjoint x.mask y.mask)
(hc : ∀ i, (localMask i x.mask).card = c i)
(hhalf : ∀ i, c i + (localMask i y.mask).card = g / 2)
(hce : ∀ i, c i ≤ e i) (heg : ∀ i, e i ≤ g)
(hfeas : ∀ i, e i - c i ≤ g / 2) :
1 - 4 * Real.exp (-10) ≤ finiteUniformProbability
(fun draws : Draws g e c => (run n (sampledFamilies draws) target xs ys).accepted = true) :=
(coverage_probability e c x y heven hd hc hhalf hce heg hfeas).trans
(finiteUniformProbability_mono (fun _ h => finds_fixed_pair n _ target xs ys hsize x y hx hy hw h))
end LowSpaceSubsetSum.SeparatorCheck
end
end OAI
|