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2f887db | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import OAI.Computability.PerfectCompleteness.Machines.BinaryDualCoordinates
namespace OAI
section
namespace PerfectCompleteness.AffineLinearHeavyWitness
open scoped BigOperators
open QuarterBalance (F2)
open BinaryDualCoordinates
open UniqueGamesTheorem.Foundations.Games
noncomputable section
variable {U V K I J : Type*}
[AddCommGroup U] [Module F2 U] [Fintype U]
[AddCommGroup V] [Module F2 V]
[AddCommGroup K] [Module F2 K]
[Fintype I] [DecidableEq I] [Fintype J] [DecidableEq J]
def conditionalMatch (μ : FiniteDistribution U) (R : U →ₗ[F2] V) (target : V)
(G : U → K) (T : U →ₗ[F2] K) (offset : K) : ℝ := by
classical
exact μ.expectation (fun x => if R x = target then
(if G x = offset + T x then 1 else 0) else 0) /
μ.expectation (fun x => if R x = target then 1 else 0)
omit [DecidableEq I] [DecidableEq J] in
theorem conditionalEquality_coordinates (μ : FiniteDistribution U)
(b : Module.Basis I F2 V) (c : Module.Basis J F2 K)
(R : U →ₗ[F2] V) (target : V) (G : U → K)
(T : U →ₗ[F2] K) (offset : K) :
OutputCharacters.conditionalEquality μ (equations b R) (coordinates b target)
(fun x => coordinates c (G x)) (fun x => coordinates c (offset + T x)) =
conditionalMatch μ R target G T offset := by
classical
simp only [OutputCharacters.conditionalEquality, OutputCharacters.sliceCorrelation,
SmallBiasSlice.sliceNumerator, SmallBiasSlice.sliceMass, conditionalMatch,
equations_eq_iff, (coordinates_injective c).eq_iff]
def badMasks (c : Module.Basis J F2 K) (bad : Finset (Module.Dual F2 K)) :
Finset (J → Bool) :=
bad.map (frequencyEquiv c).symm.toEmbedding
omit [DecidableEq J] in
@[simp] theorem mem_badMasks (c : Module.Basis J F2 K)
(bad : Finset (Module.Dual F2 K)) (s : J → Bool) :
s ∈ badMasks c bad ↔ frequency c s ∈ bad := by
classical
constructor
· rintro h
obtain ⟨f, hf, heq⟩ := Finset.mem_map.mp h
have heq' : frequency c s = f := by
change (frequencyEquiv c) s = f
rw [← heq]
exact (frequencyEquiv c).apply_symm_apply f
simpa only [heq'] using hf
· intro h
apply Finset.mem_map.mpr
refine ⟨frequency c s, h, ?_⟩
exact (frequencyEquiv c).symm_apply_apply s
omit [DecidableEq J] in
@[simp] theorem card_badMasks (c : Module.Basis J F2 K)
(bad : Finset (Module.Dual F2 K)) :
(badMasks c bad).card = bad.card := Finset.card_map _
theorem exists_heavy_in_dual_coset [Fintype (Module.Dual F2 U)]
[Fintype (Module.Dual F2 K)]
(μ : FiniteDistribution U)
(b : Module.Basis I F2 V) (c : Module.Basis J F2 K)
(R : U →ₗ[F2] V) (hR : Function.Surjective R) (target : V)
(G : U → K) (T : U →ₗ[F2] K) (offset : K)
(bad : Finset (Module.Dual F2 K)) (rho h0 biasBound : ℝ)
(hrho : 0 < rho) (hh0 : 0 < h0)
(hsize : h0 ≤ 1 / (Fintype.card (I → Bool) : ℝ))
(hb : 0 ≤ biasBound) (hbsmall : biasBound ≤ h0 / 2)
(hbias : ∀ f : Module.Dual F2 U, f ≠ 0 →
|μ.expectation (character f)| ≤ biasBound)
(hbad : (bad.card : ℝ) / (Fintype.card (Module.Dual F2 K) : ℝ) < rho / 8)
(hmatch : rho ≤ conditionalMatch μ R target G T offset) :
∃ σ : Module.Dual F2 K, σ ∉ bad ∧
∃ ψ ∈ LinearMap.range R.dualMap,
σ.comp T + ψ ∈ SmallBias.heavySet μ character
(fun x => character σ (G x)) (rho * h0 / 4) := by
classical
have hcard : Fintype.card (J → Bool) = Fintype.card (Module.Dual F2 K) :=
Fintype.card_congr (frequencyEquiv c)
have hbad' : ((badMasks c bad).card : ℝ) /
(Fintype.card (J → Bool) : ℝ) < rho / 8 := by
simpa only [card_badMasks, hcard] using hbad
have hmatch' : rho ≤ OutputCharacters.conditionalEquality μ (equations b R)
(coordinates b target) (fun x => coordinates c (G x))
(fun x => coordinates c (offset + T x)) := by
rwa [conditionalEquality_coordinates]
obtain ⟨s, hs, mask, hmask⟩ :=
OutputCharacters.equality_implies_affine_heavy_witness μ
character character_zero character_add (equationRows b R) (equations b R)
(character_equationRows b R) (equationRows_independent b R hR)
(coordinates b target) (fun x => coordinates c (G x))
(fun x => coordinates c (offset + T x))
(fun s => (frequency c s).comp T) (fun s => character (frequency c s) offset)
(fun s => abs_character _ _) (character_affine_target c T offset)
(badMasks c bad) rho h0 biasBound hrho hh0 hsize hb hbsmall hbias hbad' hmatch'
refine ⟨frequency c s, (mem_badMasks c bad s).not.mp hs,
SmallBiasSlice.rowCombination (equationRows b R) mask,
rowCombination_mem_dualRange b R mask, ?_⟩
simpa only [← character_frequency] using hmask
end
end PerfectCompleteness.AffineLinearHeavyWitness
end
end OAI
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