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92416a2 | 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 | import OAI.Computability.UniqueGames.Reduction.ActualGame
import OAI.Computability.UniqueGames.Reduction.GapSemanticsLemmas
import OAI.Computability.UniqueGames.Reduction.NoiseEnumerationLemmas
namespace OAI
namespace UniqueGamesTheorem.Integration.TableReduction
open BinaryLinear NoiseTables UniqueGamesTheorem.Reduction.ActualSource
open UniqueGamesTheorem.Reduction.ActualGame UniqueGamesTheorem.Foundations.Target
variable {s d : Nat}
def tableEnumeration (T : Table s d) (f : Ambient s d → Alphabet s)
(hf : ∀ x c, f (x + (c, 0)) = f x + c) : NoiseEnumeration (T.gadget f hf) where
indices := List.finRange T.vectors.length
nodup := List.nodup_finRange _
complete i := List.mem_finRange i
/-- The projection is used only to fill an erased witness field. It is never
used when generating constraints or evaluating their permutation tables. -/
def tableSkeleton (T : Table s d) : SplitGadget s d :=
T.gadget Prod.fst (fun _ _ => rfl)
/-- A total executable map with finite vector data and no nonlinear-map input. -/
def output (S : Source) (k : Nat) (T : Table s d) : Instance (2 ^ s) :=
outputInstanceWithEnumeration S k (tableSkeleton T)
(tableEnumeration T Prod.fst (fun _ _ => rfl))
/-- Every correcting-map witness generates exactly the same numbered instance,
including the ordered constraint list and all permutation entries. -/
theorem output_eq_with_witness (S : Source) (k : Nat) (T : Table s d)
(f : Ambient s d → Alphabet s) (hf : ∀ x c, f (x + (c, 0)) = f x + c) :
output S k T =
outputInstanceWithEnumeration S k (T.gadget f hf) (tableEnumeration T f hf) := by
rfl
theorem completeAt_output_iff (error : RationalError) (S : Source) (k : Nat)
(T : Table s d) (f : Ambient s d → Alphabet s)
(hf : ∀ x c, f (x + (c, 0)) = f x + c) :
CompleteAt error (output S k T) ↔
CompleteAt error (outputInstance S k (T.gadget f hf)) := by
rw [output_eq_with_witness S k T f hf]
exact completeAt_outputInstanceWithEnumeration_iff _ _ _ _ _
theorem soundAt_output_iff (error : RationalError) (S : Source) (k : Nat)
(T : Table s d) (f : Ambient s d → Alphabet s)
(hf : ∀ x c, f (x + (c, 0)) = f x + c) :
SoundAt error (output S k T) ↔
SoundAt error (outputInstance S k (T.gadget f hf)) := by
rw [output_eq_with_witness S k T f hf]
exact soundAt_outputInstanceWithEnumeration_iff _ _ _ _ _
theorem edge_ofEnumeration (S : Source) (k : Nat) (g : SplitGadget s d)
(en : NoiseEnumeration g)
(ω : Outcome S k ((Table.ofEnumeration g en).gadget g.f g.equivariant)) :
edge S k ((Table.ofEnumeration g en).gadget g.f g.equivariant) ω =
edge S k g (ω.1, Table.positionEquiv g en ω.2.1, ω.2.2) := by
simp only [edge, leftQuery, rightQuery]
erw [Table.noise_positionEquiv g en ω.2.1]
theorem acceptance_ofEnumeration (S : Source) (k : Nat) (g : SplitGadget s d)
(en : NoiseEnumeration g) (label : Fin (vertexCount S k s d) → Fin (2 ^ s)) :
acceptanceProbability S k ((Table.ofEnumeration g en).gadget g.f g.equivariant) label =
acceptanceProbability S k g label := by
apply Fintype.expect_equiv ((Equiv.refl (Query S k s d)).prodCongr
((Table.positionEquiv g en).prodCongr (Equiv.refl (Dual k))))
intro ω
exact congrArg
(fun c : Constraint (vertexCount S k s d) (2 ^ s) =>
if c.satisfied label then (1 : ℚ) else 0)
(edge_ofEnumeration S k g en ω)
theorem rate_ofEnumeration (S : Source) (k : Nat) (g : SplitGadget s d)
(en : NoiseEnumeration g) (label : Fin (vertexCount S k s d) → Fin (2 ^ s)) :
GapSemantics.satisfactionRate
(outputInstance S k ((Table.ofEnumeration g en).gadget g.f g.equivariant)) label =
GapSemantics.satisfactionRate (outputInstance S k g) label := by
exact (acceptanceProbability_eq_count S k _ label).symm.trans
((acceptance_ofEnumeration S k g en label).trans
(acceptanceProbability_eq_count S k g label))
/-- Normalization to a concrete table preserves the original completeness
threshold exactly; there is no additional approximation loss. -/
theorem completeAt_ofEnumeration_iff (error : RationalError)
(S : Source) (k : Nat) (g : SplitGadget s d) (en : NoiseEnumeration g) :
CompleteAt error (output S k (Table.ofEnumeration g en)) ↔
CompleteAt error (outputInstance S k g) := by
rw [completeAt_output_iff error S k _ g.f g.equivariant]
rw [GapSemantics.completeAt_iff, GapSemantics.completeAt_iff]
constructor
· rintro ⟨label, h⟩
exact ⟨label, h.trans_eq (rate_ofEnumeration S k g en label)⟩
· rintro ⟨label, h⟩
exact ⟨label, h.trans_eq (rate_ofEnumeration S k g en label).symm⟩
theorem soundAt_ofEnumeration_iff (error : RationalError)
(S : Source) (k : Nat) (g : SplitGadget s d) (en : NoiseEnumeration g) :
SoundAt error (output S k (Table.ofEnumeration g en)) ↔
SoundAt error (outputInstance S k g) := by
rw [soundAt_output_iff error S k _ g.f g.equivariant]
rw [GapSemantics.soundAt_iff, GapSemantics.soundAt_iff]
constructor
· intro h label
exact (rate_ofEnumeration S k g en label).symm.trans_le (h label)
· intro h label
exact (rate_ofEnumeration S k g en label).trans_le (h label)
end UniqueGamesTheorem.Integration.TableReduction
end OAI
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