| import OAI.Computability.UniqueGames.Reduction.ActualGame |
|
|
| namespace OAI |
|
|
| section |
|
|
| namespace UniqueGamesTheorem.Decoder.TableKeysGame |
|
|
| open UniqueGamesTheorem.Integration.BinaryLinear UniqueGamesTheorem.Reduction |
| open ActualSource Foundations.Target |
| open scoped BigOperators |
|
|
| abbrev Question (S : Source) (k : Nat) := ActualGame.Question S k |
| abbrev Map (k s d : Nat) := ActualGame.Map k s d |
| abbrev Dual (k : Nat) := ActualGame.Dual k |
| abbrev Query (S : Source) (k s d : Nat) := ActualGame.Query S k s d |
| abbrev canonical (S : Source) (k s d : Nat) := ActualGame.canonical S k s d |
|
|
| / |
| abbrev Vertex (S : Source) (k s d : Nat) := ActualGame.Vertex S k s d |
| abbrev vertex (S : Source) (k s d : Nat) := ActualGame.vertex S k s d |
| abbrev offset (S : Source) (k s d : Nat) := ActualGame.offset S k s d |
| abbrev shiftQuery (S : Source) (k s d : Nat) := ActualGame.shiftQuery S k s d |
|
|
| noncomputable def vertexCount (S : Source) (k s d : Nat) : Nat := |
| Fintype.card (Vertex S k s d) |
|
|
| noncomputable def vertexEncoding (S : Source) (k s d : Nat) : |
| Vertex S k s d ≃ Fin (vertexCount S k s d) := Fintype.equivFin _ |
|
|
| / |
| abbrev Outcome (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) := |
| Query S k s d × g.NoiseIndex × Dual k |
|
|
| abbrev leftQuery (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) := |
| ActualGame.leftQuery S k g |
| abbrev rightQuery (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) := |
| ActualGame.rightQuery S k g |
|
|
| noncomputable def edge (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) |
| (ω : Outcome S k g) : Constraint (vertexCount S k s d) (2^s) where |
| source := vertexEncoding S k s d (vertex S k s d (leftQuery S k g ω)) |
| target := vertexEncoding S k s d (vertex S k s d (rightQuery S k g ω)) |
| permutation := Encoding.translationTable |
| (offset S k s d (leftQuery S k g ω)) (offset S k s d (rightQuery S k g ω)) |
|
|
| noncomputable def outcomes (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) := |
| ActualGame.outcomes S k g |
|
|
| noncomputable def outputInstance (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) : |
| Instance (2^s) where |
| vertices := vertexCount S k s d |
| constraints := (outcomes S k g).map (edge S k g) |
| nonempty := by |
| intro h |
| let ω : Outcome S k g := Classical.choice inferInstance |
| have hm : edge S k g ω ∈ (outcomes S k g).map (edge S k g) := |
| List.mem_map.mpr ⟨ω, ActualGame.mem_outcomes S k g ω, rfl⟩ |
| rw [h] at hm |
| exact List.not_mem_nil hm |
|
|
| @[simp] theorem outputInstance_constraints_length (S : Source) (k : Nat) |
| {s d : Nat} (g : SplitGadget s d) : |
| (outputInstance S k g).constraints.length = Fintype.card (Outcome S k g) := by |
| simp [outputInstance, outcomes] |
|
|
| noncomputable def vertexLabel (S : Source) (k s d : Nat) |
| (labeling : Fin (vertexCount S k s d) → Fin (2^s)) : |
| Vertex S k s d → Alphabet s := fun v => |
| (Encoding.alphabetEquiv s).symm (labeling (vertexEncoding S k s d v)) |
|
|
| noncomputable def labelingOf (S : Source) (k s d : Nat) |
| (labels : Vertex S k s d → Alphabet s) : |
| Fin (vertexCount S k s d) → Fin (2^s) := fun i => |
| Encoding.alphabetEquiv s (labels ((vertexEncoding S k s d).symm i)) |
|
|
| @[simp] theorem vertexLabel_labelingOf (S : Source) (k s d : Nat) |
| (labels : Vertex S k s d → Alphabet s) : |
| vertexLabel S k s d (labelingOf S k s d labels) = labels := by |
| funext v |
| simp [vertexLabel, labelingOf] |
|
|
| / |
| noncomputable def unfolded (S : Source) (k s d : Nat) |
| (labeling : Fin (vertexCount S k s d) → Fin (2^s)) |
| (q : Query S k s d) : Alphabet s := |
| ActualOrbit.unfold (canonical S k s d) (vertexLabel S k s d labeling) q |
|
|
| theorem unfolded_equivariant (S : Source) (k s d : Nat) |
| (labeling : Fin (vertexCount S k s d) → Fin (2^s)) |
| (q : Query S k s d) (c : Alphabet s) : |
| unfolded S k s d labeling (shiftQuery S k s d c q) = |
| unfolded S k s d labeling q + c := by |
| change vertexLabel S k s d labeling |
| (ActualGame.vertex S k s d (ActualGame.shiftQuery S k s d c q)) + |
| ActualGame.offset S k s d (ActualGame.shiftQuery S k s d c q) = _ |
| rw [ActualGame.vertex_shiftQuery, ActualGame.offset_shiftQuery] |
| exact (add_assoc _ _ _).symm |
|
|
| theorem edge_satisfied_iff (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) |
| (labeling : Fin (vertexCount S k s d) → Fin (2^s)) (ω : Outcome S k g) : |
| (edge S k g ω).satisfied labeling = true ↔ |
| unfolded S k s d labeling (leftQuery S k g ω) = |
| unfolded S k s d labeling (rightQuery S k g ω) := by |
| rw [Constraint.satisfied, decide_eq_true_eq] |
| have h := Encoding.translationTable_satisfied_iff |
| (offset S k s d (leftQuery S k g ω)) (offset S k s d (rightQuery S k g ω)) |
| (vertexLabel S k s d labeling (vertex S k s d (leftQuery S k g ω))) |
| (vertexLabel S k s d labeling (vertex S k s d (rightQuery S k g ω))) |
| simp only [vertexLabel, Encoding.alphabetEquiv_apply_symm_apply] at h |
| exact h.trans (ActualOrbit.constraint_iff_unfolded (canonical S k s d) |
| (vertexLabel S k s d labeling) (vertexLabel S k s d labeling) _ _) |
|
|
| noncomputable def acceptanceProbability (S : Source) (k : Nat) {s d : Nat} |
| (g : SplitGadget s d) (labeling : Fin (vertexCount S k s d) → Fin (2^s)) : ℚ := |
| 𝔼 ω : Outcome S k g, if (edge S k g ω).satisfied labeling then (1 : ℚ) else 0 |
|
|
| theorem acceptanceProbability_eq_test (S : Source) (k : Nat) {s d : Nat} |
| (g : SplitGadget s d) (labeling : Fin (vertexCount S k s d) → Fin (2^s)) : |
| acceptanceProbability S k g labeling = |
| 𝔼 ω : Outcome S k g, |
| if unfolded S k s d labeling (leftQuery S k g ω) = |
| unfolded S k s d labeling (rightQuery S k g ω) |
| then (1 : ℚ) else 0 := by |
| unfold acceptanceProbability |
| apply Finset.expect_congr rfl |
| intro ω _ |
| simp only [edge_satisfied_iff] |
|
|
| private theorem countSatisfied_map {n q : Nat} {I : Type*} |
| (labeling : Fin n → Fin q) (edges : I → Constraint n q) (xs : List I) : |
| countSatisfied labeling (xs.map edges) = |
| (xs.map (fun x => if (edges x).satisfied labeling then 1 else 0)).sum := by |
| induction xs with |
| | nil => rfl |
| | cons x xs ih => simp only [List.map_cons, countSatisfied, List.sum_cons, ih] |
|
|
| theorem countSatisfied_eq_sum (S : Source) (k : Nat) {s d : Nat} |
| (g : SplitGadget s d) (labeling : Fin (vertexCount S k s d) → Fin (2^s)) : |
| countSatisfied labeling (outputInstance S k g).constraints = |
| ∑ ω : Outcome S k g, if (edge S k g ω).satisfied labeling then 1 else 0 := by |
| change countSatisfied labeling ((outcomes S k g).map (edge S k g)) = _ |
| rw [countSatisfied_map] |
| exact Finset.sum_map_toList _ _ |
|
|
| / |
| theorem acceptanceProbability_eq_count (S : Source) (k : Nat) {s d : Nat} |
| (g : SplitGadget s d) (labeling : Fin (vertexCount S k s d) → Fin (2^s)) : |
| acceptanceProbability S k g labeling = |
| (countSatisfied labeling (outputInstance S k g).constraints : ℚ) / |
| (outputInstance S k g).constraints.length := by |
| rw [acceptanceProbability, Fintype.expect_eq_sum_div_card, |
| outputInstance_constraints_length, countSatisfied_eq_sum] |
| congr 1 |
| simp |
|
|
| / |
| def explicitVertexCount (S : Source) (k s d : Nat) : Nat := |
| (ActualGame.explicitBodies S k s d).length |
|
|
| def explicitVertexEncoding (S : Source) (k s d : Nat) : |
| Vertex S k s d ≃ Fin (explicitVertexCount S k s d) := by |
| letI := ActualGame.orbitBodyDecidableEq S k s d |
| exact ActualOrbit.explicitVertexEquiv (canonical S k s d) |
| (ActualEnumeration.queries S.occurrences k s d) ActualEnumeration.mem_queries |
|
|
| noncomputable def semanticToExplicitVertices (S : Source) (k s d : Nat) : |
| Fin (vertexCount S k s d) ≃ Fin (explicitVertexCount S k s d) := |
| (vertexEncoding S k s d).symm.trans (explicitVertexEncoding S k s d) |
|
|
| @[simp] theorem semanticToExplicitVertices_apply (S : Source) (k s d : Nat) |
| (v : Vertex S k s d) : |
| semanticToExplicitVertices S k s d (vertexEncoding S k s d v) = |
| explicitVertexEncoding S k s d v := by |
| simp [semanticToExplicitVertices] |
|
|
| theorem explicitVertexCount_eq (S : Source) (k s d : Nat) : |
| explicitVertexCount S k s d = vertexCount S k s d := by |
| simpa using (Fintype.card_congr (semanticToExplicitVertices S k s d)).symm |
|
|
| def explicitEdge (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) |
| (ω : Outcome S k g) : Constraint (explicitVertexCount S k s d) (2^s) where |
| source := explicitVertexEncoding S k s d (vertex S k s d (leftQuery S k g ω)) |
| target := explicitVertexEncoding S k s d (vertex S k s d (rightQuery S k g ω)) |
| permutation := Encoding.translationTable |
| (offset S k s d (leftQuery S k g ω)) (offset S k s d (rightQuery S k g ω)) |
|
|
| theorem explicitEdge_eq_rename (S : Source) (k : Nat) {s d : Nat} |
| (g : SplitGadget s d) (ω : Outcome S k g) : |
| explicitEdge S k g ω = Integration.InstanceEquivalences.renameConstraint |
| (semanticToExplicitVertices S k s d) (edge S k g ω) := by |
| simp only [explicitEdge, Integration.InstanceEquivalences.renameConstraint, edge, |
| semanticToExplicitVertices_apply] |
|
|
| / |
| def outputInstanceWithEnumeration (S : Source) (k : Nat) {s d : Nat} |
| (g : SplitGadget s d) (en : NoiseEnumeration g) : Instance (2^s) where |
| vertices := explicitVertexCount S k s d |
| constraints := (ActualGame.explicitOutcomes S k g en).map (explicitEdge S k g) |
| nonempty := by |
| intro h |
| let ω : Outcome S k g := Classical.choice inferInstance |
| have hm : explicitEdge S k g ω ∈ |
| (ActualGame.explicitOutcomes S k g en).map (explicitEdge S k g) := |
| List.mem_map.mpr ⟨ω, ActualGame.mem_explicitOutcomes S k g en ω, rfl⟩ |
| rw [h] at hm |
| exact List.not_mem_nil hm |
|
|
| @[simp] theorem outputInstanceWithEnumeration_length (S : Source) (k : Nat) |
| {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) : |
| (outputInstanceWithEnumeration S k g en).constraints.length = |
| Explicit.edgeCount S.occurrences k (s+d) en.indices.length := by |
| simp only [outputInstanceWithEnumeration, List.length_map, |
| ActualGame.explicitOutcomes, ActualEnumeration.length_indexedOutcomes] |
|
|
| theorem explicitConstraints_perm_renamed (S : Source) (k : Nat) {s d : Nat} |
| (g : SplitGadget s d) (en : NoiseEnumeration g) : |
| (outputInstanceWithEnumeration S k g en).constraints.Perm |
| ((outputInstance S k g).constraints.map |
| (Integration.InstanceEquivalences.renameConstraint (semanticToExplicitVertices S k s d))) := by |
| simpa only [outputInstanceWithEnumeration, outputInstance, List.map_map, |
| Function.comp_def, ← explicitEdge_eq_rename, outcomes] using |
| (ActualGame.explicitOutcomes_perm_outcomes S k g en).map (explicitEdge S k g) |
|
|
| theorem completeAt_outputInstanceWithEnumeration_iff (error : RationalError) |
| (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) : |
| CompleteAt error (outputInstanceWithEnumeration S k g en) ↔ |
| CompleteAt error (outputInstance S k g) := |
| Integration.InstanceEquivalences.completeAt_iff_of_rename_perm error |
| (outputInstance S k g) (outputInstanceWithEnumeration S k g en) |
| (semanticToExplicitVertices S k s d) (explicitConstraints_perm_renamed S k g en) |
|
|
| theorem soundAt_outputInstanceWithEnumeration_iff (error : RationalError) |
| (S : Source) (k : Nat) {s d : Nat} (g : SplitGadget s d) (en : NoiseEnumeration g) : |
| SoundAt error (outputInstanceWithEnumeration S k g en) ↔ |
| SoundAt error (outputInstance S k g) := |
| Integration.InstanceEquivalences.soundAt_iff_of_rename_perm error |
| (outputInstance S k g) (outputInstanceWithEnumeration S k g en) |
| (semanticToExplicitVertices S k s d) (explicitConstraints_perm_renamed S k g en) |
|
|
| end UniqueGamesTheorem.Decoder.TableKeysGame |
|
|
| end |
|
|
| end OAI |
|
|
