| import Mathlib |
| import OAI.Computability.VertexCover.PCP.CayleySampling |
| import OAI.Computability.VertexCover.PCP.CayleyTailCount |
| import OAI.Computability.VertexCover.PCP.CayleySpectral |
|
|
| namespace OAI |
|
|
| |
|
|
| noncomputable section |
|
|
| namespace UniqueGames.Foundations.PCP.Expanders |
|
|
| open scoped BigOperators |
| open UniqueGames.Foundations.Hastad |
| open CayleySpectral |
|
|
| abbrev Generators (n m : Nat) := Fin (4 * m) → Cube (Fin n) |
|
|
| def badFrequency {n m : Nat} (g : Generators n m) (s : Cube (Fin n)) : ℝ := |
| if s = zeroFrequency then 0 else |
| if CayleyTailCount.badWord m (fun d => AssignmentTester.parity s (g d)) then 1 else 0 |
|
|
| def badMass {n m : Nat} (g : Generators n m) : ℝ := |
| ∑ s : Cube (Fin n), badFrequency g s |
|
|
| theorem badFrequency_nonnegative {n m : Nat} |
| (g : Generators n m) (s : Cube (Fin n)) : 0 ≤ badFrequency g s := by |
| unfold badFrequency |
| split_ifs <;> norm_num |
|
|
| theorem mean_badFrequency_le {n m : Nat} (s : Cube (Fin n)) : |
| (𝔼 g : Generators n m, badFrequency g s) ≤ 2 * (2 / 3 : ℝ) ^ m := by |
| classical |
| by_cases hs : s = zeroFrequency |
| · simp only [badFrequency, ite_eq_left hs, Finset.expect_const_zero] |
| positivity |
| · have hs' : s ≠ (fun _ => false) := hs |
| simp only [badFrequency, ite_eq_right hs] |
| rw [CayleySampling.expect_parity_samples s hs' |
| (fun w : Fin (4 * m) → Bool => if CayleyTailCount.badWord m w then (1 : ℝ) else 0)] |
| exact CayleyTailCount.twoTail_probability m (by simp) |
|
|
| theorem mean_badMass_le (n m : Nat) (hm : 3 * (n + 2) ≤ m) : |
| (𝔼 g : Generators n m, badMass g) ≤ 1 / 2 := by |
| calc |
| _ = ∑ s : Cube (Fin n), 𝔼 g : Generators n m, badFrequency g s := by |
| unfold badMass |
| rw [Finset.expect_sum_comm] |
| _ ≤ ∑ _s : Cube (Fin n), 2 * (2 / 3 : ℝ) ^ m := |
| Finset.sum_le_sum (fun s _ => mean_badFrequency_le s) |
| _ = (2 : ℝ) ^ n * (2 * (2 / 3 : ℝ) ^ m) := by |
| simp [Cube] |
| _ ≤ 1 / 2 := CayleyTailCount.union_bound_le_half n m hm |
|
|
| theorem exists_generators (n m : Nat) (hm : 3 * (n + 2) ≤ m) : |
| ∃ g : Generators n m, ∀ s : Cube (Fin n), |
| s ≠ zeroFrequency → |eigenvalue g s| ≤ (1 / 2 : ℝ) := by |
| classical |
| have hmpos : 0 < m := by omega |
| let : Nonempty (Fin (4 * m)) := ⟨⟨0, by omega⟩⟩ |
| have havg : (𝔼 g : Generators n m, badMass g) < (1 : ℝ) := |
| lt_of_le_of_lt (mean_badMass_le n m hm) (by norm_num) |
| obtain ⟨g, _, hg⟩ := Finset.exists_lt_of_expect_lt Finset.univ_nonempty havg |
| refine ⟨g, fun s hs => ?_⟩ |
| have hgood : ¬ CayleyTailCount.badWord m |
| (fun d => AssignmentTester.parity s (g d)) := by |
| intro hbad |
| have hsingle : badFrequency g s ≤ badMass g := |
| Finset.single_le_sum (fun t _ => badFrequency_nonnegative g t) (Finset.mem_univ s) |
| simp only [badFrequency, ite_eq_right hs, ite_eq_left hbad] at hsingle |
| exact (not_le_of_gt hg) hsingle |
| unfold eigenvalue |
| simp_rw [← AssignmentTester.bitSign_parity] |
| exact CayleyTailCount.not_badWord_bias m (by simp) hmpos _ hgood |
|
|
| def baseDimension : Nat := 448 |
| def initialDegree : Nat := 2 ^ 16 |
| def initialQuarterDegree : Nat := 2 ^ 14 |
| def basePower : Nat := 7 |
|
|
| theorem base_arithmetic : |
| 4 * initialQuarterDegree = initialDegree ∧ |
| 3 * (baseDimension + 2) ≤ initialQuarterDegree ∧ |
| (2 : Nat) ^ baseDimension = (initialDegree ^ basePower) ^ 4 ∧ |
| (1 / 2 : ℝ) ^ basePower ≤ 1 / 100 := by |
| refine ⟨?_, ?_, ?_, ?_⟩ |
| · norm_num [initialDegree, initialQuarterDegree] |
| · norm_num [baseDimension, initialQuarterDegree] |
| · unfold baseDimension initialDegree basePower |
| exact CayleyTailCount.seventh_power_size |
| · norm_num [basePower] |
|
|
| theorem exists_initial_generators : |
| ∃ g : Generators baseDimension initialQuarterDegree, |
| ∀ s : Cube (Fin baseDimension), s ≠ zeroFrequency → |
| |eigenvalue g s| ≤ (1 / 2 : ℝ) := |
| exists_generators _ _ base_arithmetic.2.1 |
|
|
| abbrev BaseVertex := Cube (Fin baseDimension) |
| abbrev BasePort := Fin basePower → Fin (4 * initialQuarterDegree) |
| def baseDegree : Nat := initialDegree ^ basePower |
|
|
| theorem card_basePort : Fintype.card BasePort = baseDegree := by |
| simp only [BasePort, Fintype.card_fun, Fintype.card_fin] |
| rw [base_arithmetic.1] |
| rfl |
|
|
| theorem card_baseVertex : Fintype.card BaseVertex = baseDegree ^ 4 := by |
| simpa only [BaseVertex, Cube, Fintype.card_fun, Fintype.card_bool, |
| Fintype.card_fin, baseDegree] using base_arithmetic.2.2.1 |
|
|
| theorem exists_baseGraph : |
| ∃ H : PoweringWalks.PortGraph BaseVertex BasePort, |
| SpectralReturn.SpectralCertificate H (1 / 100 : ℝ) := by |
| obtain ⟨g, hg⟩ := exists_initial_generators |
| refine ⟨cayleyGraph (powerGenerators g basePower), ?_⟩ |
| apply cayley_spectralCertificate |
| · norm_num |
| · norm_num |
| · intro s hs |
| exact (power_eigenvalue_bound g (1 / 2) (by norm_num) hg basePower s hs).trans |
| base_arithmetic.2.2.2 |
|
|
| end UniqueGames.Foundations.PCP.Expanders |
| end |
|
|
| end OAI |
|
|
