| import OAI.NumberTheory.TwoPoint.Circuits.CircuitDyadicBudgets |
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| namespace OAI |
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| /-! Specialize the proved approximation construction to dyadic size |
| budgets and circuit depth twenty-two. -/ |
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| namespace TwoPointCorrelations |
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| open scoped Classical |
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| lemma braverman_approx_degree_bound {m j d : ℕ} (hm : m ≤ 2 ^ j) (hd : d ≤ 22) : |
| (bravermanSamples j * (Nat.log 2 m + 3)) ^ d ≤ bravermanBase j ^ 22 := by |
| have hbase : bravermanSamples j * (Nat.log 2 m + 3) ≤ bravermanBase j := |
| Nat.mul_le_mul_left _ (Nat.add_le_add_right (nat_log_le_of_dyadic hm) 3) |
| exact (Nat.pow_le_pow_left hbase _).trans |
| (Nat.pow_le_pow_right (bravermanBase_pos j) hd) |
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| lemma braverman_norm_power_bound {m j d : ℕ} (hm : m ≤ 2 ^ j) (hd : d ≤ 22) : |
| (2 * (bravermanSamples j * (Nat.log 2 m + 3)) + 2) ^ d ≤ |
| (2 * bravermanBase j + 2) ^ 22 := by |
| have hbase : bravermanSamples j * (Nat.log 2 m + 3) ≤ bravermanBase j := |
| Nat.mul_le_mul_left _ (Nat.add_le_add_right (nat_log_le_of_dyadic hm) 3) |
| exact (Nat.pow_le_pow_left (Nat.add_le_add_right (Nat.mul_le_mul_left 2 hbase) 2) _).trans |
| (Nat.pow_le_pow_right (by omega) hd) |
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| lemma braverman_norm_base_bound {m j : ℕ} (hm : m ≤ 2 ^ j) : |
| 2 * ((m : ℝ) + 1) ^ 2 ≤ (2 : ℝ) ^ (2 * j + 3) := by |
| have hmreal : (m : ℝ) ≤ (2 : ℝ) ^ j := by exact_mod_cast hm |
| have htwo : (1 : ℝ) ≤ 2 ^ j := one_le_pow₀ (by norm_num) |
| have hsum : (m : ℝ) + 1 ≤ 2 * (2 : ℝ) ^ j := by linarith only [hmreal, htwo] |
| calc |
| _ ≤ 2 * (2 * (2 : ℝ) ^ j) ^ 2 := |
| mul_le_mul_of_nonneg_left |
| (pow_le_pow_left₀ (add_nonneg (Nat.cast_nonneg m) zero_le_one) hsum 2) (by norm_num) |
| _ = _ := by |
| rw [pow_add, show 2 * j = j * 2 by omega, pow_mul] |
| ring |
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|
| theorem AC0Circuit.dyadic_approximation {n j : ℕ} |
| (ν : FiniteLaw (BooleanCube n)) (c : AC0Circuit n) |
| (hc : c.depth ≤ 22) (hm : c.size ≤ 2 ^ j) : |
| ∃ (P : BooleanCube n → ℝ) (E : AC0Circuit n), |
| WalshDegreeLE P (bravermanBase j ^ 22) ∧ |
| E.depth ≤ 89 ∧ E.size ≤ 2 ^ bravermanErrorExponent j ∧ |
| ν.probability (fun x => E.eval x = true) ≤ |
| (2 : ℝ) ^ j * (7 / 8 : ℝ) ^ bravermanSamples j ∧ |
| (∀ x, E.eval x ≠ true → P x = c.indicator x) ∧ |
| ∀ x, |P x| ≤ (2 : ℝ) ^ bravermanNormExponent j := by |
| obtain ⟨P, E, hP, hd, hsize, hprob, he, hnorm⟩ := |
| c.polynomial_error_approximation ν (bravermanSamples j) (bravermanSamples_pos j) |
| have hmreal : (c.size : ℝ) ≤ (2 : ℝ) ^ j := by exact_mod_cast hm |
| have h89 : E.depth ≤ 89 := hd.trans (Nat.add_le_add_right (Nat.mul_le_mul_left 4 hc) 1) |
| refine ⟨P, E, hP.mono (braverman_approx_degree_bound hm hc), h89, |
| hsize.trans (braverman_error_size_budget hm), ?_, he, ?_⟩ |
| · exact hprob.trans (mul_le_mul_of_nonneg_right hmreal (by positivity)) |
| · intro x |
| calc |
| |P x| ≤ _ := hnorm x |
| _ ≤ ((2 : ℝ) ^ (2 * j + 3)) ^ |
| ((2 * (bravermanSamples j * (Nat.log 2 c.size + 3)) + 2) ^ c.depth) := |
| pow_le_pow_left₀ (by positivity) (braverman_norm_base_bound hm) _ |
| _ ≤ ((2 : ℝ) ^ (2 * j + 3)) ^ ((2 * bravermanBase j + 2) ^ 22) := |
| pow_le_pow_right₀ (one_le_pow₀ (by norm_num)) (braverman_norm_power_bound hm hc) |
| _ = _ := by rw [← pow_mul]; rfl |
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| end TwoPointCorrelations |
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| end OAI |
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