| import OAI.Computability.DepthThree.CircuitLowerBound |
| import OAI.Computability.DepthThree.CircuitDensity |
| import OAI.Computability.DepthThree.AsymptoticBounds |
| import Mathlib.Tactic.Convert |
|
|
| namespace OAI |
|
|
| namespace DepthThreeLowerBound |
|
|
| theorem circuit_lower_bound_of_small_cnf_correlations |
| (d k : ℕ) (s : ℝ) (hs : 0 ≤ s) (hd : 4 ≤ d) |
| (hk : 3 * (s + 1) * Real.sqrt (d : ℝ) ≤ (k : ℝ)) |
| (f : Cube (Fin d) → Bool) |
| (hcorr : ∀ H : CNF (Fin d), H.WidthAtMost k → |
| |finiteAvg (fun x : Cube (Fin d) => sign (f x) * indicator (H.eval x))| ≤ |
| 6 * (2 : ℝ) ^ (-(64 * (s + 1) * Real.sqrt (d : ℝ)) / 16)) |
| (C : Circuit3 (Fin d)) (hC : C.Computes f) : |
| (2 : ℝ) ^ (s * Real.sqrt (d : ℝ)) < (C.gateCount : ℝ) := by |
| by_contra h |
| have hsize : (C.gateCount : ℝ) ≤ (2 : ℝ) ^ (s * Real.sqrt (d : ℝ)) := |
| le_of_not_gt h |
| let ε : ℝ := 6 * (2 : ℝ) ^ (-(64 * (s + 1) * Real.sqrt (d : ℝ)) / 16) |
| have hε : 0 ≤ ε := mul_nonneg (by norm_num) (Real.rpow_nonneg (by norm_num) _) |
| have hdReal : (4 : ℝ) ≤ (d : ℝ) := by exact_mod_cast hd |
| have hexponent : -(64 * (s + 1) * Real.sqrt (d : ℝ)) / 16 = |
| -(4 * (s + 1) * Real.sqrt (d : ℝ)) := by ring |
| have hsmall : ε ≤ 3 / 8 := by |
| dsimp only [ε] |
| rw [hexponent] |
| exact correlation_error_le s (d : ℝ) hs (by linarith) |
| have hlower : (1 - ε) / 2 ≤ |
| finiteAvg (fun x : Cube (Fin d) => indicator (f x)) := by |
| convert acceptance_density_lower_of_width_correlation f k ε |
| (fun H hH => by convert hcorr H hH) |
| have hupper : finiteAvg (fun x : Cube (Fin d) => indicator (f x)) ≤ |
| (C.gateCount : ℝ) * ε + (C.gateCount : ℝ) ^ 2 * ((2 : ℝ) ^ k)⁻¹ := by |
| convert C.acceptance_density_le f hC k ε hε |
| (fun H hH => by convert hcorr H hH) |
| have hinverse : (2 : ℝ) ^ (-(k : ℝ)) = ((2 : ℝ) ^ k)⁻¹ := by |
| rw [Real.rpow_neg (by norm_num : (0 : ℝ) ≤ 2), Real.rpow_natCast] |
| have hfinal := final_error_le s (d : ℝ) (C.gateCount : ℝ) hs hdReal |
| (Nat.cast_nonneg _) hsize k hk |
| have hupper' : finiteAvg (fun x : Cube (Fin d) => indicator (f x)) ≤ 5 / 128 := by |
| calc |
| _ ≤ (C.gateCount : ℝ) * ε + (C.gateCount : ℝ) ^ 2 * ((2 : ℝ) ^ k)⁻¹ := hupper |
| _ = 6 * (C.gateCount : ℝ) * |
| (2 : ℝ) ^ (-(64 * (s + 1) * Real.sqrt (d : ℝ)) / 16) + |
| (C.gateCount : ℝ) ^ 2 * (2 : ℝ) ^ (-(k : ℝ)) := by |
| rw [hinverse] |
| dsimp only [ε] |
| ring |
| _ ≤ 5 / 128 := hfinal |
| linarith |
|
|
| end DepthThreeLowerBound |
|
|
| end OAI |
|
|
