| import OAI.Computability.DepthThree.HardSliceEventual |
| import OAI.Computability.DepthThree.WideCorrelation |
|
|
| namespace OAI |
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| noncomputable section |
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|
| open scoped BigOperators Classical |
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| namespace DepthThreeLowerBound |
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| theorem exists_language_slice_small_correlations (s : ℝ) (hs : 0 ≤ s) : |
| ∃ N : ℕ, 20480 ≤ N ∧ ∀ n : ℕ, N ≤ n → |
| ∃ α : Fin n → Sum (Fin (dataDimension n)) Bool, |
| ∀ H : CNF (Fin (dataDimension n)), |
| H.WidthAtMost (degreeCutoff s (dataDimension n)) → |
| |finiteAvg (fun x => |
| sign (language (List.ofFn (GateInput.assignment α x))) * indicator (H.eval x))| ≤ |
| 6 * (2 : ℝ) ^ (-(64 * (s + 1) * Real.sqrt (dataDimension n : ℝ)) / 16) := by |
| let B : ℝ := 64 * (s + 1) |
| have hB : 0 < B := by dsimp [B]; linarith |
| obtain ⟨b, hb, D, hD, hparams⟩ := exists_final_parameter_width B s hB hs |
| obtain ⟨Ng, hNg, hgood⟩ := exists_good_language_slice_eventually b hb B hB |
| refine ⟨max Ng (5 * D), hNg.trans (le_max_left _ _), ?_⟩ |
| intro n hn |
| have hng : Ng ≤ n := (le_max_left _ _).trans hn |
| have hnd : 5 * D ≤ n := (le_max_right _ _).trans hn |
| have hDd : D ≤ dataDimension n := by unfold dataDimension; omega |
| obtain ⟨hp0, hphalf, hθ⟩ := hparams (dataDimension n) hDd |
| have hp1 : liveProbability B (dataDimension n) < 1 := by linarith |
| have hθ' : restrictionTheta (degreeCutoff s (dataDimension n)) b |
| (liveProbability B (dataDimension n)) ≤ 1 / 2 := by |
| simpa only [restrictionTheta_eq_reverseTheta] using hθ |
| obtain ⟨α, hα⟩ := hgood n hng |
| refine ⟨α, ?_⟩ |
| exact wide_correlation_of_good_restrictions hp0 hp1 |
| (Real.rpow_nonneg (by norm_num) _) b (degreeCutoff s (dataDimension n)) hb hθ' |
| (fun x => language (List.ofFn (GateInput.assignment α x))) hα |
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| end DepthThreeLowerBound |
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| end |
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| end OAI |
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