| import OAI.NumberTheory.TwoPoint.Circuits.CircuitGateComposition |
|
|
| namespace OAI |
|
|
| /-! Degree bookkeeping for the random polynomial construction. The recursive |
| degree uses the largest child degree, rather than the number of leaves. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| lemma WalshDegreeLE.mono {n p q : ℕ} {F : BooleanCube n → ℝ} |
| (hF : WalshDegreeLE F p) (hpq : p ≤ q) : WalshDegreeLE F q := by |
| obtain ⟨a, ha⟩ := hF |
| refine ⟨fun S => if S.card ≤ p then a S else 0, ?_⟩ |
| intro x |
| rw [ha] |
| rw [sum_filter, sum_filter] |
| apply sum_congr rfl |
| intro S _ |
| by_cases hS : S.card ≤ p |
| · simp [hS, hS.trans hpq] |
| · simp [hS] |
|
|
| lemma literal_indicator_degree {n : ℕ} (i : Fin n) (b : Bool) : |
| WalshDegreeLE (AC0Circuit.literal i b).indicator 1 := by |
| have hw : WalshDegreeLE (walsh {i}) 1 := WalshDegreeLE.of_walsh {i} (by simp) |
| have hp := (WalshDegreeLE.const (n := n) (1 / 2) 1).sub (hw.smul (1 / 2)) |
| have hn := (WalshDegreeLE.const (n := n) 1 1).sub hp |
| cases b |
| · convert hn using 1 |
| funext x |
| simp only [AC0Circuit.indicator, AC0Circuit.eval, Bool.false_eq_true, ite_false, |
| walsh, prod_singleton] |
| cases x i <;> norm_num [booleanSign] |
| · convert hp using 1 |
| funext x |
| simp only [AC0Circuit.indicator, AC0Circuit.eval, ite_true, walsh, prod_singleton] |
| cases x i <;> norm_num [booleanSign] |
|
|
| namespace AC0Circuit |
|
|
| def approximationDegree {n : ℕ} (s : ℕ) : AC0Circuit n → ℕ |
| | .literal _ _ => 1 |
| | .andGate (k := k) children => |
| (univ.sup fun i => approximationDegree s (children i)) * s * (Nat.log 2 k + 3) |
| | .orGate (k := k) children => |
| (univ.sup fun i => approximationDegree s (children i)) * s * (Nat.log 2 k + 3) |
|
|
| end AC0Circuit |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
