| import OAI.NumberTheory.TwoPoint.Circuits.CircuitApproximationDegree |
|
|
| namespace OAI |
|
|
| /-! A decision tree making at most `d` queries on any branch has Walsh |
| degree at most `d`. This is the spectral consequence of circuit switching. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open scoped Classical |
|
|
| lemma WalshDegreeLE.add {n d : ℕ} {F G : BooleanCube n → ℝ} |
| (hF : WalshDegreeLE F d) (hG : WalshDegreeLE G d) : |
| WalshDegreeLE (fun x => F x + G x) d := by |
| have h := hF.sub (hG.smul (-1)) |
| convert h using 1 |
| funext x |
| ring |
|
|
| inductive BooleanDecisionTree (n : ℕ) where |
| | leaf (value : Bool) |
| | query (i : Fin n) (low high : BooleanDecisionTree n) |
|
|
| namespace BooleanDecisionTree |
|
|
| def eval {n : ℕ} : BooleanDecisionTree n → BooleanCube n → Bool |
| | .leaf b, _ => b |
| | .query i low high, x => if x i then high.eval x else low.eval x |
|
|
| def depth {n : ℕ} : BooleanDecisionTree n → ℕ |
| | .leaf _ => 0 |
| | .query _ low high => max low.depth high.depth + 1 |
|
|
| noncomputable def indicator {n : ℕ} (c : BooleanDecisionTree n) (x : BooleanCube n) : ℝ := |
| if c.eval x then 1 else 0 |
|
|
| theorem indicator_degree {n : ℕ} (c : BooleanDecisionTree n) : |
| WalshDegreeLE c.indicator c.depth := by |
| induction c with |
| | leaf b => |
| cases b <;> exact WalshDegreeLE.const _ _ |
| | query i low high hl hh => |
| let d := max low.depth high.depth |
| have hlow : WalshDegreeLE low.indicator d := hl.mono (Nat.le_max_left _ _) |
| have hhigh : WalshDegreeLE high.indicator d := hh.mono (Nat.le_max_right _ _) |
| have hbit := literal_indicator_degree i true |
| have hnot := literal_indicator_degree i false |
| have h := (hnot.mul hlow).add (hbit.mul hhigh) |
| have heq : (fun x => (AC0Circuit.literal i false).indicator x * low.indicator x + |
| (AC0Circuit.literal i true).indicator x * high.indicator x) = |
| (query i low high).indicator := by |
| funext x |
| cases hx : x i <;> |
| simp [indicator, eval, AC0Circuit.indicator, AC0Circuit.eval, hx] |
| rw [heq] at h |
| simpa only [depth, d, Nat.add_comm 1] using h |
| |
| end BooleanDecisionTree |
| |
| lemma WalshDegreeLE.high_coefficient_zero {n d : ℕ} {F : BooleanCube n → ℝ} |
| (hF : WalshDegreeLE F d) (S : Finset (Fin n)) (hS : d < S.card) : |
| walshCoefficient F S = 0 := by |
| obtain ⟨a, ha⟩ := hF |
| unfold walshCoefficient |
| simp_rw [ha, Finset.sum_mul] |
| rw [cubeAverage_sum] |
| simp_rw [mul_assoc, cubeAverage_mul_const, cubeAverage_walsh_mul] |
| simp [Nat.not_le.mpr hS] |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
