File size: 2,496 Bytes
8769036 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | 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
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