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namespace OAI
noncomputable section
open scoped BigOperators
universe u
namespace Problem335
inductive D5Leaf (K : Type u) (n : ℕ) where
| scalar : K → D5Leaf K n
| variable : (Fin n × Fin n × Fin n) → D5Leaf K n
def d5LeafDegree {K : Type u} {n : ℕ} : D5Leaf K n → ℕ
| D5Leaf.scalar _ => 0
| D5Leaf.variable _ => 1
structure Depth5Circuit (K : Type u) [CommSemiring K] (n : ℕ) where
leafCount : ℕ
leaves : Fin leafCount → D5Leaf K n
bottomCount : ℕ
bottomInputs : Fin bottomCount → List (K × Fin leafCount)
bottomDegree : Fin bottomCount → ℕ
bottomHomogeneous :
∀ i entry, entry ∈ bottomInputs i → d5LeafDegree (leaves entry.2) = bottomDegree i
bottomEmpty :
∀ i, bottomInputs i = [] → bottomDegree i = 0
lowerCount : ℕ
lowerInputs : Fin lowerCount → List (Fin bottomCount)
middleCount : ℕ
middleInputs : Fin middleCount → List (K × Fin lowerCount)
middleDegree : Fin middleCount → ℕ
middleHomogeneous :
∀ i entry, entry ∈ middleInputs i →
((lowerInputs entry.2).map bottomDegree).sum = middleDegree i
middleEmpty :
∀ i, middleInputs i = [] → middleDegree i = 0
upperCount : ℕ
upperInputs : Fin upperCount → List (Fin middleCount)
outputInputs : List (K × Fin upperCount)
outputDegree : ℕ
outputHomogeneous :
∀ entry, entry ∈ outputInputs →
((upperInputs entry.2).map middleDegree).sum = outputDegree
outputEmpty :
outputInputs = [] → outputDegree = 0
def d5LeafValue {K : Type u} [CommSemiring K] {n : ℕ}
(x : D5Leaf K n) : MvPolynomial (Fin n × Fin n × Fin n) K :=
match x with
| D5Leaf.scalar a => MvPolynomial.C a
| D5Leaf.variable i => MvPolynomial.X i
def bottomValue {K : Type u} [CommSemiring K] {n : ℕ}
(c : Depth5Circuit K n) (i : Fin c.bottomCount) :
MvPolynomial (Fin n × Fin n × Fin n) K := by
classical
exact ((c.bottomInputs i).map (fun entry =>
MvPolynomial.C entry.1 * d5LeafValue (c.leaves entry.2))).sum
def lowerValue {K : Type u} [CommSemiring K] {n : ℕ}
(c : Depth5Circuit K n) (i : Fin c.lowerCount) :
MvPolynomial (Fin n × Fin n × Fin n) K := by
classical
exact ((c.lowerInputs i).map (fun j => bottomValue c j)).prod
def middleValue {K : Type u} [CommSemiring K] {n : ℕ}
(c : Depth5Circuit K n) (i : Fin c.middleCount) :
MvPolynomial (Fin n × Fin n × Fin n) K := by
classical
exact ((c.middleInputs i).map (fun entry =>
MvPolynomial.C entry.1 * lowerValue c entry.2)).sum
def upperValue {K : Type u} [CommSemiring K] {n : ℕ}
(c : Depth5Circuit K n) (i : Fin c.upperCount) :
MvPolynomial (Fin n × Fin n × Fin n) K := by
classical
exact ((c.upperInputs i).map (fun j => middleValue c j)).prod
def circuitValue {K : Type u} [CommSemiring K] {n : ℕ}
(c : Depth5Circuit K n) : MvPolynomial (Fin n × Fin n × Fin n) K := by
classical
exact (c.outputInputs.map (fun entry =>
MvPolynomial.C entry.1 * upperValue c entry.2)).sum
def circuitSize {K : Type u} [CommSemiring K] {n : ℕ}
(c : Depth5Circuit K n) : ℕ :=
c.leafCount + c.bottomCount + c.lowerCount + c.middleCount + c.upperCount + 1
def immLayer (K : Type u) [CommSemiring K] (n : ℕ) (t : Fin n) :
Matrix (Fin n) (Fin n) (MvPolynomial (Fin n × Fin n × Fin n) K) :=
fun i j => MvPolynomial.X (t, i, j)
def imm (K : Type u) [CommSemiring K] (n : ℕ) :
MvPolynomial (Fin n × Fin n × Fin n) K := by
classical
by_cases h : 0 < n
· let matrices :
List (Matrix (Fin n) (Fin n)
(MvPolynomial (Fin n × Fin n × Fin n) K)) :=
List.ofFn (fun t : Fin n => immLayer K n t)
exact matrices.prod ⟨0, h⟩ ⟨0, h⟩
· exact 0
def ceilSqrt (n : ℕ) : ℕ :=
Nat.sqrt n + if (Nat.sqrt n) ^ 2 = n then 0 else 1
def ceilDiv (n t : ℕ) : ℕ :=
(n + t - 1) / t
def upperGateBound (n : ℕ) : ℕ :=
let t := ceilSqrt n
let r := ceilDiv n t
2 * n ^ 3 + r * n ^ 2 + r * n ^ (t + 1) + n ^ (r - 1) + 1
end Problem335
end
end OAI
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