File size: 3,348 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 74 75 76 77 78 79 80 81 82 83 84 85 | import OAI.NumberTheory.TwoPoint.Circuits.CircuitEncoding
namespace OAI
/-! Relabel, complement and conjoin the residue circuits used in a
closed-word expansion without increasing their internal depth. -/
namespace TwoPointCorrelations.AC0Circuit
open Finset
def negate {n : ℕ} : AC0Circuit n → AC0Circuit n
| .literal i b => .literal i (!b)
| .andGate c => .orGate (fun i => negate (c i))
| .orGate c => .andGate (fun i => negate (c i))
lemma negate_eval {n : ℕ} (c : AC0Circuit n) (x : BooleanCube n) :
c.negate.eval x = true ↔ c.eval x ≠ true := by
induction c with
| literal i b => cases b <;> cases hx : x i <;> simp [negate, eval, hx]
| andGate c ih => simp only [negate, eval, ne_eq, decide_eq_true_eq, ih, not_forall]
| orGate c ih => simp only [negate, eval, ne_eq, decide_eq_true_eq, ih, not_exists]
lemma negate_depth {n : ℕ} (c : AC0Circuit n) : c.negate.depth = c.depth := by
induction c with
| literal i b => rfl
| andGate c ih => simp only [negate, depth, ih]
| orGate c ih => simp only [negate, depth, ih]
lemma negate_size {n : ℕ} (c : AC0Circuit n) : c.negate.size = c.size := by
induction c with
| literal i b => rfl
| andGate c ih => simp only [negate, size, ih]
| orGate c ih => simp only [negate, size, ih]
def relabel {n m : ℕ} (f : Fin n → Fin m) (c : AC0Circuit n) : AC0Circuit m :=
substitute (fun i b => .literal (f i) b) c
lemma relabel_eval {n m : ℕ} (f : Fin n → Fin m) (c : AC0Circuit n) (x : BooleanCube m) :
(relabel f c).eval x = c.eval (fun i => x (f i)) :=
substitute_eval _ (fun x i => x (f i)) (fun _ _ _ => rfl) c x
lemma relabel_depth {n m : ℕ} (f : Fin n → Fin m) (c : AC0Circuit n) :
(relabel f c).depth ≤ c.depth := by
simpa only [relabel, Nat.add_zero] using substitute_depth (fun i b => literal (f i) b)
(d := 0) (fun _ _ => Nat.le_refl 0) c
lemma relabel_size {n m : ℕ} (f : Fin n → Fin m) (c : AC0Circuit n) :
(relabel f c).size ≤ c.size := by
simpa only [relabel, Nat.one_mul] using substitute_size (fun i b => literal (f i) b)
(K := 1) (by omega) (fun _ _ => Nat.le_refl 1) c
noncomputable def conjunction {ι : Type*} [Fintype ι] {n : ℕ}
(c : ι → AC0Circuit n) : AC0Circuit n :=
.andGate (fun i : Fin (Fintype.card ι) => c ((Fintype.equivFin ι).symm i))
lemma conjunction_eval {ι : Type*} [Fintype ι] {n : ℕ}
(c : ι → AC0Circuit n) (x : BooleanCube n) :
(conjunction c).eval x = true ↔ ∀ i, (c i).eval x = true := by
simp only [conjunction, eval, decide_eq_true_eq]
constructor
· intro h i
simpa only [Equiv.symm_apply_apply] using h ((Fintype.equivFin ι) i)
· intro h i
exact h _
lemma conjunction_depth {ι : Type*} [Fintype ι] {n d : ℕ}
(c : ι → AC0Circuit n) (hc : ∀ i, (c i).depth ≤ d) :
(conjunction c).depth ≤ d + 1 := by
simp only [conjunction, depth]
have hh : (univ.sup fun i : Fin (Fintype.card ι) =>
(c ((Fintype.equivFin ι).symm i)).depth) ≤ d :=
Finset.sup_le (fun i _ => hc _)
omega
lemma conjunction_size {ι : Type*} [Fintype ι] {n : ℕ}
(c : ι → AC0Circuit n) : (conjunction c).size = 1 + ∑ i, (c i).size := by
simp only [conjunction, size]
congr 1
exact (Fintype.equivFin ι).symm.sum_comp (fun i => (c i).size)
end TwoPointCorrelations.AC0Circuit
end OAI
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