File size: 3,803 Bytes
2c7d975 | 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 | import OAI.Algebra.DepthFive.ComplexEmbedding
import OAI.Algebra.DepthFive.CoefficientMap
import OAI.Algebra.DepthFive.ImmMap
import OAI.Algebra.DepthFive.CoefficientDescent
namespace OAI
noncomputable section
universe u
namespace Problem335
/-- A circuit over a countable characteristic-zero field can be realized over
`ℂ`, with exactly the same number of gates and the same IMM identity. -/
theorem exists_complex_circuit_of_countable {K : Type u} [Field K] [CharZero K]
[Countable K] {n : ℕ} (c : Depth5Circuit K n)
(hc : circuitValue c = imm K n) :
∃ d : Depth5Circuit ℂ n, circuitValue d = imm ℂ n ∧ circuitSize d = circuitSize c := by
obtain ⟨f⟩ := countable_field_embeds_complex K
refine ⟨c.mapCoefficients f, ?_, circuitSize_mapCoefficients f c⟩
rw [circuitValue_mapCoefficients, hc, map_imm]
/-- Any pointwise real gate-count lower bound over `ℂ` transfers to every
countable field of characteristic zero. -/
theorem countable_circuit_bound_of_complex {K : Type u} [Field K] [CharZero K]
[Countable K] {n : ℕ} {b : ℝ}
(h : ∀ d : Depth5Circuit ℂ n,
circuitValue d = imm ℂ n → b ≤ (circuitSize d : ℝ))
(c : Depth5Circuit K n) (hc : circuitValue c = imm K n) :
b ≤ (circuitSize c : ℝ) := by
obtain ⟨d, hd, hs⟩ := exists_complex_circuit_of_countable c hc
simpa only [hs] using h d hd
/-- Restriction to the coefficient subfield preserves and reflects the IMM identity. -/
theorem restrictCircuit_computes_imm {K : Type u} [Field K] {n : ℕ}
(c : Depth5Circuit K n) :
circuitValue (restrictCircuit c) = imm (circuitCoefficientField c) n ↔
circuitValue c = imm K n := by
have hi := MvPolynomial.map_injective (σ := Fin n × Fin n × Fin n)
(circuitCoefficientField c).subtype
(circuitCoefficientField c).subtype.injective
rw [← hi.eq_iff, map_circuitValue_restrictCircuit, map_imm]
/-- Every characteristic-zero IMM circuit has a same-size complex realization.
Only the finitely many occurring coefficients are embedded, not the entire field. -/
theorem exists_complex_circuit {K : Type u} [Field K] [CharZero K]
{n : ℕ} (c : Depth5Circuit K n) (hc : circuitValue c = imm K n) :
∃ d : Depth5Circuit ℂ n, circuitValue d = imm ℂ n ∧ circuitSize d = circuitSize c := by
obtain ⟨d, hd, hs⟩ := exists_complex_circuit_of_countable (restrictCircuit c)
((restrictCircuit_computes_imm c).mpr hc)
exact ⟨d, hd, hs.trans (circuitSize_restrictCircuit c)⟩
/-- Pointwise complex gate lower bounds hold over any characteristic-zero field. -/
theorem circuit_bound_of_complex {K : Type u} [Field K] [CharZero K]
{n : ℕ} {b : ℝ}
(h : ∀ d : Depth5Circuit ℂ n,
circuitValue d = imm ℂ n → b ≤ (circuitSize d : ℝ))
(c : Depth5Circuit K n) (hc : circuitValue c = imm K n) :
b ≤ (circuitSize c : ℝ) := by
obtain ⟨d, hd, hs⟩ := exists_complex_circuit c hc
simpa only [hs] using h d hd
/-- The extension to characteristic zero uses the same absolute threshold as
its complex counterpart. This implication has no lower-bound assumptions other
than the precise complex conclusion. -/
theorem characteristic_zero_depth_five_lower_bound_of_complex
(h : ∃ n0 : ℕ, ∀ n : ℕ, n0 ≤ n → ∀ c : Depth5Circuit ℂ n,
circuitValue c = imm ℂ n →
(n : ℝ) ^ (Real.sqrt (n : ℝ) / 400) ≤ (circuitSize c : ℝ)) :
∃ n0 : ℕ, ∀ (K : Type u) [Field K] [CharZero K], ∀ n : ℕ, n0 ≤ n →
∀ c : Depth5Circuit K n, circuitValue c = imm K n →
(n : ℝ) ^ (Real.sqrt (n : ℝ) / 400) ≤ (circuitSize c : ℝ) := by
obtain ⟨n0, h0⟩ := h
refine ⟨n0, ?_⟩
intro K _ _ n hn c hc
exact circuit_bound_of_complex (h0 n hn) c hc
end Problem335
end
end OAI
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