Skip to main content

D1-8-formal: φ-表示系统的形式化定义

机器验证元数据​

type: definition
verification: machine_ready
dependencies: ["D1-2-formal.md", "D1-3-formal.md"]
verification_points:
  - fibonacci_generation
  - bijection_property
  - no11_constraint_preservation
  - zeckendorf_uniqueness

核心定义​

定义 D1-8（φ-表示系统）​

PhiRepresentationSystem : Type ≡
  (F, B, encode_φ, decode_φ)

where
  F : Sequence[ℕ]           // Fibonacci sequence
  B : Set[BinaryString]     // Binary strings satisfying no-11 constraint
  encode_φ : B → ℕ⁺         // Encoding function
  decode_φ : ℕ⁺ → B         // Decoding function

Fibonacci数列定义​

修改的Fibonacci序列​

FibonacciSequence : Sequence[ℕ] ≡
  F₁ = 1
  F₂ = 2
  Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 3

// Sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Fibonacci性质​

FibonacciProperties : Prop ≡
  RecursiveRelation(F) ∧
  BinetFormula(F) ∧
  AsymptoticBehavior(F)

where
  RecursiveRelation(F) := ∀n≥3 . Fₙ = Fₙ₋₁ + Fₙ₋₂
  BinetFormula(F) := ∀n . Fₙ = (φⁿ - ψⁿ)/√5
  AsymptoticBehavior(F) := limₙ→∞ Fₙ/φⁿ = 1/√5

编码与解码函数​

编码函数​

encode_φ : B → ℕ⁺ ≡
  encode_φ(b₁b₂...bₙ) = Σᵢ₌₁ⁿ bᵢ · Fᵢ

where
  bᵢ ∈ {0,1}
  ∀i . bᵢ = 1 → bᵢ₊₁ ≠ 1  // no-11 constraint

解码函数（贪心算法）​

decode_φ : ℕ⁺ → B ≡
  GreedyDecode(n) where
    GreedyDecode(n):
      1. Find max k such that Fₖ ≤ n
      2. Initialize result[1..k] = 0
      3. remaining := n
      4. For i from k down to 1:
         If Fᵢ ≤ remaining:
           result[i] := 1
           remaining := remaining - Fᵢ
      5. Return result

系统性质​

性质1：双射性​

Bijection(encode_φ, decode_φ) : Prop ≡
  ∀b ∈ B . decode_φ(encode_φ(b)) = b ∧
  ∀n ∈ ℕ⁺ . encode_φ(decode_φ(n)) = n

性质2：Zeckendorf唯一性​

ZeckendorfUniqueness : Prop ≡
  ∀n ∈ ℕ⁺ . ∃!b ∈ B . encode_φ(b) = n

性质3：保序性​

OrderPreserving : Prop ≡
  ∀n₁, n₂ ∈ ℕ⁺ . n₁ < n₂ ⟺ 
    decode_φ(n₁) <_lex decode_φ(n₂)

性质4：紧致性​

Compactness : Prop ≡
  ∀n ∈ ℕ⁺ . |decode_φ(n)| = ⌊log_φ n⌋ + 1

信息容量​

渐近容量​

AsymptoticCapacity : Real ≡
  C_φ = limₙ→∞ log₂(Fₙ₊₂)/n = log₂(φ)
  
where
  φ = (1 + √5)/2  // Golden ratio

效率度量​

Efficiency : Real ≡
  η_φ = C_φ/1 = log₂(φ) ≈ 0.694

Zeckendorf表示​

Zeckendorf定理​

ZeckendorfTheorem : Prop ≡
  ∀n ∈ ℕ⁺ . ∃!I ⊂ ℕ . 
    n = Σᵢ∈I Fᵢ ∧
    ∀i,j ∈ I . |i-j| ≥ 2

等价关系​

PhiZeckendorfEquivalence : Prop ≡
  B ↔ ZeckendorfRepresentations ↔ ℕ⁺

类型定义​

Type BinaryString := List[Bit]
Type Bit := {0, 1}
Type ℕ⁺ := {n ∈ ℕ | n > 0}
Type Real := ℝ

机器验证检查点​

检查点1：Fibonacci生成验证​

def verify_fibonacci_generation(n):
    F = [1, 2]  # F[0] = F₁, F[1] = F₂
    for i in range(2, n):
        F.append(F[i-1] + F[i-2])
    return all(F[i] == F[i-1] + F[i-2] for i in range(2, n))

检查点2：双射性验证​

def verify_bijection_property(test_range):
    for n in test_range:
        b = decode_phi(n)
        if encode_phi(b) != n:
            return False
    return True

检查点3：no-11约束保持验证​

def verify_no11_constraint_preservation(n):
    binary_string = decode_phi(n)
    return "11" not in binary_string

检查点4：Zeckendorf唯一性验证​

def verify_zeckendorf_uniqueness(n):
    representation = decode_phi(n)
    # 验证表示中没有连续的1（满足Zeckendorf条件）
    return is_valid_zeckendorf(representation)

形式化验证状态​

  • 定义语法正确
  • 算法描述完整
  • 性质相互独立
  • 类型系统清晰
  • 最小完备