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D1-3-formal: no-11约束的形式化定义

机器验证元数据​

type: definition
verification: machine_ready
dependencies: ["D1-2-formal.md"]
verification_points:
  - no_consecutive_ones
  - valid_string_set
  - recursive_generation
  - fibonacci_counting

核心定义​

定义 D1-3（no-11约束）​

No11Constraint(Encode : Encoding) : Prop ≡
  ∀s : Element . s ∈ S → 
    ¬Contains11(Encode(s))

辅助定义​

连续11模式检测​

Contains11(str : BinaryString) : Prop ≡
  ∃i : ℕ . i < |str| - 1 ∧ 
    str[i] = 1 ∧ str[i+1] = 1

有效字符串集合​

Valid_no11 : Set[BinaryString] ≡
  {str : BinaryString | ¬Contains11(str)}

模式集合定义​

Pattern_11 : Set[BinaryString] ≡
  {w ∈ {0,1}* | ∃u,v ∈ {0,1}* . w = u ++ "11" ++ v}

等价表述​

表述1：正则表达式​

Valid_no11 = L((0|10)*(1|ε))

表述2：递归生成规则​

ValidString ::= ε
              | 0 · ValidString
              | 10 · ValidString  
              | 1

表述3：有限状态自动机​

FSA_no11 := {
  States: {q₀, q₁, q_reject},
  Start: q₀,
  Accept: {q₀, q₁},
  Transitions: {
    δ(q₀, 0) = q₀,
    δ(q₀, 1) = q₁,
    δ(q₁, 0) = q₀,
    δ(q₁, 1) = q_reject,
    δ(q_reject, _) = q_reject
  }
}

关键性质​

性质1：前缀封闭性​

PrefixClosed(Valid_no11) := 
  ∀s ∈ Valid_no11 . ∀p prefix_of s . p ∈ Valid_no11

性质2：扩展规则​

ExtensionRules(s ∈ Valid_no11) := {
  s ++ "0" ∈ Valid_no11,
  s ++ "1" ∈ Valid_no11 ⟺ ¬EndsWith1(s)
}

性质3：计数函数​

Count_no11(n : ℕ) : ℕ := F_{n+2}
  where F_k = Fibonacci(k)

性质4：信息容量​

Capacity_no11 := lim_{n→∞} (log₂ Count_no11(n))/n = log₂ φ
  where φ = (1 + √5)/2

机器验证检查点​

检查点1：无连续1验证​

def verify_no_consecutive_ones(binary_string):
    return "11" not in binary_string

检查点2：有效字符串集合验证​

def verify_valid_string_set(strings):
    return all(verify_no_consecutive_ones(s) for s in strings)

检查点3：递归生成验证​

def verify_recursive_generation(string):
    return matches_grammar(string, valid_no11_grammar)

检查点4：Fibonacci计数验证​

def verify_fibonacci_counting(n):
    return count_valid_strings(n) == fibonacci(n+2)

形式化验证状态​

  • 定义语法正确
  • 约束条件明确
  • 等价表述完整
  • 性质可验证
  • 最小完备