L1-3-formal: 约束必然性的形式化证明

机器验证元数据

type: lemma
verification: machine_ready
dependencies: ["L1-2-formal.md", "D1-2-formal.md", "A1-formal.md"]
verification_points:
  - unique_decodability
  - prefix_ambiguity
  - constraint_classification
  - minimal_constraint_length

核心引理

引理 L1-3（约束的必然性）

ConstraintNecessity : Prop ≡
  ∀E : BinaryEncodingSystem .
    UniqueDecodable(E) →
    ∃F ⊂ {0,1}* . Forbidden(E, F)

where
  BinaryEncodingSystem : Type
  UniqueDecodable(E) : Prop
  Forbidden(E, F) ≡ ∀f ∈ F . f ∉ Codewords(E)

辅助定义

定义：唯一可解码性

UniqueDecodability : Prop ≡
  ∀w ∈ L . ∃! decomposition : List[Codeword] .
    w = concatenate(decomposition) ∧
    ∀c ∈ decomposition . c ∈ Codewords(E)

where
  L : Set[BinaryString]      // Language of all concatenations
  concatenate : List[String] → String

定义：前缀自由性

PrefixFree : Prop ≡
  ∀c₁, c₂ ∈ Codewords(E) .
    c₁ ≠ c₂ → ¬IsPrefix(c₁, c₂) ∧ ¬IsPrefix(c₂, c₁)

where
  IsPrefix(a, b) ≡ ∃s . b = a ++ s

辅助引理

引理 L1-3.1（无约束导致前缀歧义）

UnconstrainedPrefixAmbiguity : Prop ≡
  Let E_unconstrained = AllBinaryStrings in
    ¬UniqueDecodable(E_unconstrained)

Proof:
  Consider codewords {1, 11, 0}
  String "11" has two decompositions:
    - [1, 1]
    - [11]
  Therefore not uniquely decodable ∎

引理 L1-3.2（约束长度分类）

ConstraintLengthClassification : Prop ≡
  ∀F : Set[BinaryString] . Let ℓ_min = min{|f| : f ∈ F} in
    Case ℓ_min of
      | 1 → SystemDegenerates(E)
      | 2 → PossibleNonDegenerate(E)
      | ≥3 → InsufficientForPrefixFree(E)

where
  SystemDegenerates(E) ≡ |Alphabet(E)| = 1
  InsufficientForPrefixFree(E) ≡ ∃ prefix conflicts

引理 L1-3.3（长度2是最小有效约束）

MinimalEffectiveConstraint : Prop ≡
  ∀E : BinaryEncodingSystem .
    UniqueDecodable(E) ∧ NonDegenerate(E) →
    ∃f ∈ ForbiddenPatterns(E) . |f| = 2

Proof by cases:
  Case |f| = 1:
    Forbid "0" → only "1" → H = 0
    Forbid "1" → only "0" → H = 0
    Contradicts entropy increase
    
  Case |f| ≥ 3:
    All strings of length < 3 are valid
    {1, 11, 111, ...} all valid
    But 1 is prefix of 11
    Not prefix-free → not uniquely decodable
    
  Therefore |f| = 2 is necessary ∎

引理 L1-3.4（约束与容量关系）

ConstraintCapacityTradeoff : Prop ≡
  ∀F : ConstraintSet .
    Let C(F) = lim_{n→∞} log(N_F(n))/n in
      C(∅) = 1 ∧ ¬UniqueDecodable(∅) ∧
      C({0}) = 0 ∧ Degenerate({0}) ∧
      0 < C({pattern_2}) < 1 ∧ UniqueDecodable({pattern_2})

where
  N_F(n) = |{s ∈ {0,1}ⁿ : s satisfies F}|
  pattern_2 : BinaryString with |pattern_2| = 2

证明结构

步骤1：前缀问题的必然性

Proof of PrefixProblem:
  For unconstrained binary strings:
  1. ∀n ∃s₁, s₂ . |s₁| = n ∧ |s₂| = 2n ∧ s₁ is prefix of s₂
  2. Probability of prefix relation = 2^{-n}
  3. For large codeword sets, prefix conflicts inevitable
  4. Prefix conflicts → decoding ambiguity

步骤2：约束分类证明

Proof of ConstraintClassification:
  Length 1 constraints:
    Forbid "0" → Alphabet = {1}
    Forbid "1" → Alphabet = {0}
    Both cases: entropy = 0
    
  Length ≥3 constraints:
    {1, 11} both valid (no length-2 forbidden)
    Prefix conflict exists
    Cannot achieve prefix-free property
    
  Length 2 constraints:
    Can break prefix chains
    Maintain binary nature
    Enable unique decodability

步骤3：最小性证明

Proof of Minimality:
  Assume only constraints of length k > 2
  
  Then ∀s . |s| < k → s is valid codeword
  In particular: 1, 11, ..., 1^{k-1} all valid
  
  But 1 is prefix of 11
  String "11" can be decoded as:
    - [1, 1]
    - [11]
    
  Contradiction with unique decodability
  Therefore need constraint with length ≤ 2
  Combined with length-1 analysis: must be length 2 ∎

自指系统要求

引理 L1-3.5（自指编码的约束要求）

SelfReferentialConstraints : Prop ≡
  ∀E : SelfReferentialEncodingSystem .
    Let F = Constraints(E) in
      Describable(F) ∧ Simple(F) ∧ Recursive(F)

where
  Describable(F) ≡ Desc(F) ∈ L
  Simple(F) ≡ |Desc(F)| = O(1)
  Recursive(F) ≡ F preserves recursive structure

信息论性质

Kraft-McMillan推广

GeneralizedKraftInequality : Prop ≡
  ∀E : ConstrainedPrefixFreeCode .
    ∑_{c ∈ Codewords(E)} λ^{-|c|} ≤ 1

where
  λ = largest eigenvalue of constraint transfer matrix

容量定理

CapacityTheorem : Prop ≡
  ∀F : ConstraintSet .
    C(F) = log(λ_F)

where
  λ_F = growth rate of valid strings under F

机器验证检查点

检查点1：唯一可解码性验证

def verify_unique_decodability(codewords):
    # 检查是否存在解码歧义
    for length in range(2, max_test_length):
        strings = generate_strings(length)
        for s in strings:
            decompositions = find_all_decompositions(s, codewords)
            if len(decompositions) > 1:
                return False, s, decompositions
    return True, None, None

检查点2：前缀歧义验证

def verify_prefix_ambiguity(codewords):
    # 检查前缀冲突
    for c1 in codewords:
        for c2 in codewords:
            if c1 != c2:
                if c1.startswith(c2) or c2.startswith(c1):
                    return True, (c1, c2)
    return False, None

检查点3：约束分类验证

def verify_constraint_classification(forbidden_patterns):
    min_length = min(len(p) for p in forbidden_patterns)
    
    if min_length == 1:
        # 检查是否退化
        return "degenerate", calculate_entropy()
    elif min_length == 2:
        return "possibly_valid", check_unique_decodability()
    else:
        return "insufficient", find_prefix_conflicts()

检查点4：最小约束长度验证

def verify_minimal_constraint_length(encoding_system):
    # 验证长度2是最小有效约束
    constraints = encoding_system.get_constraints()
    
    if all(len(c) > 2 for c in constraints):
        # 应该找到前缀冲突
        return find_prefix_conflicts(encoding_system)
    
    if any(len(c) == 1 for c in constraints):
        # 应该是退化系统
        return check_degeneration(encoding_system)
        
    # 长度2约束应该有效
    return check_effectiveness(encoding_system)

形式化验证状态

• 引理语法正确
• 证明步骤完整
• 包含必要性证明
• 最小性证明完整
• 最小完备