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D1-6-formal: 系统熵的形式化定义

机器验证元数据​

type: definition
verification: machine_ready
dependencies: ["D1-1-formal.md", "D1-2-formal.md"]
verification_points:
  - entropy_non_negativity
  - entropy_monotonicity
  - entropy_additivity
  - information_equivalence

核心定义​

定义 D1-6（系统熵）​

SystemEntropy(S : SelfReferentialComplete) : Prop ≡
  ∃H : Function[SystemState → Real⁺] .
    NonNegative(H) ∧
    StrictlyMonotonic(H) ∧
    Additive(H) ∧
    InformationMeasure(H)

熵的形式化表述​

基本定义​

H(S_t) : Real⁺ ≡
  log |{d ∈ L | ∃s ∈ S_t . d = Desc_t(s)}|

where
  S_t : SystemState at time t
  L : FormalLanguage (set of finite symbol strings)
  Desc_t : S_t → L (description function at time t)
  log : natural logarithm

等价表述​

表述1：描述多样性​

H_desc(S_t) : Real⁺ ≡
  log |D_t|

where
  D_t := {Desc_t(s) | s ∈ S_t}

表述2：信息容量​

H_info(S_t) : Real⁺ ≡
  max_P Σ(s ∈ S_t) P(s) · log(1/P(s))

where
  P : Probability distribution over S_t

表述3：编码长度界​

H_encode(S_t) : Real⁺ ≡
  lower_bound {avg_length(Encode) | Encode is valid encoding}

熵的基本性质​

性质1：非负性​

NonNegative(H) : Prop ≡
  ∀S_t . H(S_t) ≥ 0 ∧
  (H(S_t) = 0 ⟺ |S_t| = 1)

性质2：严格单调性​

StrictlyMonotonic(H) : Prop ≡
  ∀t ∈ ℕ . H(S_{t+1}) > H(S_t)

性质3：可加性​

Additive(H) : Prop ≡
  ∀S₁, S₂ . S₁ ∩ S₂ = ∅ →
    H(S₁ ∪ S₂) = H(S₁) + H(S₂)

性质4：信息度量​

InformationMeasure(H) : Prop ≡
  ∀S . H(S) = log |EquivalenceClasses(S, InfoEquiv)|

信息等价关系​

信息等价定义​

InfoEquiv(s₁, s₂ : Element) : Prop ≡
  Desc(s₁) = Desc(s₂)

等价类划分​

EquivalenceClasses(S, ~) : Set[Set[Element]] ≡
  {[s] | s ∈ S}

where
  [s] := {s' ∈ S | s ~ s'}

熵增机制​

描述展开熵增​

ΔH_desc(S_t) : Real⁺ ≡
  H(S_t ∪ {Desc^(t+1)(S_t)}) - H(S_t)

递归深化熵增​

ΔH_recursive(S, k) : Real⁺ ≡
  log(|Desc^(k+1)(S)| / |Desc^(k)(S)|)

观察反作用熵增​

ΔH_measurement(S, result) : Real⁺ ≡
  H(S ∪ {result}) - H(S)

熵的边界​

下界​

LowerBound(H, S_t) : Prop ≡
  H(S_t) ≥ log(t)

增长率界​

GrowthRateBound(H) : Prop ≡
  ∀t . dH/dt ≤ log(φ)
  where φ = (1 + √5)/2

类型定义​

Type Real⁺ := {r ∈ ℝ | r ≥ 0}
Type FormalLanguage := Set[String]
Type Description := Element → String
Type ProbabilityDist := Element → [0,1]

机器验证检查点​

检查点1：熵非负性验证​

def verify_entropy_non_negativity(system):
    return entropy(system) >= 0 and (entropy(system) == 0 iff len(system) == 1)

检查点2：熵单调性验证​

def verify_entropy_monotonicity(system_sequence):
    return all(entropy(s[i+1]) > entropy(s[i]) for i in range(len(s)-1))

检查点3：熵可加性验证​

def verify_entropy_additivity(subsystem1, subsystem2):
    if disjoint(subsystem1, subsystem2):
        return entropy(union(subsystem1, subsystem2)) == entropy(subsystem1) + entropy(subsystem2)

检查点4：信息等价验证​

def verify_information_equivalence(system):
    equiv_classes = compute_equivalence_classes(system)
    return entropy(system) == log(len(equiv_classes))

形式化验证状态​

  • 定义语法正确
  • 性质相互独立
  • 熵增机制明确
  • 边界条件完整
  • 最小完备