D1-5-formal: 观察者的形式化定义

机器验证元数据

type: definition
verification: machine_ready
dependencies: ["D1-1-formal.md", "D1-4-formal.md"]
verification_points:
  - read_function
  - compute_function
  - update_function
  - measurement_effect

核心定义

定义 D1-5（观察者）

Observer(S : SelfReferentialComplete) : Prop ≡
  ∃O = (S_O, A_O, M_O) .
    S_O ⊆ S ∧
    ReadFunction(O) ∧
    ComputeFunction(O) ∧
    UpdateFunction(O) ∧
    InternalObserver(O)

三重功能结构

功能1：读取能力

ReadFunction(O : Observer) : Prop ≡
  ∃read : S → I_O .
    Distinguishing(read) ∧
    Complete(read) ∧
    SelfReferential(read)

where
  Distinguishing(read) := ∀s₁, s₂ ∈ S . s₁ ≠ s₂ → read(s₁) ≠ read(s₂)
  Complete(read) := read(S) = I_O
  SelfReferential(read) := O ∈ S → read(O) ∈ I_O

功能2：计算能力

ComputeFunction(O : Observer) : Prop ≡
  ∃compute : I_O → D_O .
    Deterministic(compute) ∧
    Consistent(compute) ∧
    Recursive(compute)

where
  Deterministic(compute) := ∀i ∈ I_O . ∃!d ∈ D_O . compute(i) = d
  Consistent(compute) := ∀s₁, s₂ . compute(read(s₁)) = compute(read(s₂)) ⟺ s₁ ~ s₂
  Recursive(compute) := ∀i . compute(read(compute(i))) is well-defined

功能3：更新能力

UpdateFunction(O : Observer) : Prop ≡
  ∃update : S × D_O → S .
    StateChange(update) ∧
    DeterministicEvolution(update) ∧
    EntropyIncrease(update)

where
  StateChange(update) := ∀s ∈ S, d ∈ D_O . update(s, d) ≠ s
  DeterministicEvolution(update) := ∀s . update(s, compute(read(s))) is unique
  EntropyIncrease(update) := ∀s, d . H(update(s, d)) > H(s)

观察者内在性

内生条件

InternalObserver(O : Observer) : Prop ≡
  O ⊆ S ∧
  SelfDescribing(O) ∧
  RecursiveClosure(O)

where
  SelfDescribing(O) := ∃self_desc ∈ D_O . self_desc = compute(read(O))
  RecursiveClosure(O) := read ∘ update ∘ compute ∘ read : S → I_O

测量操作

测量映射

Measurement(O : Observer) : Type ≡
  measure : S × O → R × S

satisfying:
  UniqueResult(measure) ∧
  Irreversible(measure) ∧
  BackAction(measure)

where
  UniqueResult(measure) := ∀s, o . ∃!r, s' . measure(s, o) = (r, s')
  Irreversible(measure) := ∀s, o, s' . measure(s, o) = (_, s') → ¬∃f . f(s') = s
  BackAction(measure) := ∀s, o, s' . measure(s, o) = (_, s') → s' = update(s, compute(read(s)))

类型定义

Type ObserverState := Set[Element]
Type ActionSet := Set[Action]
Type MeasurementSet := Set[Measurement]
Type InformationSpace := Set[Information]
Type DecisionSpace := Set[Decision]
Type ResultSpace := Set[Result]

观察者类型分类

ObserverType := Enum {
  Complete,    // read : S → S
  Partial,     // read : S → Projection(S)
  Recursive    // read(read(s)) is well-defined
}

机器验证检查点

检查点1：读取功能验证

def verify_read_function(observer, system):
    return (
        can_distinguish_states(observer.read, system) and
        covers_information_space(observer.read, system) and
        can_read_self(observer.read, observer)
    )

检查点2：计算功能验证

def verify_compute_function(observer):
    return (
        is_deterministic(observer.compute) and
        is_consistent(observer.compute, observer.read) and
        supports_recursion(observer.compute, observer.read)
    )

检查点3：更新功能验证

def verify_update_function(observer, system):
    return (
        always_changes_state(observer.update) and
        evolution_is_deterministic(observer.update) and
        entropy_increases(observer.update)
    )

检查点4：测量效应验证

def verify_measurement_effect(observer, system):
    return (
        measurement_produces_unique_result(observer.measure) and
        measurement_is_irreversible(observer.measure) and
        measurement_has_backaction(observer.measure)
    )

形式化验证状态

• 定义语法正确
• 功能结构完整
• 内在性条件明确
• 测量操作规范
• 最小完备