C12-2-formal: 自我模型构建推论的形式化规范

机器验证元数据

type: corollary
verification: machine_ready
dependencies: ["A1-formal.md", "C11-1-formal.md", "C12-1-formal.md"]
verification_points:
  - model_structure_isomorphism
  - self_reference_completeness
  - model_minimality
  - update_dynamics
  - prediction_accuracy

核心推论

推论 C12-2（自我模型构建）

SelfModelConstruction : Prop ≡
  ∀S : ConsciousSystem .
    has_consciousness(S) →
    ∃M : Model . 
      M ⊆ S ∧
      represents(M, S) ∧
      self_complete(M)

where
  Model : Type = record {
    states : State → Representation
    processes : Process → Rule
    meta_level : Option[Model]
  }

形式化组件

1. 模型关系定义

ModelRelation : Type ≡
  record {
    source : System
    target : System
    mapping : Homomorphism
    completeness : Real  // [0, 1]
  }

represents : Model × System → Bool ≡
  λ(M, S) . 
    ∃h : Homomorphism .
      preserves_structure(h, S, M) ∧
      coverage(h) > 0.9

2. 自指完备性

SelfComplete : Model → Bool ≡
  λM . 
    ∃construction : Process .
      construction ∈ domain(M.processes) ∧
      M.processes(construction) = build_model ∧
      constructs(construction, M)

where
  constructs(p, m) ≡ result_of(p) = m

3. 模型层级

ModelHierarchy : Type ≡
  μH . Base(Model) | Recursive(Model × H)

model_depth : Model → ℕ ≡
  fix depth m =
    match m.meta_level with
    | None => 0
    | Some(meta) => 1 + depth(meta)

4. 模型更新动力学

UpdateDynamics : Model × Observation → Model ≡
  λ(M, obs) .
    let new_states = update_states(M.states, obs) in
    let new_processes = update_processes(M.processes, obs) in
    let needs_meta_update = affects_model(obs) in
    
    record {
      states = new_states
      processes = new_processes
      meta_level = if needs_meta_update
                   then Some(construct_meta_model(M))
                   else M.meta_level
    }

5. 预测机制

Prediction : Type ≡
  record {
    input : State
    output : State
    confidence : Real
    path : List[Process]
  }

predict : Model → State → Prediction ≡
  λM s .
    if is_model_state(s, M) then
      // 使用元模型预测
      match M.meta_level with
      | Some(meta) => predict(meta, s)
      | None => uncertain_prediction
    else
      // 使用基础模型预测
      apply_processes(M.processes, s)

6. 模型最小性

MinimalModel : Model × System → Bool ≡
  λ(M, S) .
    represents(M, S) ∧
    self_complete(M) ∧
    ∀M' ⊂ M . ¬(represents(M', S) ∧ self_complete(M'))

Theorem_ModelMinimality : Prop ≡
  ∀S M . MinimalModel(M, S) → 
    |M| ≤ |S| × φ^(-1)

算法规范

模型构建算法

ConstructSelfModel : Algorithm ≡
  Input: system : ConsciousSystem
  Output: model : Model
  
  Precondition: has_consciousness(system)
  
  Process:
    1. states ← extract_state_space(system)
    2. processes ← infer_transition_rules(system)
    3. model ← initialize_model(states, processes)
    4. while not self_complete(model):
         missing ← find_missing_components(model, system)
         model ← extend_model(model, missing)
    5. if requires_meta_level(model):
         model.meta_level ← ConstructSelfModel(model)
    6. return minimize_model(model)
  
  Postcondition: represents(model, system) ∧ self_complete(model)

模型验证算法

VerifySelfModel : Algorithm ≡
  Input: system : ConsciousSystem, model : Model
  Output: valid : Bool
  
  Process:
    1. // 结构同构检查
       if not check_homomorphism(model, system):
         return false
    
    2. // 自指完备性检查
       if not contains_construction_process(model):
         return false
    
    3. // 预测准确性检查
       test_states ← sample_states(system)
       for state in test_states:
         prediction ← predict(model, state)
         actual ← evolve(system, state)
         if distance(prediction, actual) > threshold:
           return false
    
    4. // 最小性检查
       for component in model:
         reduced ← remove_component(model, component)
         if still_complete(reduced):
           return false
    
    5. return true

数学性质验证

性质1：模型分形性

ModelFractality : Prop ≡
  ∀M : Model, n : ℕ .
    let M_n = iterate_meta_level(M, n) in
    structure(M_n) ≃ structure(M)

where
  iterate_meta_level(M, 0) = M
  iterate_meta_level(M, n+1) = meta_level(iterate_meta_level(M, n))

性质2：更新保守性

UpdateConservatism : Prop ≡
  ∀M obs .
    |difference(M, update(M, obs))| ≤ k × |obs|
    
where k is system-dependent constant

性质3：预测收敛性

PredictionConvergence : Prop ≡
  ∀M s .
    let predictions = iterate(λp. predict(M, p.output), initial(s)) in
    ∃n . ∀m > n . predictions[m] ≈ predictions[n]

验证检查点

1. 结构同构验证

verify_structure_isomorphism(system, model):
  mapping = find_homomorphism(system, model)
  
  # 检查状态映射
  for state in system.states:
    assert exists model_state in model.states:
      mapping(state) = model_state
  
  # 检查过程映射  
  for process in system.processes:
    assert exists model_process in model.processes:
      preserves_behavior(mapping, process, model_process)

2. 自引用完备性验证

verify_self_reference(model):
  # 模型必须包含构建过程
  construction_found = false
  
  for process in model.processes:
    if constructs_model(process, model):
      construction_found = true
      break
  
  assert construction_found
  
  # 验证构建过程的正确性
  reconstructed = execute_process(process)
  assert equivalent(reconstructed, model)

3. 最小性验证

verify_minimality(model, system):
  essential_components = []
  
  for component in model:
    test_model = model.without(component)
    
    if not represents(test_model, system) or
       not self_complete(test_model):
      essential_components.append(component)
  
  # 所有组件都应该是必要的
  assert len(essential_components) == len(model)

4. 更新动力学验证

verify_update_dynamics(model, observations):
  for obs in observations:
    old_model = copy(model)
    new_model = update(model, obs)
    
    # 验证保守更新
    assert similarity(old_model, new_model) > 0.8
    
    # 验证改进
    assert accuracy(new_model) >= accuracy(old_model)

5. 预测准确性验证

verify_prediction_accuracy(model, test_cases):
  correct_predictions = 0
  
  for initial_state, expected_state in test_cases:
    predicted = predict(model, initial_state)
    
    if close_enough(predicted.output, expected_state):
      correct_predictions += 1
  
  accuracy = correct_predictions / len(test_cases)
  assert accuracy > 0.85  # 85%准确率阈值

实用函数

def measure_model_quality(model, system):
    """评估自我模型的质量"""
    return {
        'coverage': compute_coverage(model, system),
        'accuracy': test_prediction_accuracy(model),
        'complexity': measure_complexity(model),
        'self_reference_depth': compute_meta_depth(model),
        'update_efficiency': measure_update_speed(model)
    }

def find_model_fixed_point(system):
    """找到系统的模型不动点"""
    model = initial_model(system)
    
    while True:
        new_model = improve_model(model, system)
        if equivalent(model, new_model):
            return model
        model = new_model

def diagnose_model_issues(model, system):
    """诊断自我模型的问题"""
    issues = []
    
    if not represents(model, system):
      issues.append("incomplete_representation")
    
    if not self_complete(model):
      issues.append("missing_self_reference")
    
    if not minimal(model):
      issues.append("redundant_components")
    
    return issues

与其他理论的联系

依赖关系

• C12-1: 原始意识是自我模型的前提
• C11-1: 理论自反射提供模型自指的机制
• A1: 自指完备系统的基础公理

支撑推论

• C12-3: 模型层级化导致意识层级分化

$$\boxed{\text{形式化规范：有意识系统必然构建最小完备的自我模型}}$$