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C10-4 元数学结构可判定性形式化规范

1. 判定性层次结构​

1.1 直接可判定类​

class DirectlyDecidable:
    """直接可判定性质（多项式时间）"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def verify_no_11_constraint(self, state: str) -> bool:
        """O(n)时间验证no-11约束"""
        return '11' not in state
        
    def compute_entropy(self, state: 'State') -> float:
        """O(n)时间计算熵"""
        if not state.binary:
            return 0.0
            
        ones = state.binary.count('1')
        zeros = state.binary.count('0')
        total = ones + zeros
        
        if ones == 0 or zeros == 0:
            return 0.0
            
        p1 = ones / total
        p0 = zeros / total
        return -(p1 * np.log2(p1) + p0 * np.log2(p0)) * total
        
    def compute_recursive_depth(self, state: 'State') -> int:
        """O(n)时间计算递归深度"""
        h = self.compute_entropy(state)
        return int(np.log(h + 1) / np.log(self.phi))
        
    def phi_distance(self, s1: 'State', s2: 'State') -> float:
        """O(n)时间计算φ-距离"""
        distance = 0.0
        max_len = max(len(s1.binary), len(s2.binary))
        
        for i in range(max_len):
            bit1 = int(s1.binary[i]) if i < len(s1.binary) else 0
            bit2 = int(s2.binary[i]) if i < len(s2.binary) else 0
            distance += abs(bit1 - bit2) / (self.phi ** (i + 1))
            
        return distance

1.2 轨道可判定类​

class OrbitDecidable:
    """轨道性质可判定（指数时间）"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.visited_states = {}
        
    def detect_period(self, initial: 'State') -> Tuple[int, int]:
        """检测轨道周期 O(F_{n+2})"""
        self.visited_states.clear()
        current = initial
        step = 0
        
        while current not in self.visited_states:
            self.visited_states[current] = step
            current = current.collapse()
            step += 1
            
            # 防止无限循环的保护
            if step > self.max_steps(initial):
                raise RuntimeError("Exceeded maximum steps")
                
        pre_period = self.visited_states[current]
        period = step - pre_period
        
        return pre_period, period
        
    def max_steps(self, state: 'State') -> int:
        """最大步数界限"""
        n = len(state.binary)
        # Fibonacci界限
        return self.fibonacci(n + 2)
        
    def is_same_orbit(self, s1: 'State', s2: 'State') -> bool:
        """判断两个状态是否在同一轨道"""
        # 获取两个状态的轨道
        orbit1 = self.compute_orbit(s1)
        orbit2 = self.compute_orbit(s2)
        
        # 检查是否有交集
        return bool(orbit1.intersection(orbit2))
        
    def compute_orbit(self, initial: 'State') -> Set['State']:
        """计算完整轨道"""
        orbit = set()
        current = initial
        
        while current not in orbit:
            orbit.add(current)
            current = current.collapse()
            
            if len(orbit) > self.max_steps(initial):
                break
                
        return orbit
        
    def find_attractor(self, initial: 'State') -> List['State']:
        """找到吸引子（周期轨道）"""
        pre_period, period = self.detect_period(initial)
        
        # 前进到周期部分
        current = initial
        for _ in range(pre_period):
            current = current.collapse()
            
        # 收集周期轨道
        attractor = []
        for _ in range(period):
            attractor.append(current)
            current = current.collapse()
            
        return attractor

1.3 临界可判定类​

class CriticalDecidable:
    """临界深度内可判定"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def critical_depth(self, n: int) -> int:
        """计算临界深度"""
        return int(np.log(n) / np.log(self.phi)) + 1
        
    def is_decidable_at_depth(self, property: 'Property', depth: int, n: int) -> bool:
        """判断性质在给定深度是否可判定"""
        return depth < self.critical_depth(n)
        
    def bounded_search(self, source: 'State', target: 'State', 
                      max_depth: int) -> Optional[List['State']]:
        """有界深度搜索"""
        if max_depth == 0:
            return [source] if source == target else None
            
        # BFS搜索
        queue = [(source, [source])]
        visited = {source}
        
        while queue:
            current, path = queue.pop(0)
            
            if len(path) > max_depth:
                continue
                
            if current == target:
                return path
                
            # 扩展后继
            successors = self.get_successors(current)
            for next_state in successors:
                if next_state not in visited:
                    visited.add(next_state)
                    queue.append((next_state, path + [next_state]))
                    
        return None
        
    def decide_within_critical(self, property: Callable, 
                             state_space: 'StateSpace') -> Optional[bool]:
        """在临界深度内判定性质"""
        n = state_space.max_length
        critical = self.critical_depth(n)
        
        # 只检查深度小于临界值的状态
        for state in state_space.iterate_by_depth(max_depth=critical):
            if not property(state):
                return False
                
        return True

2. 判定算法实现​

2.1 核心判定器​

class DecidabilityChecker:
    """可判定性检查器"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.direct = DirectlyDecidable()
        self.orbit = OrbitDecidable()
        self.critical = CriticalDecidable()
        
    def classify_property(self, property: 'Property') -> str:
        """分类性质的可判定性"""
        # 分析性质的计算需求
        if property.is_local():
            return "DIRECTLY_DECIDABLE"
        elif property.involves_orbit():
            return "ORBIT_DECIDABLE"
        elif property.depth_bounded():
            return "CRITICALLY_DECIDABLE"
        else:
            return "UNDECIDABLE"
            
    def decide(self, property: 'Property', input_size: int) -> Optional[bool]:
        """主判定函数"""
        classification = self.classify_property(property)
        
        if classification == "DIRECTLY_DECIDABLE":
            return self.decide_directly(property, input_size)
        elif classification == "ORBIT_DECIDABLE":
            return self.decide_orbit(property, input_size)
        elif classification == "CRITICALLY_DECIDABLE":
            return self.decide_critical(property, input_size)
        else:
            return None  # 不可判定
            
    def decide_directly(self, property: 'Property', input_size: int) -> bool:
        """直接判定（多项式时间）"""
        # 生成测试用例
        test_cases = self.generate_test_cases(input_size, sample_size=100)
        
        for test in test_cases:
            result = property.evaluate(test)
            if result is False:
                return False
                
        # 如果所有测试通过，返回真
        return True
        
    def decide_orbit(self, property: 'Property', input_size: int) -> bool:
        """轨道判定（指数时间）"""
        # 枚举所有可能的初始状态
        max_states = self.fibonacci(input_size + 2)
        checked = 0
        
        for state in self.enumerate_states(input_size):
            if checked >= max_states:
                break
                
            orbit = self.orbit.compute_orbit(state)
            if not property.holds_on_orbit(orbit):
                return False
                
            checked += 1
            
        return True
        
    def decide_critical(self, property: 'Property', input_size: int) -> Optional[bool]:
        """临界深度判定"""
        critical_depth = self.critical.critical_depth(input_size)
        
        # 使用深度限制搜索
        for depth in range(critical_depth + 1):
            partial_result = property.check_at_depth(depth)
            
            if partial_result is False:
                return False
            elif partial_result is None:
                # 达到不可判定边界
                return None
                
        return True

2.2 复杂度分析器​

class ComplexityAnalyzer:
    """判定复杂度分析"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def time_complexity(self, property: 'Property', n: int) -> str:
        """分析时间复杂度"""
        if property.is_syntactic():
            return f"O({n})"
        elif property.is_semantic():
            if property.depth_bound:
                d = property.depth_bound
                if d < np.log(n) / np.log(self.phi):
                    return f"O({n}^{d})"
                else:
                    return f"O(φ^{n})"
            else:
                return "UNDECIDABLE"
                
    def space_complexity(self, property: 'Property', n: int) -> str:
        """分析空间复杂度"""
        if property.requires_full_orbit():
            return f"O(F_{n+2})"  # Fibonacci界限
        elif property.is_local():
            return f"O({n})"
        else:
            return f"O({n}^2)"
            
    def is_tractable(self, property: 'Property', n: int) -> bool:
        """判断是否易处理"""
        time = self.time_complexity(property, n)
        
        # 检查是否多项式时间
        if "^" not in time or ("^" in time and self.extract_exponent(time) <= 3):
            return True
        return False
        
    def suggest_algorithm(self, property: 'Property', n: int) -> str:
        """建议使用的算法"""
        if self.is_tractable(property, n):
            return "EXACT_ALGORITHM"
        elif property.allows_approximation():
            return "APPROXIMATION_ALGORITHM"
        elif property.allows_randomization():
            return "RANDOMIZED_ALGORITHM"
        else:
            return "HEURISTIC_ALGORITHM"

2.3 性质分类器​

class PropertyClassifier:
    """性质分类器"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def analyze_property(self, property_expr: str) -> 'PropertyClass':
        """分析性质表达式"""
        # 解析逻辑结构
        parsed = self.parse_expression(property_expr)
        
        # 检查量词
        quantifiers = self.extract_quantifiers(parsed)
        
        # 检查递归深度
        max_depth = self.extract_max_depth(parsed)
        
        # 分类
        if not quantifiers:
            return PropertyClass.QUANTIFIER_FREE
        elif all(q == 'exists' for q in quantifiers):
            return PropertyClass.EXISTENTIAL
        elif all(q == 'forall' for q in quantifiers):
            return PropertyClass.UNIVERSAL
        elif max_depth is not None:
            return PropertyClass.BOUNDED_QUANTIFIER
        else:
            return PropertyClass.ALTERNATING_QUANTIFIER
            
    def decidability_class(self, prop_class: 'PropertyClass', n: int) -> str:
        """根据性质类别判断可判定性"""
        critical_depth = int(np.log(n) / np.log(self.phi))
        
        if prop_class == PropertyClass.QUANTIFIER_FREE:
            return "P"  # 多项式时间
        elif prop_class == PropertyClass.EXISTENTIAL:
            return "NP"  # 非确定多项式时间
        elif prop_class == PropertyClass.UNIVERSAL:
            return "coNP"  # 补NP
        elif prop_class == PropertyClass.BOUNDED_QUANTIFIER:
            return "PSPACE"  # 多项式空间
        else:
            return "UNDECIDABLE"  # 不可判定

3. 判定界限定理​

3.1 临界深度定理​

class CriticalDepthTheorem:
    """临界深度定理"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def critical_depth(self, n: int, c: float = 1.0) -> int:
        """计算临界深度 d_critical = c * log_φ(n)"""
        return int(c * np.log(n) / np.log(self.phi))
        
    def is_below_critical(self, depth: int, n: int) -> bool:
        """判断是否在临界深度以下"""
        return depth < self.critical_depth(n)
        
    def complexity_at_depth(self, depth: int) -> str:
        """给定深度的复杂度类"""
        if depth <= 1:
            return "P"
        elif depth <= 2:
            return "NP"
        elif depth <= 3:
            return "PSPACE"
        else:
            return "EXPTIME"
            
    def prove_decidability_boundary(self, n: int) -> Dict[str, Any]:
        """证明可判定性边界"""
        critical = self.critical_depth(n)
        
        return {
            'critical_depth': critical,
            'decidable_range': (0, critical),
            'undecidable_above': critical,
            'state_space_size': self.fibonacci(n + 2),
            'search_space_at_critical': self.phi ** critical,
            'polynomial_bound': n ** 3,
            'exponential_threshold': critical
        }

3.2 不可判定性证明​

class UndecidabilityProof:
    """不可判定性证明"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def encode_turing_machine(self, tm: 'TuringMachine') -> 'State':
        """将图灵机编码为高深度状态"""
        # 编码转移函数
        transitions = self.encode_transitions(tm.delta)
        
        # 编码为满足no-11约束的二进制串
        encoding = self.to_no_11_encoding(transitions)
        
        # 确保深度足够高
        depth = self.compute_depth(encoding)
        if depth < self.critical_depth(len(encoding)):
            encoding = self.increase_depth(encoding)
            
        return State(encoding)
        
    def reduce_halting_problem(self, property: 'Property') -> bool:
        """将停机问题归约到性质判定"""
        def halting_property(state: 'State') -> bool:
            # 解码为图灵机
            tm = self.decode_turing_machine(state)
            
            # 模拟有限步
            steps = self.phi ** state.recursive_depth()
            
            # 检查是否停机
            return tm.halts_within(steps)
            
        # 如果可以判定此性质，则可以解决停机问题
        # 但停机问题不可判定，因此性质也不可判定
        return property.equivalent_to(halting_property)
        
    def construct_undecidable_property(self) -> 'Property':
        """构造不可判定性质"""
        def undecidable_prop(state: 'State') -> Optional[bool]:
            # 对于深度超过临界值的状态
            if state.recursive_depth() >= self.critical_depth(len(state.binary)):
                # 等价于停机问题
                return None  # 不可判定
            else:
                # 在临界深度内可判定
                return self.bounded_decision(state)
                
        return Property(undecidable_prop)

4. 实用判定策略​

4.1 分层判定器​

class LayeredDecider:
    """分层判定策略"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.cache = {}
        
    def decide_by_layer(self, property: 'Property', input_size: int) -> Optional[bool]:
        """按层次判定"""
        # 第1层：语法检查
        if not self.syntax_check(property, input_size):
            return False
            
        # 第2层：局部性质
        if property.is_local():
            return self.local_decision(property, input_size)
            
        # 第3层：有界深度
        depth = property.max_depth()
        if depth and depth < self.critical_depth(input_size):
            return self.bounded_decision(property, input_size, depth)
            
        # 第4层：启发式
        return self.heuristic_decision(property, input_size)
        
    def incremental_decision(self, property: 'Property', 
                           initial_size: int, target_size: int) -> Optional[bool]:
        """增量判定"""
        # 从小规模开始
        for size in range(initial_size, target_size + 1):
            result = self.decide_by_layer(property, size)
            
            if result is False:
                return False  # 找到反例
            elif result is None:
                # 达到不可判定界限
                return self.extrapolate(size, target_size)
                
        return True
        
    def cached_decision(self, property: 'Property', state: 'State') -> Optional[bool]:
        """带缓存的判定"""
        key = (property.id, state.hash())
        
        if key in self.cache:
            return self.cache[key]
            
        result = self.compute_decision(property, state)
        self.cache[key] = result
        
        return result

4.2 概率判定器​

class ProbabilisticDecider:
    """概率判定策略"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def monte_carlo_decision(self, property: 'Property', 
                           input_size: int, samples: int = 1000) -> Tuple[float, float]:
        """蒙特卡洛判定"""
        positive = 0
        
        for _ in range(samples):
            # 随机采样
            state = self.random_state(input_size)
            
            if property.evaluate(state):
                positive += 1
                
        # 计算概率和置信区间
        p = positive / samples
        confidence = 1.96 * np.sqrt(p * (1 - p) / samples)
        
        return p, confidence
        
    def las_vegas_search(self, target_property: 'Property', 
                        input_size: int, timeout: int) -> Optional['State']:
        """Las Vegas搜索"""
        start_time = time.time()
        
        while time.time() - start_time < timeout:
            # 随机生成候选
            candidate = self.random_state(input_size)
            
            # 验证性质
            if target_property.evaluate(candidate):
                return candidate
                
        return None  # 超时未找到

5. 判定复杂度界限​

5.1 复杂度类层次​

class ComplexityHierarchy:
    """复杂度类层次"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def define_classes(self) -> Dict[str, 'ComplexityClass']:
        """定义φ-复杂度类"""
        return {
            'P_φ': ComplexityClass(
                name='P_φ',
                time_bound=lambda n: n ** 3,
                decidable=True
            ),
            'NP_φ': ComplexityClass(
                name='NP_φ',
                time_bound=lambda n: self.phi ** (np.log(n)),
                verifier_time=lambda n: n ** 2,
                decidable=True
            ),
            'PSPACE_φ': ComplexityClass(
                name='PSPACE_φ',
                space_bound=lambda n: n ** 2,
                decidable=True
            ),
            'EXP_φ': ComplexityClass(
                name='EXP_φ',
                time_bound=lambda n: self.phi ** n,
                decidable=True
            ),
            'UNDEC_φ': ComplexityClass(
                name='UNDEC_φ',
                decidable=False
            )
        }
        
    def classify_problem(self, problem: 'Problem') -> str:
        """分类问题的复杂度"""
        if problem.has_polynomial_algorithm():
            return 'P_φ'
        elif problem.has_polynomial_verifier():
            return 'NP_φ'
        elif problem.has_polynomial_space_algorithm():
            return 'PSPACE_φ'
        elif problem.decidable:
            return 'EXP_φ'
        else:
            return 'UNDEC_φ'

6. 验证函数​

6.1 可判定性测试​

def test_decidability_hierarchy() -> Dict[str, bool]:
    """测试可判定性层次"""
    results = {}
    
    # 测试直接可判定性质
    direct = DirectlyDecidable()
    results['no_11_constraint'] = direct.verify_no_11_constraint("1010")
    results['entropy_computation'] = direct.compute_entropy(State("1010")) > 0
    
    # 测试轨道可判定性质
    orbit = OrbitDecidable()
    pre, period = orbit.detect_period(State("1"))
    results['period_detection'] = period > 0
    
    # 测试临界可判定性质
    critical = CriticalDecidable()
    results['critical_depth'] = critical.critical_depth(100) == 3
    
    # 测试不可判定边界
    undec = UndecidabilityProof()
    prop = undec.construct_undecidable_property()
    results['undecidability'] = prop.decidable is False
    
    return results

6.2 算法正确性验证​

def verify_decision_algorithms() -> bool:
    """验证判定算法的正确性"""
    checker = DecidabilityChecker()
    
    # 测试用例
    test_cases = [
        # (property, input_size, expected_decidable)
        (LocalProperty(), 10, True),
        (OrbitProperty(), 20, True),
        (DeepProperty(depth=10), 5, False),
    ]
    
    for prop, size, expected in test_cases:
        result = checker.decide(prop, size)
        if (result is not None) != expected:
            return False
            
    return True

7. 关键常数​

# 基础常数
PHI = (1 + np.sqrt(5)) / 2  # 黄金分割率

# 判定性参数
CRITICAL_DEPTH_FACTOR = 1.0  # 临界深度系数
POLYNOMIAL_DEGREE_BOUND = 3  # 多项式度数界限
EXPONENTIAL_BASE = PHI  # 指数基数

# 算法参数
MAX_CACHE_SIZE = 10000  # 缓存大小限制
MONTE_CARLO_SAMPLES = 1000  # 蒙特卡洛采样数
TIMEOUT_SECONDS = 60  # 超时限制

# 复杂度界限
P_TIME_BOUND = lambda n: n ** 3
NP_TIME_BOUND = lambda n: PHI ** (np.log(n) / np.log(PHI))
PSPACE_BOUND = lambda n: n ** 2
EXP_TIME_BOUND = lambda n: PHI ** n

8. 错误处理​

class DecidabilityError(Exception):
    """可判定性错误基类"""
    
class UndecidableError(DecidabilityError):
    """不可判定错误"""
    
class TimeoutError(DecidabilityError):
    """判定超时错误"""
    
class DepthExceededError(DecidabilityError):
    """超过深度限制错误"""
    
class StateSpaceExhaustedError(DecidabilityError):
    """状态空间耗尽错误"""