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C13-1 φ-计算复杂性分类形式化规范

1. 基础复杂性类定义​

1.1 φ-P类​

class PhiPClass:
    """φ-P复杂性类"""
    def __init__(self, depth: int):
        self.phi = (1 + np.sqrt(5)) / 2
        self.depth = depth
        
    def time_bound(self, n: int) -> float:
        """时间复杂度界限"""
        return (n ** self.depth) * (self.phi ** self.recursive_depth(n))
        
    def recursive_depth(self, n: int) -> int:
        """输入的递归深度"""
        # 基于输入大小的对数
        return int(np.log(n + 1) / np.log(self.phi))
        
    def contains(self, problem: 'Problem') -> bool:
        """判断问题是否属于此类"""
        # 检查是否存在满足时间界限的算法
        for algorithm in problem.get_algorithms():
            if self.verify_algorithm(algorithm, problem):
                return True
        return False
        
    def verify_algorithm(self, algorithm: 'Algorithm', problem: 'Problem') -> bool:
        """验证算法是否满足复杂度要求"""
        # 采样测试
        test_sizes = [10, 100, 1000]
        
        for n in test_sizes:
            instance = problem.generate_instance(n)
            time = algorithm.measure_time(instance)
            bound = self.time_bound(n)
            
            if time > bound * 1.1:  # 允许10%误差
                return False
                
        return True

1.2 φ-NP类​

class PhiNPClass:
    """φ-NP复杂性类"""
    def __init__(self, depth: int):
        self.phi = (1 + np.sqrt(5)) / 2
        self.depth = depth
        self.p_class = PhiPClass(depth + int(np.log(depth) / np.log(self.phi)))
        
    def time_bound(self, n: int) -> float:
        """非确定性时间界限"""
        return (n ** self.depth) * (self.phi ** self.recursive_depth(n))
        
    def verifier_time(self, n: int) -> float:
        """验证器时间界限"""
        return n ** 2  # 多项式验证
        
    def recursive_depth(self, n: int) -> int:
        """递归深度"""
        return int(np.log(n + 1) / np.log(self.phi))
        
    def contains(self, problem: 'Problem') -> bool:
        """判断问题是否属于此类"""
        # 检查是否有多项式验证器
        if not problem.has_polynomial_verifier():
            return False
            
        # 检查搜索空间
        search_space = problem.search_space_size()
        bound = self.phi ** (self.depth * np.log(problem.input_size))
        
        return search_space <= bound
        
    def collapse_to_p(self, problem: 'Problem') -> Optional['Algorithm']:
        """尝试将NP问题塌缩到P"""
        depth = problem.recursive_depth()
        
        if depth < self.critical_depth(problem.input_size):
            # 可以塌缩
            return self.construct_p_algorithm(problem)
        else:
            return None
            
    def critical_depth(self, n: int) -> int:
        """临界深度"""
        return int(np.log(n) / np.log(self.phi))
        
    def construct_p_algorithm(self, problem: 'Problem') -> 'Algorithm':
        """构造P算法"""
        # 利用φ-分解
        return PhiDecompositionAlgorithm(problem)

1.3 φ-PSPACE类​

class PhiPSPACEClass:
    """φ-PSPACE复杂性类"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def space_bound(self, n: int) -> float:
        """空间复杂度界限"""
        return n * np.log(n) / np.log(self.phi)
        
    def contains(self, problem: 'Problem') -> bool:
        """判断问题是否属于PSPACE_φ"""
        # 检查空间需求
        space = problem.space_requirement()
        bound = self.space_bound(problem.input_size)
        
        return space <= bound
        
    def complete_problems(self) -> List['Problem']:
        """PSPACE_φ完全问题"""
        return [
            PhiQuantifiedBooleanFormula(),
            PhiGameOfLife(),
            PhiPlanningProblem()
        ]

2. 复杂性类层次​

2.1 层次结构​

class ComplexityHierarchy:
    """φ-复杂性类层次"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.classes = self.build_hierarchy()
        
    def build_hierarchy(self) -> Dict[str, 'ComplexityClass']:
        """构建复杂性类层次"""
        hierarchy = {}
        
        # P层次
        for d in range(10):
            hierarchy[f'P_φ^({d})'] = PhiPClass(d)
            
        # NP层次
        for d in range(10):
            hierarchy[f'NP_φ^({d})'] = PhiNPClass(d)
            
        # 空间类
        hierarchy['PSPACE_φ'] = PhiPSPACEClass()
        hierarchy['L_φ'] = PhiLogSpaceClass()
        
        # 指数类
        hierarchy['EXP_φ'] = PhiEXPClass()
        hierarchy['NEXP_φ'] = PhiNEXPClass()
        
        return hierarchy
        
    def inclusion_relations(self) -> List[Tuple[str, str]]:
        """包含关系"""
        relations = []
        
        # P层次包含
        for d in range(9):
            relations.append((f'P_φ^({d})', f'P_φ^({d+1})'))
            
        # NP塌缩
        for d in range(5):
            log_d = int(np.log(d + 1) / np.log(self.phi))
            relations.append((f'NP_φ^({d})', f'P_φ^({d + log_d})'))
            
        # P ⊆ NP ⊆ PSPACE
        relations.append(('P_φ^(1)', 'NP_φ^(1)'))
        relations.append(('NP_φ^(1)', 'PSPACE_φ'))
        
        return relations
        
    def separation_oracle(self, class1: str, class2: str) -> Optional['Oracle']:
        """构造分离两个类的oracle"""
        if self.provably_different(class1, class2):
            return self.construct_separating_oracle(class1, class2)
        return None

2.2 完全问题​

class CompleteProblem:
    """完全问题基类"""
    def __init__(self, complexity_class: str):
        self.phi = (1 + np.sqrt(5)) / 2
        self.complexity_class = complexity_class
        
    def reduce_from(self, problem: 'Problem') -> 'Reduction':
        """从其他问题归约"""
        pass
        
    def is_complete(self) -> bool:
        """验证完全性"""
        # 1. 属于该复杂性类
        # 2. 所有该类问题都可归约到此问题
        return True

class PhiSAT(CompleteProblem):
    """φ-SAT问题"""
    def __init__(self):
        super().__init__('NP_φ^(1)')
        
    def encode_formula(self, variables: int, clauses: List[List[int]]) -> str:
        """编码SAT公式为φ-二进制"""
        encoding = ""
        
        for clause in clauses:
            # 确保满足no-11约束
            clause_encoding = self.encode_clause(clause)
            encoding += clause_encoding + "0"  # 分隔符
            
        return self.normalize_no_11(encoding)
        
    def solve(self, formula: str) -> Optional[Dict[int, bool]]:
        """求解φ-SAT"""
        n = self.count_variables(formula)
        depth = int(np.log(n) / np.log(self.phi))
        
        if depth < self.critical_depth(n):
            # 可以高效求解
            return self.phi_decomposition_solve(formula)
        else:
            # 使用标准SAT求解器
            return self.standard_sat_solve(formula)
            
    def phi_decomposition_solve(self, formula: str) -> Optional[Dict[int, bool]]:
        """φ-分解求解算法"""
        # 将问题分解为子问题
        subproblems = self.decompose_by_phi(formula)
        
        # 递归求解
        solutions = []
        for sub in subproblems:
            sol = self.solve(sub)
            if sol is None:
                return None
            solutions.append(sol)
            
        # 合并解
        return self.merge_solutions(solutions)

3. 熵增复杂性类​

3.1 熵增类定义​

class EntropyComplexityClass:
    """基于熵增的复杂性类"""
    def __init__(self, entropy_rate: float):
        self.phi = (1 + np.sqrt(5)) / 2
        self.entropy_rate = entropy_rate
        
    def contains(self, algorithm: 'Algorithm') -> bool:
        """判断算法是否属于此熵增类"""
        # 测量算法的熵增率
        test_inputs = self.generate_test_inputs()
        
        total_entropy_increase = 0
        for input_data in test_inputs:
            initial_entropy = self.compute_entropy(input_data)
            
            # 运行算法
            trace = algorithm.run_with_trace(input_data)
            final_entropy = self.compute_entropy(trace.final_state)
            
            entropy_increase = final_entropy - initial_entropy
            total_entropy_increase += entropy_increase / len(input_data)
            
        avg_rate = total_entropy_increase / len(test_inputs)
        
        return avg_rate >= self.entropy_rate
        
    def compute_entropy(self, state: Any) -> float:
        """计算状态的熵"""
        if isinstance(state, str):
            # 二进制串的熵
            ones = state.count('1')
            zeros = state.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
        else:
            # 其他类型的熵计算
            return 0.0
            
    def optimize_by_entropy(self, algorithm: 'Algorithm') -> 'Algorithm':
        """通过熵增优化算法"""
        # 分析算法的熵增瓶颈
        bottlenecks = self.analyze_entropy_bottlenecks(algorithm)
        
        # 优化低熵增部分
        optimized = algorithm.copy()
        for bottleneck in bottlenecks:
            optimized = self.optimize_component(optimized, bottleneck)
            
        return optimized

3.2 熵增层次​

class EntropyHierarchy:
    """熵增复杂性层次"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.levels = self.build_entropy_hierarchy()
        
    def build_entropy_hierarchy(self) -> List[EntropyComplexityClass]:
        """构建熵增层次"""
        levels = []
        
        # 线性熵增
        levels.append(EntropyComplexityClass(entropy_rate=1.0))
        
        # φ-熵增
        levels.append(EntropyComplexityClass(entropy_rate=self.phi))
        
        # 平方熵增
        levels.append(EntropyComplexityClass(entropy_rate=self.phi ** 2))
        
        # 指数熵增
        for k in range(1, 5):
            rate = self.phi ** k
            levels.append(EntropyComplexityClass(entropy_rate=rate))
            
        return levels
        
    def classify_algorithm(self, algorithm: 'Algorithm') -> int:
        """分类算法的熵增等级"""
        for i, level in enumerate(self.levels):
            if not level.contains(algorithm):
                return i - 1 if i > 0 else 0
                
        return len(self.levels) - 1

4. 相变现象​

4.1 复杂度相变​

class ComplexityPhaseTransition:
    """复杂度相变"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.critical_ratio = 1 / self.phi  # ≈ 0.618
        
    def detect_phase_transition(self, problem_family: 'ProblemFamily', 
                               parameter: str) -> float:
        """检测相变点"""
        # 二分搜索相变点
        low, high = 0.0, 1.0
        epsilon = 0.001
        
        while high - low > epsilon:
            mid = (low + high) / 2
            
            # 生成该参数下的问题实例
            instances = problem_family.generate_instances(
                parameter_value=mid,
                count=100
            )
            
            # 测量复杂度
            avg_complexity = self.measure_average_complexity(instances)
            
            if avg_complexity < self.polynomial_threshold():
                low = mid
            else:
                high = mid
                
        return (low + high) / 2
        
    def measure_average_complexity(self, instances: List['Instance']) -> float:
        """测量平均复杂度"""
        total_time = 0
        
        for instance in instances:
            solver = self.get_solver(instance)
            start_time = time.time()
            solver.solve(instance)
            elapsed = time.time() - start_time
            
            total_time += elapsed
            
        return total_time / len(instances)
        
    def analyze_sat_threshold(self, n: int) -> float:
        """分析SAT相变阈值"""
        # 理论预测：m/n ≈ φ² - 1/φ
        theoretical = self.phi ** 2 - 1 / self.phi  # ≈ 2.236
        
        # 实验验证
        experimental = self.experimental_sat_threshold(n)
        
        return {
            'theoretical': theoretical,
            'experimental': experimental,
            'error': abs(theoretical - experimental)
        }

4.2 可满足性阈值​

class SatisfiabilityThreshold:
    """可满足性阈值分析"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def compute_threshold(self, formula_type: str) -> float:
        """计算不同类型公式的阈值"""
        if formula_type == "3-SAT":
            return self.phi ** 2 - 1 / self.phi  # ≈ 2.236
        elif formula_type == "k-SAT":
            k = self.extract_k(formula_type)
            return (2 ** k) / (self.phi ** (k - 2))
        elif formula_type == "φ-SAT":
            return self.phi  # 特殊的φ-SAT阈值
            
    def probability_satisfiable(self, n: int, m: int, formula_type: str) -> float:
        """计算可满足概率"""
        ratio = m / n
        threshold = self.compute_threshold(formula_type)
        
        if ratio < threshold * 0.9:
            return 0.99  # 几乎必然可满足
        elif ratio > threshold * 1.1:
            return 0.01  # 几乎必然不可满足
        else:
            # 相变区域，使用S型函数
            x = (ratio - threshold) / (threshold * 0.1)
            return 1 / (1 + np.exp(10 * x))

5. 近似复杂性​

5.1 φ-APX类​

class PhiAPXClass:
    """φ-近似类"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def contains(self, problem: 'OptimizationProblem') -> bool:
        """判断问题是否属于φ-APX"""
        # 检查是否存在φ-近似算法
        algorithm = problem.get_approximation_algorithm()
        
        if algorithm is None:
            return False
            
        # 验证近似比
        test_instances = problem.generate_test_instances(100)
        
        for instance in test_instances:
            approx_solution = algorithm.solve(instance)
            optimal_solution = problem.get_optimal_solution(instance)
            
            ratio = approx_solution.value / optimal_solution.value
            
            if ratio < 1 / self.phi:
                return False
                
        return True
        
    def phi_approximation_algorithm(self, problem: 'OptimizationProblem') -> 'Algorithm':
        """构造φ-近似算法"""
        class PhiApproximation:
            def __init__(self, problem):
                self.problem = problem
                self.phi = (1 + np.sqrt(5)) / 2
                
            def solve(self, instance):
                # φ-贪心策略
                solution = self.phi_greedy(instance)
                
                # 局部改进
                solution = self.local_improvement(solution)
                
                return solution
                
            def phi_greedy(self, instance):
                # 每步选择φ-最优的局部决策
                current = instance.initial_state()
                
                while not instance.is_complete(current):
                    choices = instance.get_choices(current)
                    
                    # 评估每个选择
                    best_choice = None
                    best_value = -float('inf')
                    
                    for choice in choices:
                        value = self.evaluate_choice(current, choice)
                        if value > best_value:
                            best_value = value
                            best_choice = choice
                            
                    current = instance.apply_choice(current, best_choice)
                    
                return current
                
        return PhiApproximation(problem)

5.2 不可近似性​

class InapproximabilityResult:
    """不可近似性结果"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def prove_hardness(self, problem: 'Problem', factor: float) -> 'Proof':
        """证明不可近似到给定因子"""
        if factor >= 1 / self.phi:
            # 可能可以φ-近似
            return None
            
        # 构造归约
        reduction = self.construct_gap_reduction(problem, factor)
        
        # 从已知困难问题归约
        if self.reduces_from_hard_problem(reduction):
            return Proof(
                statement=f"{problem} cannot be approximated within {factor}",
                technique="gap-preserving reduction",
                assumption="P ≠ NP"
            )
            
        return None
        
    def phi_unique_games_conjecture(self) -> 'Conjecture':
        """φ-唯一博弈猜想"""
        return Conjecture(
            statement="For every ε > 0, it is NP-hard to approximate "
                     f"φ-UniqueGames within {1/self.phi - ε}",
            implications=[
                "Many problems have φ-approximation threshold",
                "φ is fundamental approximation barrier"
            ]
        )

6. 算法分类器​

6.1 自动分类​

class AlgorithmClassifier:
    """算法复杂性自动分类"""
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.hierarchy = ComplexityHierarchy()
        
    def classify(self, algorithm: 'Algorithm') -> str:
        """分类算法的复杂性"""
        # 运行基准测试
        profile = self.profile_algorithm(algorithm)
        
        # 分析时间复杂度
        time_class = self.analyze_time_complexity(profile)
        
        # 分析空间复杂度
        space_class = self.analyze_space_complexity(profile)
        
        # 分析熵增特性
        entropy_class = self.analyze_entropy_behavior(profile)
        
        # 综合判断
        return self.synthesize_classification(time_class, space_class, entropy_class)
        
    def profile_algorithm(self, algorithm: 'Algorithm') -> 'Profile':
        """分析算法性能"""
        profile = Profile()
        
        # 不同规模的测试
        test_sizes = [10, 20, 50, 100, 200, 500, 1000]
        
        for size in test_sizes:
            # 生成测试输入
            inputs = self.generate_inputs(size, count=10)
            
            times = []
            spaces = []
            entropies = []
            
            for input_data in inputs:
                # 运行并测量
                start_time = time.time()
                start_memory = self.get_memory_usage()
                
                trace = algorithm.run_with_trace(input_data)
                
                elapsed = time.time() - start_time
                memory = self.get_memory_usage() - start_memory
                entropy_increase = trace.entropy_increase()
                
                times.append(elapsed)
                spaces.append(memory)
                entropies.append(entropy_increase)
                
            profile.add_measurement(size, {
                'time': np.mean(times),
                'space': np.mean(spaces),
                'entropy': np.mean(entropies)
            })
            
        return profile
        
    def analyze_time_complexity(self, profile: 'Profile') -> str:
        """分析时间复杂度"""
        # 拟合复杂度函数
        sizes = profile.get_sizes()
        times = profile.get_times()
        
        # 尝试不同的复杂度模型
        models = {
            'O(n)': lambda n: n,
            'O(n log n)': lambda n: n * np.log(n),
            'O(n²)': lambda n: n ** 2,
            'O(n³)': lambda n: n ** 3,
            f'O(φⁿ)': lambda n: self.phi ** n,
            f'O(n^log_φ(n))': lambda n: n ** (np.log(n) / np.log(self.phi))
        }
        
        best_fit = None
        best_error = float('inf')
        
        for name, model in models.items():
            error = self.fit_error(sizes, times, model)
            if error < best_error:
                best_error = error
                best_fit = name
                
        return best_fit

7. 验证函数​

7.1 复杂性类测试​

def test_complexity_classification() -> Dict[str, bool]:
    """测试复杂性分类"""
    results = {}
    
    # 测试P_φ层次
    for d in range(5):
        p_class = PhiPClass(d)
        # 创建一个d次多项式时间算法
        algorithm = PolynomialTimeAlgorithm(degree=d)
        problem = algorithm.get_problem()
        
        results[f'P_φ^({d})_contains'] = p_class.contains(problem)
        
    # 测试NP_φ塌缩
    for d in range(3):
        np_class = PhiNPClass(d)
        sat_problem = PhiSAT()
        sat_problem.set_depth(d)
        
        p_algorithm = np_class.collapse_to_p(sat_problem)
        results[f'NP_φ^({d})_collapse'] = p_algorithm is not None
        
    # 测试相变
    transition = ComplexityPhaseTransition()
    sat_threshold = transition.analyze_sat_threshold(100)
    results['phase_transition'] = abs(sat_threshold['error']) < 0.1
    
    return results

7.2 算法验证​

def verify_algorithm_complexity(algorithm: 'Algorithm', 
                              expected_class: str) -> bool:
    """验证算法的复杂性类"""
    classifier = AlgorithmClassifier()
    actual_class = classifier.classify(algorithm)
    
    return actual_class == expected_class

8. 关键常数​

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

# 复杂性参数
CRITICAL_DEPTH_FACTOR = 1.0  # 临界深度系数
PHASE_TRANSITION_RATIO = 1 / PHI  # 相变比率 ≈ 0.618
SAT_THRESHOLD = PHI ** 2 - 1 / PHI  # SAT阈值 ≈ 2.236

# 近似因子
APPROXIMATION_RATIO = 1 / PHI  # φ-近似比 ≈ 0.618
INAPPROXIMABILITY_GAP = PHI - 1  # 不可近似间隙 ≈ 0.618

# 熵增率
LINEAR_ENTROPY_RATE = 1.0
PHI_ENTROPY_RATE = PHI
QUADRATIC_ENTROPY_RATE = PHI ** 2

# 算法参数
GREEDY_SELECTION_RATIO = PHI  # 贪心选择比率
DECOMPOSITION_RATIO = PHI  # 分解比率
LOCAL_SEARCH_RADIUS = int(PHI ** 3)  # 局部搜索半径

9. 错误处理​

class ComplexityError(Exception):
    """复杂性错误基类"""
    
class ClassificationError(ComplexityError):
    """分类错误"""
    
class ReductionError(ComplexityError):
    """归约错误"""
    
class ApproximationError(ComplexityError):
    """近似错误"""
    
class PhaseTransitionError(ComplexityError):
    """相变检测错误"""