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T10-5 NP-P Collapse转化形式化规范

1. 基础数学对象​

1.1 φ-计算模型​

class PhiTuringMachine:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.tape = []  # No-11 constrained binary tape
        self.state = 0
        self.head_position = 0
        
    def verify_no_11_constraint(self) -> bool:
        """验证带上无连续11"""
        tape_string = ''.join(map(str, self.tape))
        return '11' not in tape_string
        
    def step(self) -> bool:
        """单步计算，返回是否继续"""
        if not self.verify_no_11_constraint():
            raise ValueError("No-11 constraint violated")
        # 状态转移逻辑
        
    def compute_time_complexity(self, input_size: int) -> 'PhiNumber':
        """计算时间复杂度"""

1.2 复杂度类定义​

class ComplexityClass:
    def __init__(self, name: str):
        self.name = name
        self.phi = (1 + np.sqrt(5)) / 2
        
class P_phi(ComplexityClass):
    def __init__(self):
        super().__init__("P_φ")
        
    def contains(self, problem: 'Problem', depth: int) -> bool:
        """判断问题在深度d时是否属于P_φ"""
        time_bound = problem.size ** depth
        return problem.solvable_in_time(time_bound)
        
class NP_phi(ComplexityClass):
    def __init__(self):
        super().__init__("NP_φ")
        
    def contains(self, problem: 'Problem', depth: int) -> bool:
        """判断问题在深度d时是否属于NP_φ"""
        verify_time = problem.size ** depth
        return problem.verifiable_in_time(verify_time)

1.3 递归深度函数​

class RecursiveDepth:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def compute(self, problem_instance: 'Instance') -> int:
        """计算问题实例的递归深度"""
        entropy = self.compute_entropy(problem_instance)
        return int(np.log(entropy + 1) / np.log(self.phi))
        
    def critical_depth(self, problem_size: int) -> 'PhiNumber':
        """计算临界深度"""
        return PhiNumber(self.phi ** np.sqrt(problem_size))
        
    def is_collapsible(self, instance: 'Instance') -> bool:
        """判断实例是否可坍缩"""
        depth = self.compute(instance)
        critical = self.critical_depth(instance.size)
        return depth < critical.value

2. 搜索空间压缩​

2.1 约束搜索空间​

class ConstrainedSearchSpace:
    def __init__(self, n: int):
        self.n = n
        self.phi = (1 + np.sqrt(5)) / 2
        self.compression_factor = 0.306  # γ ≈ 0.306
        
    def classical_size(self) -> int:
        """经典搜索空间大小"""
        return 2 ** self.n
        
    def phi_constrained_size(self) -> 'PhiNumber':
        """φ约束搜索空间大小"""
        # |Ω_NP^φ| = |Ω_NP| * φ^(-γn)
        classical = self.classical_size()
        compression = self.phi ** (-self.compression_factor * self.n)
        return PhiNumber(classical * compression)
        
    def enumerate_valid_strings(self) -> List[str]:
        """枚举所有满足no-11约束的串"""
        valid = []
        for i in range(2 ** self.n):
            binary = format(i, f'0{self.n}b')
            if '11' not in binary:
                valid.append(binary)
        return valid
        
    def compression_ratio(self) -> float:
        """计算压缩比"""
        return len(self.enumerate_valid_strings()) / self.classical_size()

2.2 Fibonacci搜索模式​

class FibonacciSearchPattern:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.fib_cache = {}
        
    def fibonacci(self, n: int) -> int:
        """计算第n个Fibonacci数"""
        if n in self.fib_cache:
            return self.fib_cache[n]
        if n <= 1:
            return n
        result = self.fibonacci(n-1) + self.fibonacci(n-2)
        self.fib_cache[n] = result
        return result
        
    def valid_configurations(self, length: int) -> int:
        """长度为n的有效配置数（无11）"""
        # 等于F_{n+2}
        return self.fibonacci(length + 2)
        
    def search_tree_pruning(self, depth: int) -> float:
        """搜索树剪枝比例"""
        full_tree = 2 ** depth
        pruned_tree = self.valid_configurations(depth)
        return 1 - (pruned_tree / full_tree)

3. 坍缩机制​

3.1 深度诱导坍缩​

class DepthInducedCollapse:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def time_complexity_reduction(self, n: int, depth: int) -> 'PhiNumber':
        """计算时间复杂度降低"""
        # TIME_φ(2^n) ⊆ TIME_φ(n^(d*log φ))
        if depth >= np.log(n) / np.log(self.phi):
            # 无坍缩
            return PhiNumber(2 ** n)
        else:
            # 坍缩到多项式
            exponent = depth * np.log(self.phi)
            return PhiNumber(n ** exponent)
            
    def decompose_problem(self, problem: 'Problem', depth: int) -> List['Problem']:
        """根据深度分解问题"""
        num_subproblems = int(self.phi ** depth)
        subproblem_size = problem.size / num_subproblems
        
        subproblems = []
        for i in range(num_subproblems):
            sub = problem.create_subproblem(i, subproblem_size)
            subproblems.append(sub)
            
        return subproblems
        
    def solve_by_decomposition(self, problem: 'Problem', depth: int) -> 'Solution':
        """通过分解求解"""
        if problem.size < threshold:
            return problem.solve_directly()
            
        subproblems = self.decompose_problem(problem, depth)
        subsolutions = [self.solve_by_decomposition(sub, depth-1) 
                       for sub in subproblems]
        
        return problem.combine_solutions(subsolutions)

3.2 验证-搜索对称性​

class VerifySearchSymmetry:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def verify_certificate(self, instance: 'Instance', 
                         certificate: 'Certificate') -> bool:
        """验证证书"""
        # φ系统中的验证过程
        return self.phi_verify(instance, certificate)
        
    def search_from_verification(self, instance: 'Instance',
                               verification_trace: List) -> 'Solution':
        """从验证过程重构搜索"""
        # 利用验证轨迹反向构造解
        search_path = self.reverse_verification_trace(verification_trace)
        return self.follow_search_path(instance, search_path)
        
    def entropy_guided_search(self, instance: 'Instance') -> 'Solution':
        """熵导向搜索"""
        current_state = instance.initial_state()
        
        while not self.is_solution(current_state):
            # 选择熵增方向
            next_states = self.get_successors(current_state)
            next_state = max(next_states, 
                           key=lambda s: self.compute_entropy(s))
            current_state = next_state
            
        return self.extract_solution(current_state)

4. 具体问题实现​

4.1 φ-SAT求解器​

class PhiSATSolver:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def solve(self, formula: 'CNFFormula') -> Optional[Dict[str, bool]]:
        """求解SAT问题"""
        depth = self.compute_formula_depth(formula)
        
        if self.is_easy_case(formula, depth):
            return self.polynomial_solve(formula)
        else:
            return self.exponential_solve(formula)
            
    def is_easy_case(self, formula: 'CNFFormula', depth: int) -> bool:
        """判断是否为易解情况"""
        n = formula.num_variables
        m = formula.num_clauses
        
        # m < n * φ 时多项式可解
        return m < n * self.phi and depth < self.critical_depth(n)
        
    def polynomial_solve(self, formula: 'CNFFormula') -> Optional[Dict[str, bool]]:
        """多项式时间求解"""
        # 利用no-11约束和自相似性
        constraint_graph = self.build_constraint_graph(formula)
        
        if self.is_tree_like(constraint_graph):
            return self.tree_solve(constraint_graph)
        else:
            return self.phi_propagation(formula)
            
    def phi_propagation(self, formula: 'CNFFormula') -> Optional[Dict[str, bool]]:
        """φ-传播算法"""
        assignment = {}
        unit_clauses = formula.get_unit_clauses()
        
        while unit_clauses:
            # 单元传播
            for lit in unit_clauses:
                var = abs(lit)
                value = lit > 0
                assignment[var] = value
                
            # φ-推理规则
            formula = self.phi_simplify(formula, assignment)
            unit_clauses = formula.get_unit_clauses()
            
        if formula.is_satisfied():
            return assignment
        elif formula.has_empty_clause():
            return None
        else:
            # 递归分解
            return self.phi_branch(formula, assignment)

4.2 图着色问题​

class PhiGraphColoring:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def can_color_in_polynomial_time(self, graph: 'Graph', k: int) -> bool:
        """判断是否可在多项式时间内k-着色"""
        return k >= self.phi ** 2  # k ≥ φ² ≈ 2.618
        
    def phi_coloring(self, graph: 'Graph', k: int) -> Optional[Dict[int, int]]:
        """φ-着色算法"""
        if not self.can_color_in_polynomial_time(graph, k):
            return self.exponential_coloring(graph, k)
            
        # 利用自相似性递归着色
        if graph.num_vertices < 10:
            return self.greedy_coloring(graph, k)
            
        # φ-分解
        components = self.phi_decompose_graph(graph)
        colorings = []
        
        for comp in components:
            sub_coloring = self.phi_coloring(comp, k)
            if sub_coloring is None:
                return None
            colorings.append(sub_coloring)
            
        # 合并着色
        return self.merge_colorings(colorings, graph)
        
    def phi_decompose_graph(self, graph: 'Graph') -> List['Graph']:
        """按φ比例分解图"""
        n = graph.num_vertices
        split_size = int(n / self.phi)
        
        # 找最小割
        cut = self.find_balanced_cut(graph, split_size)
        return self.split_by_cut(graph, cut)

4.3 TSP问题​

class PhiTSP:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def is_polynomial_solvable(self, cities: List['City']) -> bool:
        """判断TSP实例是否多项式可解"""
        fractal_dim = self.compute_fractal_dimension(cities)
        return fractal_dim < np.log(self.phi)
        
    def phi_tsp_solve(self, cities: List['City']) -> List[int]:
        """φ-TSP求解"""
        if not self.is_polynomial_solvable(cities):
            return self.branch_and_bound(cities)
            
        # 利用分形结构
        clusters = self.fractal_clustering(cities)
        
        # 递归求解子问题
        sub_tours = []
        for cluster in clusters:
            sub_tour = self.phi_tsp_solve(cluster)
            sub_tours.append(sub_tour)
            
        # 连接子回路
        return self.connect_subtours(sub_tours)
        
    def fractal_clustering(self, cities: List['City']) -> List[List['City']]:
        """分形聚类"""
        # 按φ-比例递归划分空间
        return self.recursive_phi_partition(cities)

5. 临界现象​

5.1 临界深度计算​

class CriticalDepth:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def compute_critical_depth(self, problem_size: int) -> 'PhiNumber':
        """计算临界深度 d_c = φ^√n"""
        return PhiNumber(self.phi ** np.sqrt(problem_size))
        
    def phase_transition_point(self, problem: 'Problem') -> float:
        """计算相变点"""
        n = problem.size
        d_c = self.compute_critical_depth(n)
        
        # 在d_c附近复杂度发生突变
        return d_c.value
        
    def complexity_at_depth(self, problem: 'Problem', depth: int) -> str:
        """给定深度下的复杂度类"""
        critical = self.compute_critical_depth(problem.size)
        
        if depth < critical.value * 0.9:
            return "P"
        elif depth > critical.value * 1.1:
            return "NP-complete"
        else:
            return "Phase transition region"

5.2 坍缩检测器​

class CollapseDetector:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def detect_collapse(self, problem: 'Problem') -> bool:
        """检测问题是否会坍缩"""
        # 检查三个条件
        space_compressed = self.check_space_compression(problem)
        decomposable = self.check_decomposability(problem)
        self_similar = self.check_self_similarity(problem)
        
        return space_compressed and decomposable and self_similar
        
    def check_space_compression(self, problem: 'Problem') -> bool:
        """检查搜索空间压缩"""
        original_space = problem.search_space_size()
        phi_space = problem.phi_search_space_size()
        
        ratio = phi_space / original_space
        return ratio < 1 / problem.size  # 多项式压缩
        
    def check_decomposability(self, problem: 'Problem') -> bool:
        """检查递归可分解性"""
        try:
            decomposition = problem.phi_decompose()
            return len(decomposition) <= self.phi ** problem.recursive_depth
        except:
            return False
            
    def check_self_similarity(self, problem: 'Problem') -> bool:
        """检查自相似结构"""
        sub_structures = problem.extract_substructures()
        
        if len(sub_structures) < 2:
            return False
            
        # 比较子结构的相似性
        reference = sub_structures[0]
        for sub in sub_structures[1:]:
            if not self.is_isomorphic(reference, sub):
                return False
                
        return True

6. 算法框架​

6.1 自适应求解器​

class AdaptiveSolver:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.solvers = {
            'polynomial': PolynomialSolver(),
            'exponential': ExponentialSolver(),
            'hybrid': HybridSolver()
        }
        
    def solve(self, problem: 'Problem') -> 'Solution':
        """自适应求解"""
        # 分析问题特征
        depth = self.compute_depth(problem)
        critical = self.critical_depth(problem.size)
        
        # 选择求解策略
        if depth < critical.value * 0.8:
            solver = self.solvers['polynomial']
        elif depth > critical.value * 1.2:
            solver = self.solvers['exponential']
        else:
            solver = self.solvers['hybrid']
            
        # 求解
        return solver.solve(problem)
        
    def compute_depth(self, problem: 'Problem') -> int:
        """计算问题的递归深度"""
        entropy = problem.compute_entropy()
        return int(np.log(entropy + 1) / np.log(self.phi))
        
    def critical_depth(self, size: int) -> 'PhiNumber':
        """计算临界深度"""
        return PhiNumber(self.phi ** np.sqrt(size))

6.2 φ-分解框架​

class PhiDecompositionFramework:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def decompose(self, problem: 'Problem', depth: int) -> List['Problem']:
        """φ-分解问题"""
        if problem.size < self.threshold:
            return [problem]
            
        # 计算分解数
        num_parts = min(int(self.phi ** depth), problem.size)
        
        # 按黄金比例分割
        parts = []
        remaining = problem
        
        for i in range(num_parts - 1):
            size = int(remaining.size / self.phi)
            part, remaining = remaining.split_at(size)
            parts.append(part)
            
        parts.append(remaining)
        return parts
        
    def solve_recursively(self, problem: 'Problem') -> 'Solution':
        """递归求解"""
        depth = self.compute_depth(problem)
        
        if self.is_base_case(problem):
            return self.solve_directly(problem)
            
        # 分解
        subproblems = self.decompose(problem, depth)
        
        # 递归求解
        subsolutions = []
        for sub in subproblems:
            sol = self.solve_recursively(sub)
            subsolutions.append(sol)
            
        # 合并
        return self.combine_by_similarity(subsolutions, problem)

7. 验证函数​

7.1 坍缩验证​

def verify_collapse(problem_class: type, size: int) -> bool:
    """验证特定规模的问题类是否坍缩"""
    # 创建测试实例
    instances = generate_test_instances(problem_class, size, count=100)
    
    collapsed_count = 0
    for instance in instances:
        detector = CollapseDetector()
        if detector.detect_collapse(instance):
            # 验证多项式可解
            solver = AdaptiveSolver()
            start_time = time.time()
            solution = solver.solve(instance)
            solve_time = time.time() - start_time
            
            # 检查是否真的是多项式时间
            if solve_time < size ** 10:  # 宽松的多项式界
                collapsed_count += 1
                
    # 如果大部分实例坍缩，则认为该类坍缩
    return collapsed_count > len(instances) * 0.8

7.2 复杂度测量​

def measure_complexity_transition(problem_class: type) -> Dict[int, str]:
    """测量复杂度相变"""
    results = {}
    
    for size in range(10, 100, 10):
        problem = problem_class(size)
        depth = RecursiveDepth().compute(problem)
        critical = RecursiveDepth().critical_depth(size)
        
        if depth < critical.value * 0.9:
            results[size] = "P"
        elif depth > critical.value * 1.1:
            results[size] = "NP"
        else:
            results[size] = "Transition"
            
    return results

8. 关键常数​

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

# 压缩参数
COMPRESSION_FACTOR = 0.306  # no-11约束的压缩因子
FIBONACCI_LIMIT_RATIO = PHI ** (-1)  # Fibonacci/2^n的极限比

# 临界参数
CRITICAL_EXPONENT = 0.5  # d_c = φ^(n^0.5)
PHASE_TRANSITION_WIDTH = 0.2  # 相变区域宽度

# 算法参数
DECOMPOSITION_THRESHOLD = 10  # 直接求解阈值
MAX_RECURSION_DEPTH = 100  # 最大递归深度

9. 错误处理​

class CollapseError(Exception):
    """坍缩相关错误基类"""
    
class NoCollapseError(CollapseError):
    """问题不满足坍缩条件"""
    
class DepthExceededError(CollapseError):
    """超过临界深度"""
    
class DecompositionError(CollapseError):
    """分解失败"""
    
class VerificationError(CollapseError):
    """验证-搜索对称性破坏"""