T10-4 递归稳定性定理形式化规范
1. 核心稳定性判据定义
1.1 深度稳定性判据
import numpy as np
from typing import List, Tuple, Optional, Dict, Any
from dataclasses import dataclass
from enum import Enum
class StabilityLevel(Enum):
STABLE = "stable"
UNSTABLE = "unstable"
CRITICAL = "critical"
MARGINAL = "marginal"
@dataclass
class RecursiveState:
"""递归系统状态"""
def __init__(self, data: str, entropy: float, depth: int):
self.phi = (1 + np.sqrt(5)) / 2
self.data = data
self.entropy = entropy
self.depth = depth
self.phi_length = self.compute_phi_length()
def compute_phi_length(self) -> float:
"""计算φ-长度"""
if not self.data:
return 0.0
# φ-长度基于Zeckendorf表示
length = 0.0
for i, bit in enumerate(self.data):
if bit == '1':
fib_index = i + 2 # Fibonacci索引从F_2开始
length += 1.0 / (self.phi ** fib_index)
return length
def phi_distance(self, other: 'RecursiveState') -> float:
"""计算φ-距离"""
if len(self.data) != len(other.data):
# 长度不同时的φ-距离
len_diff = abs(len(self.data) - len(other.data))
return len_diff / self.phi + abs(self.phi_length - other.phi_length)
# 相同长度的比特距离
bit_distance = sum(1 for a, b in zip(self.data, other.data) if a != b)
return bit_distance / len(self.data) + abs(self.phi_length - other.phi_length)
class DepthStabilityChecker:
"""深度稳定性检查器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def compute_max_depth(self, max_entropy: float) -> int:
"""计算最大递归深度界限"""
return int(np.log(max_entropy + 1) / np.log(self.phi))
def check_depth_stability(self, state: RecursiveState, max_depth: int) -> bool:
"""检查深度稳定性"""
return state.depth <= max_depth
def compute_depth_stability_score(self, state: RecursiveState, max_depth: int) -> float:
"""计算深度稳定性分数"""
if max_depth == 0:
return 1.0 if state.depth == 0 else 0.0
return max(0.0, (max_depth - state.depth) / max_depth)
1.2 周期稳定性判据
class PeriodicOrbit:
"""周期轨道"""
def __init__(self, states: List[RecursiveState]):
self.phi = (1 + np.sqrt(5)) / 2
self.states = states
self.period = len(states)
self.center = self.compute_center()
self.lyapunov_exponent = self.compute_lyapunov_exponent()
def compute_center(self) -> RecursiveState:
"""计算周期轨道中心"""
if not self.states:
return None
# 使用平均熵和深度
avg_entropy = np.mean([s.entropy for s in self.states])
avg_depth = int(np.mean([s.depth for s in self.states]))
# 寻找最接近中心的状态作为代表
center_entropy = avg_entropy
best_state = min(self.states,
key=lambda s: abs(s.entropy - center_entropy))
return best_state
def compute_lyapunov_exponent(self) -> float:
"""计算Lyapunov指数"""
if len(self.states) < 2:
return 0.0
# 计算相邻状态间的φ-距离变化
distances = []
for i in range(self.period):
curr = self.states[i]
next_state = self.states[(i + 1) % self.period]
distances.append(curr.phi_distance(next_state))
if not distances or all(d == 0 for d in distances):
return -1.0 # 完全稳定
# Lyapunov指数近似
avg_distance = np.mean(distances)
return np.log(avg_distance) / np.log(self.phi)
class PeriodicStabilityChecker:
"""周期稳定性检查器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def detect_periodic_orbit(self, trajectory: List[RecursiveState],
max_period: int = 20) -> Optional[PeriodicOrbit]:
"""检测周期轨道"""
n = len(trajectory)
if n < 4:
return None
# 检测不同周期长度
for period in range(1, min(max_period + 1, n // 2)):
if self.is_periodic(trajectory, period):
# 提取周期轨道
orbit_states = trajectory[-period:]
return PeriodicOrbit(orbit_states)
return None
def is_periodic(self, trajectory: List[RecursiveState], period: int) -> bool:
"""检查是否存在给定周期的轨道"""
n = len(trajectory)
if n < 2 * period:
return False
# 检查最后两个周期是否匹配
for i in range(period):
state1 = trajectory[n - period + i]
state2 = trajectory[n - 2 * period + i]
# 使用φ-距离判断状态相似性
if state1.phi_distance(state2) > 1.0 / self.phi:
return False
return True
def check_periodic_stability(self, orbit: PeriodicOrbit) -> bool:
"""检查周期稳定性"""
return orbit.lyapunov_exponent < 0
def compute_periodic_stability_score(self, orbit: Optional[PeriodicOrbit]) -> float:
"""计算周期稳定性分数"""
if orbit is None:
return 0.0
# 基于Lyapunov指数的稳定性分数
return max(0.0, np.exp(orbit.lyapunov_exponent))
1.3 结构稳定性判据
class StructuralPattern:
"""结构模式"""
def __init__(self, pattern: str, frequency: int, scale: int):
self.phi = (1 + np.sqrt(5)) / 2
self.pattern = pattern
self.frequency = frequency
self.scale = scale
self.signature = self.compute_signature()
def compute_signature(self) -> float:
"""计算结构签名"""
# 基于模式长度和频率的φ-加权签名
pattern_complexity = len(self.pattern) * np.log2(len(self.pattern) + 1)
frequency_weight = self.frequency / (self.phi ** self.scale)
return pattern_complexity * frequency_weight
class StructuralStabilityChecker:
"""结构稳定性检查器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def extract_patterns(self, state: RecursiveState,
min_length: int = 2, max_length: int = 8) -> List[StructuralPattern]:
"""提取结构模式"""
patterns = []
data = state.data
for length in range(min_length, min(max_length + 1, len(data) + 1)):
pattern_counts = {}
for i in range(len(data) - length + 1):
pattern = data[i:i + length]
if self.is_valid_pattern(pattern):
pattern_counts[pattern] = pattern_counts.get(pattern, 0) + 1
for pattern, frequency in pattern_counts.items():
if frequency > 1: # 只保留重复模式
scale = int(np.log(length) / np.log(self.phi))
patterns.append(StructuralPattern(pattern, frequency, scale))
return patterns
def is_valid_pattern(self, pattern: str) -> bool:
"""检查模式有效性（no-11约束）"""
return "11" not in pattern
def compute_structural_similarity(self, patterns1: List[StructuralPattern],
patterns2: List[StructuralPattern]) -> float:
"""计算结构相似度"""
if not patterns1 and not patterns2:
return 1.0
if not patterns1 or not patterns2:
return 0.0
# 构建模式签名集合
sig1 = {p.pattern: p.signature for p in patterns1}
sig2 = {p.pattern: p.signature for p in patterns2}
all_patterns = set(sig1.keys()) | set(sig2.keys())
if not all_patterns:
return 1.0
# 计算加权相似度
similarity_sum = 0.0
weight_sum = 0.0
for pattern in all_patterns:
s1 = sig1.get(pattern, 0.0)
s2 = sig2.get(pattern, 0.0)
if s1 == 0.0 and s2 == 0.0:
continue
# 使用调和平均计算相似度
if s1 > 0 and s2 > 0:
similarity = 2 * s1 * s2 / (s1 + s2)
weight = s1 + s2
else:
similarity = 0.0
weight = max(s1, s2)
similarity_sum += similarity * weight
weight_sum += weight
return similarity_sum / weight_sum if weight_sum > 0 else 0.0
def check_structural_stability(self, original: RecursiveState,
perturbed: RecursiveState, threshold: float = 0.8) -> bool:
"""检查结构稳定性"""
patterns1 = self.extract_patterns(original)
patterns2 = self.extract_patterns(perturbed)
similarity = self.compute_structural_similarity(patterns1, patterns2)
return similarity >= threshold
def compute_structural_stability_score(self, original: RecursiveState,
perturbed: RecursiveState) -> float:
"""计算结构稳定性分数"""
patterns1 = self.extract_patterns(original)
patterns2 = self.extract_patterns(perturbed)
return self.compute_structural_similarity(patterns1, patterns2)
2. 三重稳定性综合判据
2.1 稳定性指数计算
class RecursiveStabilityAnalyzer:
"""递归稳定性分析器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.depth_checker = DepthStabilityChecker()
self.periodic_checker = PeriodicStabilityChecker()
self.structural_checker = StructuralStabilityChecker()
def compute_phi_stability_index(self, state: RecursiveState,
trajectory: List[RecursiveState],
perturbed_state: Optional[RecursiveState] = None) -> float:
"""计算φ-稳定性指数"""
# 1. 深度稳定性分数
max_depth = self.depth_checker.compute_max_depth(100.0) # 假设最大熵为100
depth_score = self.depth_checker.compute_depth_stability_score(state, max_depth)
# 2. 周期稳定性分数
orbit = self.periodic_checker.detect_periodic_orbit(trajectory)
periodic_score = self.periodic_checker.compute_periodic_stability_score(orbit)
# 3. 结构稳定性分数
if perturbed_state is not None:
structural_score = self.structural_checker.compute_structural_stability_score(
state, perturbed_state)
else:
# 如果没有扰动状态，使用轨道内的结构一致性
structural_score = self.compute_trajectory_structural_consistency(trajectory)
# φ-加权综合稳定性指数
weights = [1.0, self.phi, self.phi**2] # φ-权重
total_weight = sum(weights)
normalized_weights = [w / total_weight for w in weights]
stability_index = (normalized_weights[0] * depth_score +
normalized_weights[1] * periodic_score +
normalized_weights[2] * structural_score)
return min(1.0, max(0.0, stability_index))
def compute_trajectory_structural_consistency(self, trajectory: List[RecursiveState]) -> float:
"""计算轨道结构一致性"""
if len(trajectory) < 2:
return 1.0
consistencies = []
for i in range(len(trajectory) - 1):
consistency = self.structural_checker.compute_structural_stability_score(
trajectory[i], trajectory[i + 1])
consistencies.append(consistency)
return np.mean(consistencies)
def classify_stability(self, stability_index: float) -> StabilityLevel:
"""分类稳定性水平"""
if stability_index >= 0.8:
return StabilityLevel.STABLE
elif stability_index >= 0.6:
return StabilityLevel.MARGINAL
elif stability_index >= 0.4:
return StabilityLevel.CRITICAL
else:
return StabilityLevel.UNSTABLE
def analyze_stability(self, state: RecursiveState,
trajectory: List[RecursiveState],
perturbed_state: Optional[RecursiveState] = None) -> Dict[str, Any]:
"""综合稳定性分析"""
# 计算各项稳定性指标
max_depth = self.depth_checker.compute_max_depth(100.0)
depth_stable = self.depth_checker.check_depth_stability(state, max_depth)
depth_score = self.depth_checker.compute_depth_stability_score(state, max_depth)
orbit = self.periodic_checker.detect_periodic_orbit(trajectory)
periodic_stable = orbit is not None and self.periodic_checker.check_periodic_stability(orbit)
periodic_score = self.periodic_checker.compute_periodic_stability_score(orbit)
if perturbed_state is not None:
structural_stable = self.structural_checker.check_structural_stability(
state, perturbed_state)
structural_score = self.structural_checker.compute_structural_stability_score(
state, perturbed_state)
else:
structural_score = self.compute_trajectory_structural_consistency(trajectory)
structural_stable = structural_score >= 0.8
# 综合稳定性指数
stability_index = self.compute_phi_stability_index(state, trajectory, perturbed_state)
stability_level = self.classify_stability(stability_index)
return {
'stability_index': stability_index,
'stability_level': stability_level,
'depth_stable': depth_stable,
'depth_score': depth_score,
'periodic_stable': periodic_stable,
'periodic_score': periodic_score,
'structural_stable': structural_stable,
'structural_score': structural_score,
'orbit': orbit,
'max_depth': max_depth,
'current_depth': state.depth,
'recommendations': self.generate_recommendations(
depth_stable, periodic_stable, structural_stable, stability_index)
}
def generate_recommendations(self, depth_stable: bool, periodic_stable: bool,
structural_stable: bool, stability_index: float) -> List[str]:
"""生成稳定性改进建议"""
recommendations = []
if not depth_stable:
recommendations.append("降低递归深度，避免过度复杂化")
if not periodic_stable:
recommendations.append("寻找稳定的周期轨道，改进动力学参数")
if not structural_stable:
recommendations.append("增强结构鲁棒性，保持关键模式不变")
if stability_index < 0.6:
recommendations.append("系统处于不稳定状态，需要全面重构")
elif stability_index < 0.8:
recommendations.append("系统处于边缘稳定，建议优化关键参数")
if not recommendations:
recommendations.append("系统稳定性良好，保持当前配置")
return recommendations
3. 稳定性验证与测试框架
3.1 测试用例生成
class StabilityTestSuite:
"""稳定性测试套件"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.analyzer = RecursiveStabilityAnalyzer()
def generate_fibonacci_trajectory(self, length: int) -> List[RecursiveState]:
"""生成Fibonacci轨道测试用例"""
trajectory = []
# Fibonacci序列
fib = [1, 1]
while len(fib) < length:
fib.append(fib[-1] + fib[-2])
for i, f in enumerate(fib):
# 转换为二进制
binary = bin(f)[2:]
# 确保no-11约束
binary = self.enforce_no11_constraint(binary)
entropy = self.compute_entropy(binary)
depth = int(np.log(i + 2) / np.log(self.phi))
state = RecursiveState(binary, entropy, depth)
trajectory.append(state)
return trajectory
def enforce_no11_constraint(self, binary: str) -> str:
"""强制no-11约束"""
result = ""
i = 0
while i < len(binary):
if i < len(binary) - 1 and binary[i:i+2] == "11":
# 替换11为101
result += "101"
i += 2
else:
result += binary[i]
i += 1
return result
def compute_entropy(self, binary: str) -> float:
"""计算二进制串熵"""
if not binary:
return 0.0
counts = {'0': 0, '1': 0}
for bit in binary:
counts[bit] += 1
total = len(binary)
entropy = 0.0
for count in counts.values():
if count > 0:
p = count / total
entropy -= p * np.log2(p)
return entropy * total # 总熵
def generate_periodic_trajectory(self, period: int, cycles: int) -> List[RecursiveState]:
"""生成周期轨道测试用例"""
# 基础周期模式
base_patterns = ["10", "101", "1010", "10101"]
pattern = base_patterns[period % len(base_patterns)]
trajectory = []
for cycle in range(cycles):
for phase in range(period):
# 生成周期状态
data = pattern * ((phase + 1) % 3 + 1) # 变化长度
data = self.enforce_no11_constraint(data)
entropy = self.compute_entropy(data)
depth = phase + 1
state = RecursiveState(data, entropy, depth)
trajectory.append(state)
return trajectory
def generate_unstable_trajectory(self, length: int) -> List[RecursiveState]:
"""生成不稳定轨道测试用例"""
trajectory = []
for i in range(length):
# 生成随机不稳定模式
data_length = min(50, i + 5) # 长度快速增长
data = ""
for j in range(data_length):
# 高概率生成"11"模式（违反约束）
if np.random.random() < 0.4:
data += "11"
else:
data += "10"
# 部分强制约束（模拟不完全约束）
if i % 3 == 0:
data = self.enforce_no11_constraint(data)
entropy = self.compute_entropy(data) + i * 2 # 熵快速增长
depth = i + 1 # 深度线性增长
state = RecursiveState(data, entropy, depth)
trajectory.append(state)
return trajectory
def run_stability_test(self, trajectory: List[RecursiveState],
test_name: str) -> Dict[str, Any]:
"""运行稳定性测试"""
if not trajectory:
return {'test_name': test_name, 'error': 'Empty trajectory'}
# 分析最后状态的稳定性
final_state = trajectory[-1]
# 生成扰动状态（用于结构稳定性测试）
perturbed_state = self.generate_perturbation(final_state)
# 综合分析
analysis = self.analyzer.analyze_stability(final_state, trajectory, perturbed_state)
analysis['test_name'] = test_name
analysis['trajectory_length'] = len(trajectory)
return analysis
def generate_perturbation(self, state: RecursiveState,
perturbation_strength: float = 0.1) -> RecursiveState:
"""生成扰动状态"""
data = state.data
perturbed_data = ""
for bit in data:
if np.random.random() < perturbation_strength:
# 翻转比特
perturbed_data += '0' if bit == '1' else '1'
else:
perturbed_data += bit
# 确保约束
perturbed_data = self.enforce_no11_constraint(perturbed_data)
perturbed_entropy = self.compute_entropy(perturbed_data)
perturbed_depth = state.depth # 深度保持不变
return RecursiveState(perturbed_data, perturbed_entropy, perturbed_depth)
def run_comprehensive_test_suite(self) -> Dict[str, Any]:
"""运行综合测试套件"""
results = {}
# 测试1：Fibonacci稳定性
fib_trajectory = self.generate_fibonacci_trajectory(20)
results['fibonacci'] = self.run_stability_test(fib_trajectory, "Fibonacci")
# 测试2：周期轨道稳定性
periodic_trajectory = self.generate_periodic_trajectory(3, 5)
results['periodic'] = self.run_stability_test(periodic_trajectory, "Periodic")
# 测试3：不稳定轨道
unstable_trajectory = self.generate_unstable_trajectory(15)
results['unstable'] = self.run_stability_test(unstable_trajectory, "Unstable")
# 综合统计
stable_count = sum(1 for r in results.values()
if r.get('stability_level') == StabilityLevel.STABLE)
total_tests = len(results)
results['summary'] = {
'total_tests': total_tests,
'stable_count': stable_count,
'stability_rate': stable_count / total_tests if total_tests > 0 else 0,
'test_passed': stable_count >= total_tests * 0.6 # 60%通过率
}
return results
4. 实际应用接口
4.1 系统稳定性监控
class RecursiveSystemMonitor:
"""递归系统稳定性监控器"""
def __init__(self, monitoring_window: int = 100):
self.phi = (1 + np.sqrt(5)) / 2
self.analyzer = RecursiveStabilityAnalyzer()
self.window_size = monitoring_window
self.history = []
self.alerts = []
def add_state(self, state: RecursiveState) -> None:
"""添加新状态到监控历史"""
self.history.append(state)
# 保持监控窗口大小
if len(self.history) > self.window_size:
self.history.pop(0)
# 检查稳定性
if len(self.history) >= 10: # 最少需要10个状态
self.check_stability_alerts()
def check_stability_alerts(self) -> None:
"""检查稳定性警报"""
current_state = self.history[-1]
recent_trajectory = self.history[-20:] if len(self.history) >= 20 else self.history
analysis = self.analyzer.analyze_stability(current_state, recent_trajectory)
# 生成警报
if analysis['stability_level'] == StabilityLevel.UNSTABLE:
self.alerts.append({
'timestamp': len(self.history),
'level': 'CRITICAL',
'message': f"系统不稳定，稳定性指数: {analysis['stability_index']:.3f}",
'recommendations': analysis['recommendations']
})
elif analysis['stability_level'] == StabilityLevel.CRITICAL:
self.alerts.append({
'timestamp': len(self.history),
'level': 'WARNING',
'message': f"系统接近不稳定，稳定性指数: {analysis['stability_index']:.3f}",
'recommendations': analysis['recommendations']
})
def get_current_stability_report(self) -> Dict[str, Any]:
"""获取当前稳定性报告"""
if len(self.history) < 5:
return {'error': 'Insufficient data for stability analysis'}
current_state = self.history[-1]
analysis = self.analyzer.analyze_stability(current_state, self.history)
return {
'current_stability': analysis,
'recent_alerts': self.alerts[-5:], # 最近5个警报
'trend': self.compute_stability_trend(),
'monitoring_status': 'active',
'history_length': len(self.history)
}
def compute_stability_trend(self) -> str:
"""计算稳定性趋势"""
if len(self.history) < 10:
return 'insufficient_data'
# 计算最近几个状态的稳定性指数
recent_indices = []
for i in range(max(1, len(self.history) - 10), len(self.history)):
state = self.history[i]
trajectory = self.history[:i+1]
index = self.analyzer.compute_phi_stability_index(state, trajectory)
recent_indices.append(index)
if len(recent_indices) < 2:
return 'stable'
# 线性回归判断趋势
x = np.arange(len(recent_indices))
slope = np.polyfit(x, recent_indices, 1)[0]
if slope > 0.01:
return 'improving'
elif slope < -0.01:
return 'deteriorating'
else:
return 'stable'
这个形式化规范提供了T10-4递归稳定性定理的完整实现，包括三重稳定性判据的计算、综合分析、测试验证和实际应用接口。