T10-1 递归深度定理 - 形式化描述
1. 形式化框架
1.1 递归深度计算系统
class RecursiveDepthSystem:
"""递归深度定理的数学模型"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.fibonacci = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597]
def calculate_recursive_depth(self, binary_string: str) -> int:
"""计算二进制串的递归深度"""
if not binary_string or not self.verify_no11_constraint(binary_string):
return 0
# 计算系统熵
entropy = self.calculate_system_entropy(binary_string)
# φ-量化递归深度公式
depth = int(np.floor(np.log(entropy + 1) / np.log(self.phi)))
return max(0, depth)
def calculate_system_entropy(self, binary_string: str) -> float:
"""计算系统熵 H(S)"""
if not binary_string:
return 0
# Shannon熵计算
char_counts = {}
for char in binary_string:
char_counts[char] = char_counts.get(char, 0) + 1
total_chars = len(binary_string)
shannon_entropy = 0
for count in char_counts.values():
p = count / total_chars
shannon_entropy -= p * np.log2(p)
# φ-权重熵修正
phi_entropy = 0
for i, char in enumerate(binary_string):
if char == '1':
phi_entropy += 1 / (self.phi ** i)
# 组合熵
combined_entropy = shannon_entropy + phi_entropy * np.log2(self.phi)
return combined_entropy
def verify_no11_constraint(self, binary_str: str) -> bool:
"""验证no-11约束"""
return '11' not in binary_str
def generate_recursive_sequence(self, initial_state: str, max_depth: int) -> List[str]:
"""生成递归序列 S_0, Ξ[S_0], Ξ^2[S_0], ..."""
sequence = [initial_state]
current_state = initial_state
for depth in range(1, max_depth + 1):
# 应用collapse算子Ξ
next_state = self.apply_collapse_operator(current_state)
# 验证熵增条件
current_entropy = self.calculate_system_entropy(current_state)
next_entropy = self.calculate_system_entropy(next_state)
if next_entropy <= current_entropy:
break # 熵增条件不满足，停止递归
sequence.append(next_state)
current_state = next_state
return sequence
def apply_collapse_operator(self, state: str) -> str:
"""应用collapse算子Ξ"""
if not state:
return "0"
# Ξ算子的具体实现：自指变换
# 规则1：每个'1'后面添加其位置的φ-表示
result = ""
for i, char in enumerate(state):
result += char
if char == '1':
phi_pos = self.fibonacci_position_encoding(i + 1)
result += phi_pos
# 应用no-11约束
result = self.enforce_no11_constraint(result)
return result
def fibonacci_position_encoding(self, position: int) -> str:
"""将位置编码为Fibonacci表示"""
if position == 0:
return ""
# 使用贪心算法进行Zeckendorf分解
representation = []
remaining = position
for i in range(len(self.fibonacci) - 1, -1, -1):
if self.fibonacci[i] <= remaining:
representation.append(i)
remaining -= self.fibonacci[i]
# 转换为二进制形式
if not representation:
return "0"
max_index = max(representation)
binary = ['0'] * (max_index + 1)
for idx in representation:
binary[idx] = '1'
return ''.join(reversed(binary))
def enforce_no11_constraint(self, binary_str: str) -> str:
"""强制执行no-11约束"""
result = ""
i = 0
while i < len(binary_str):
if i < len(binary_str) - 1 and binary_str[i] == '1' and binary_str[i+1] == '1':
# 将"11"替换为"10"
result += "10"
i += 2
else:
result += binary_str[i]
i += 1
return result
def calculate_max_depth_bound(self, max_string_length: int) -> int:
"""计算最大深度界限"""
max_entropy = max_string_length * np.log2(2) # 最大可能熵
max_depth = int(np.floor(np.log(2**max_string_length + 1) / np.log(self.phi)))
return max_depth
1.2 深度分层系统
class DepthStratificationSystem:
"""递归深度分层的数学模型"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def partition_by_depth(self, state_space: List[str]) -> Dict[int, List[str]]:
"""将状态空间按递归深度分层"""
depth_system = RecursiveDepthSystem()
partitions = {}
for state in state_space:
depth = depth_system.calculate_recursive_depth(state)
if depth not in partitions:
partitions[depth] = []
partitions[depth].append(state)
return partitions
def verify_depth_transition_rule(self, state1: str, state2: str) -> bool:
"""验证深度跃迁规律：R(S_{t+1}) ∈ {R(S_t), R(S_t) + 1}"""
depth_system = RecursiveDepthSystem()
depth1 = depth_system.calculate_recursive_depth(state1)
depth2 = depth_system.calculate_recursive_depth(state2)
# 允许的跃迁：保持不变或增加1
return depth2 in [depth1, depth1 + 1]
def calculate_layer_entropy(self, layer: List[str]) -> float:
"""计算某一深度层级的平均熵"""
if not layer:
return 0
depth_system = RecursiveDepthSystem()
total_entropy = 0
for state in layer:
entropy = depth_system.calculate_system_entropy(state)
total_entropy += entropy
return total_entropy / len(layer)
def verify_layer_separation(self, partitions: Dict[int, List[str]]) -> bool:
"""验证层级间的清晰分离"""
depth_system = RecursiveDepthSystem()
for depth1, layer1 in partitions.items():
for depth2, layer2 in partitions.items():
if depth1 == depth2:
continue
# 验证不同深度层级间的状态确实有不同深度
for state1 in layer1:
calculated_depth1 = depth_system.calculate_recursive_depth(state1)
if calculated_depth1 != depth1:
return False
for state2 in layer2:
calculated_depth2 = depth_system.calculate_recursive_depth(state2)
if calculated_depth2 != depth2:
return False
return True
2. 深度不变性验证
2.1 φ-变换不变性
class DepthInvarianceVerifier:
"""验证递归深度的不变性质"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def phi_transform(self, binary_string: str) -> str:
"""应用φ-表示变换"""
if not binary_string:
return "0"
# φ-变换：每个位置的权重乘以φ
transformed = ""
for i, char in enumerate(binary_string):
if char == '1':
# 位置i的φ-权重增强
weight = int(self.phi ** (i + 1)) % 2
transformed += str(weight)
else:
transformed += char
# 确保no-11约束
return self.enforce_no11_constraint(transformed)
def verify_depth_invariance_under_transform(self, original_state: str) -> bool:
"""验证φ-变换下的深度不变性"""
depth_system = RecursiveDepthSystem()
original_depth = depth_system.calculate_recursive_depth(original_state)
transformed_state = self.phi_transform(original_state)
transformed_depth = depth_system.calculate_recursive_depth(transformed_state)
# 在适当的常数修正下，深度应该保持相对关系
depth_difference = abs(transformed_depth - original_depth)
# 允许1个单位的深度变化（由于离散化效应）
return depth_difference <= 1
def enforce_no11_constraint(self, binary_str: str) -> str:
"""强制执行no-11约束"""
result = ""
i = 0
while i < len(binary_str):
if i < len(binary_str) - 1 and binary_str[i] == '1' and binary_str[i+1] == '1':
result += "10"
i += 2
else:
result += binary_str[i]
i += 1
return result
3. 应用验证系统
3.1 Fibonacci序列递归深度验证
class FibonacciDepthVerifier:
"""验证Fibonacci序列的递归深度规律"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.fibonacci = [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987]
def verify_fibonacci_depth_pattern(self, max_index: int) -> Dict[int, Dict[str, float]]:
"""验证Fibonacci数列的递归深度模式"""
depth_system = RecursiveDepthSystem()
results = {}
for i in range(1, min(max_index + 1, len(self.fibonacci))):
fib_num = self.fibonacci[i]
fib_binary = bin(fib_num)[2:] # 转换为二进制（去掉'0b'前缀）
# 确保满足no-11约束
fib_binary = depth_system.enforce_no11_constraint(fib_binary)
calculated_depth = depth_system.calculate_recursive_depth(fib_binary)
theoretical_depth = int(np.floor(np.log(depth_system.calculate_system_entropy(fib_binary) + 1) / np.log(self.phi)))
results[i] = {
'fibonacci_number': fib_num,
'binary_representation': fib_binary,
'calculated_depth': calculated_depth,
'theoretical_depth': theoretical_depth,
'depth_match': calculated_depth == theoretical_depth
}
return results
def verify_depth_growth_pattern(self) -> Dict[str, float]:
"""验证深度增长模式"""
depth_system = RecursiveDepthSystem()
# 计算一系列Fibonacci数的深度
depths = []
for i in range(1, min(10, len(self.fibonacci))):
fib_num = self.fibonacci[i]
binary_repr = bin(fib_num)[2:]
binary_repr = depth_system.enforce_no11_constraint(binary_repr)
depth = depth_system.calculate_recursive_depth(binary_repr)
depths.append(depth)
# 分析增长模式
growth_ratios = []
for i in range(1, len(depths)):
if depths[i-1] > 0:
ratio = depths[i] / depths[i-1]
growth_ratios.append(ratio)
avg_growth_ratio = np.mean(growth_ratios) if growth_ratios else 1.0
theoretical_ratio = self.phi
return {
'depths': depths,
'average_growth_ratio': avg_growth_ratio,
'theoretical_phi_ratio': theoretical_ratio,
'ratio_consistency': abs(avg_growth_ratio - theoretical_ratio) / theoretical_ratio
}
3.2 二进制模式递归深度验证
class BinaryPatternDepthVerifier:
"""验证二进制模式的递归深度"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def verify_pattern_depth_scaling(self, base_pattern: str, repetitions: List[int]) -> Dict[int, Dict[str, any]]:
"""验证模式重复时的深度缩放"""
depth_system = RecursiveDepthSystem()
results = {}
for rep in repetitions:
# 生成重复模式
repeated_pattern = (base_pattern * rep)[:50] # 限制长度避免计算复杂度过高
# 确保no-11约束
repeated_pattern = depth_system.enforce_no11_constraint(repeated_pattern)
depth = depth_system.calculate_recursive_depth(repeated_pattern)
entropy = depth_system.calculate_system_entropy(repeated_pattern)
results[rep] = {
'pattern': repeated_pattern,
'depth': depth,
'entropy': entropy,
'pattern_length': len(repeated_pattern)
}
return results
def verify_hierarchical_structure(self, test_patterns: List[str]) -> Dict[str, Dict[str, any]]:
"""验证层级结构"""
depth_system = RecursiveDepthSystem()
stratification = DepthStratificationSystem()
# 按深度分层
partitions = stratification.partition_by_depth(test_patterns)
results = {}
for depth, layer in partitions.items():
layer_entropy = stratification.calculate_layer_entropy(layer)
layer_size = len(layer)
results[f'depth_{depth}'] = {
'layer_size': layer_size,
'average_entropy': layer_entropy,
'patterns': layer[:5] # 显示前5个模式作为示例
}
return results
4. 综合验证系统
4.1 完整验证框架
class ComprehensiveDepthVerifier:
"""T10-1递归深度定理的综合验证"""
def __init__(self):
self.depth_system = RecursiveDepthSystem()
self.stratification = DepthStratificationSystem()
self.invariance_verifier = DepthInvarianceVerifier()
self.fibonacci_verifier = FibonacciDepthVerifier()
self.pattern_verifier = BinaryPatternDepthVerifier()
def run_comprehensive_verification(self, test_cases: List[str]) -> Dict[str, any]:
"""运行全面的验证测试"""
results = {
'basic_depth_calculation': {},
'depth_stratification': {},
'invariance_properties': {},
'fibonacci_patterns': {},
'binary_patterns': {},
'theoretical_consistency': {}
}
# 1. 基础深度计算验证
for i, test_case in enumerate(test_cases):
depth = self.depth_system.calculate_recursive_depth(test_case)
entropy = self.depth_system.calculate_system_entropy(test_case)
results['basic_depth_calculation'][f'case_{i}'] = {
'input': test_case,
'depth': depth,
'entropy': entropy,
'satisfies_constraint': self.depth_system.verify_no11_constraint(test_case)
}
# 2. 深度分层验证
partitions = self.stratification.partition_by_depth(test_cases)
results['depth_stratification'] = {
'partitions': {k: len(v) for k, v in partitions.items()},
'layer_separation': self.stratification.verify_layer_separation(partitions)
}
# 3. 不变性验证
invariance_results = []
for test_case in test_cases[:5]: # 测试前5个案例
invariant = self.invariance_verifier.verify_depth_invariance_under_transform(test_case)
invariance_results.append(invariant)
results['invariance_properties'] = {
'phi_transform_invariance': invariance_results,
'invariance_rate': sum(invariance_results) / len(invariance_results)
}
# 4. Fibonacci模式验证
results['fibonacci_patterns'] = self.fibonacci_verifier.verify_fibonacci_depth_pattern(8)
# 5. 二进制模式验证
binary_patterns = ['10', '101', '1010', '10100']
results['binary_patterns'] = self.pattern_verifier.verify_hierarchical_structure(binary_patterns)
# 6. 理论一致性检验
theoretical_consistency = self.verify_theoretical_consistency(test_cases)
results['theoretical_consistency'] = theoretical_consistency
return results
def verify_theoretical_consistency(self, test_cases: List[str]) -> Dict[str, any]:
"""验证理论一致性"""
phi_quantization_matches = 0
depth_transition_valid = 0
max_depth_bounds = []
for test_case in test_cases:
# 验证φ-量化公式
depth = self.depth_system.calculate_recursive_depth(test_case)
entropy = self.depth_system.calculate_system_entropy(test_case)
theoretical_depth = int(np.floor(np.log(entropy + 1) / np.log(self.depth_system.phi)))
if depth == theoretical_depth:
phi_quantization_matches += 1
# 验证深度界限
max_bound = self.depth_system.calculate_max_depth_bound(len(test_case))
max_depth_bounds.append(depth <= max_bound)
# 生成递归序列并验证跃迁规律
for test_case in test_cases[:3]: # 限制计算量
sequence = self.depth_system.generate_recursive_sequence(test_case, 3)
for i in range(len(sequence) - 1):
valid_transition = self.stratification.verify_depth_transition_rule(sequence[i], sequence[i+1])
if valid_transition:
depth_transition_valid += 1
return {
'phi_quantization_accuracy': phi_quantization_matches / len(test_cases),
'depth_transition_validity': depth_transition_valid / max(1, 3 * len(test_cases[:3])),
'max_depth_bound_satisfaction': sum(max_depth_bounds) / len(max_depth_bounds),
'overall_consistency': (
(phi_quantization_matches / len(test_cases)) * 0.4 +
(depth_transition_valid / max(1, 3 * len(test_cases[:3]))) * 0.3 +
(sum(max_depth_bounds) / len(max_depth_bounds)) * 0.3
)
}
5. 总结
本形式化框架提供了：
- 完整的递归深度计算系统：严格实现φ-量化公式
- 深度分层机制：验证状态空间的层级结构
- 不变性验证：确认φ-变换下的深度稳定性
- 应用案例验证：Fibonacci序列和二进制模式的深度规律
- 综合验证框架：全面的理论一致性检验
这为T10-1递归深度定理提供了严格的数学基础和可验证的实现，确保理论与形式化描述的完全一致性。