P6 尺度不变性命题 - 形式化描述
1. 形式化框架
1.1 尺度变换系统
class ScaleInvariantSystem:
"""尺度不变性系统的数学模型"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.fibonacci = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144]
def phi_representation(self, n: int) -> str:
"""将自然数n表示为φ-表示（Zeckendorf表示）"""
if n == 0:
return "0"
# 使用贪心算法生成Zeckendorf表示
representation = []
remaining = n
# 从最大的Fibonacci数开始
for i in range(len(self.fibonacci) - 1, -1, -1):
if self.fibonacci[i] <= remaining:
representation.append(i)
remaining -= self.fibonacci[i]
# 转换为二进制字符串
if not representation:
return "1"
max_index = max(representation)
binary = ['0'] * (max_index + 1)
for idx in representation:
binary[idx] = '1'
return ''.join(reversed(binary))
def verify_no11_constraint(self, binary_str: str) -> bool:
"""验证φ-表示满足no-11约束"""
return '11' not in binary_str
def scale_transform(self, binary_str: str, scale_factor: int) -> str:
"""对二进制串进行尺度变换"""
if scale_factor <= 0:
return binary_str
# 每个比特重复scale_factor次
scaled = ""
for bit in binary_str:
scaled += bit * scale_factor
return scaled
def calculate_phi_complexity(self, binary_str: str) -> float:
"""计算φ-表示的复杂度"""
complexity = 0
for i, bit in enumerate(binary_str):
if bit == '1':
# 位置权重基于φ的幂次
weight = self.phi ** i
complexity += weight
return complexity
1.2 分形生成器
class PhiFractal:
"""φ-分形的生成与分析"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def generate_phi_fractal(self, depth: int, base_pattern: str = "10") -> List[str]:
"""生成φ-分形序列"""
if depth <= 0:
return [base_pattern]
patterns = [base_pattern]
for level in range(1, depth + 1):
new_patterns = []
for pattern in patterns:
# φ-分形递归规则: L_n = L_{n-1} + L_{n-2}
if level == 1:
# F_1 = "1", F_0 = "0"
new_pattern = pattern + "0"
else:
# F_n = F_{n-1} + F_{n-2}
if len(patterns) >= 2:
new_pattern = patterns[-1] + patterns[-2]
else:
new_pattern = pattern + pattern[::-1]
# 确保满足no-11约束
new_pattern = self._enforce_no11_constraint(new_pattern)
new_patterns.append(new_pattern)
patterns.extend(new_patterns)
return patterns
def _enforce_no11_constraint(self, pattern: str) -> str:
"""强制执行no-11约束"""
result = ""
i = 0
while i < len(pattern):
if i < len(pattern) - 1 and pattern[i] == '1' and pattern[i+1] == '1':
# 遇到"11"，替换为"10"
result += "10"
i += 2
else:
result += pattern[i]
i += 1
return result
def calculate_fractal_dimension(self, patterns: List[str]) -> float:
"""计算分形维数"""
if len(patterns) < 2:
return 1.0
# 计算长度比例
lengths = [len(p) for p in patterns]
# 分形维数基于增长率
if len(lengths) >= 3:
# 使用φ-比例计算维数
ratio = lengths[-1] / lengths[-2] if lengths[-2] > 0 else self.phi
dimension = np.log(ratio) / np.log(self.phi)
else:
# 默认φ-分形维数
dimension = np.log(self.phi + 1) / np.log(self.phi)
return dimension
def measure_self_similarity(self, pattern: str, scale: int) -> float:
"""测量自相似性"""
if scale <= 1 or len(pattern) < scale:
return 1.0
# 将模式分割成scale个部分
segment_length = len(pattern) // scale
segments = []
for i in range(scale):
start = i * segment_length
end = start + segment_length
if end <= len(pattern):
segments.append(pattern[start:end])
if len(segments) < 2:
return 0.0
# 计算段之间的相似性
similarities = []
for i in range(len(segments) - 1):
similarity = self._string_similarity(segments[i], segments[i+1])
similarities.append(similarity)
return np.mean(similarities)
def _string_similarity(self, s1: str, s2: str) -> float:
"""计算两个字符串的相似度"""
if len(s1) != len(s2):
min_len = min(len(s1), len(s2))
s1, s2 = s1[:min_len], s2[:min_len]
if len(s1) == 0:
return 1.0
matches = sum(1 for a, b in zip(s1, s2) if a == b)
return matches / len(s1)
2. 尺度不变性验证
2.1 结构保持验证器
class StructurePreservationVerifier:
"""验证结构在尺度变换下的保持性"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def verify_scale_invariance(self, pattern: str, scales: List[int]) -> Dict[str, float]:
"""验证模式在不同尺度下的不变性"""
results = {}
# 原始复杂度
original_complexity = self._calculate_structural_complexity(pattern)
for scale in scales:
scaled_pattern = self._apply_scale_transform(pattern, scale)
scaled_complexity = self._calculate_structural_complexity(scaled_pattern)
# 归一化复杂度（考虑尺度因子）
normalized_complexity = scaled_complexity / (scale ** self._get_scaling_dimension())
# 计算不变性度量
invariance_measure = 1 - abs(normalized_complexity - original_complexity) / original_complexity
results[f"scale_{scale}"] = max(0, invariance_measure)
return results
def _calculate_structural_complexity(self, pattern: str) -> float:
"""计算结构复杂度"""
if not pattern:
return 0
complexity = 0
# 信息熵贡献
char_counts = {}
for char in pattern:
char_counts[char] = char_counts.get(char, 0) + 1
total_chars = len(pattern)
entropy = 0
for count in char_counts.values():
p = count / total_chars
entropy -= p * np.log2(p)
complexity += entropy
# 模式复杂度贡献
transitions = 0
for i in range(len(pattern) - 1):
if pattern[i] != pattern[i+1]:
transitions += 1
pattern_complexity = transitions / len(pattern) if len(pattern) > 1 else 0
complexity += pattern_complexity
# φ-权重
phi_weight = 0
for i, char in enumerate(pattern):
if char == '1':
phi_weight += self.phi ** (-i)
complexity += phi_weight / len(pattern)
return complexity
def _apply_scale_transform(self, pattern: str, scale: int) -> str:
"""应用尺度变换"""
if scale <= 0:
return pattern
# 基本重复变换
scaled = ""
for char in pattern:
scaled += char * scale
# 应用φ-修正以保持no-11约束
scaled = self._apply_phi_correction(scaled)
return scaled
def _apply_phi_correction(self, pattern: str) -> str:
"""应用φ-修正以维持约束"""
corrected = ""
i = 0
while i < len(pattern):
if i < len(pattern) - 1 and pattern[i] == '1' and pattern[i+1] == '1':
# 插入φ-分隔符
corrected += "1" + "0" * int(self.phi) + "1"
i += 2
else:
corrected += pattern[i]
i += 1
return corrected
def _get_scaling_dimension(self) -> float:
"""获取尺度维数"""
# 基于φ的理论尺度维数
return np.log(self.phi + 1) / np.log(self.phi)
3. 信息密度分析
3.1 密度不变性分析器
class InformationDensityAnalyzer:
"""分析信息密度的尺度不变性"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def calculate_information_density(self, pattern: str) -> float:
"""计算信息密度"""
if not pattern:
return 0
# Shannon信息量
char_counts = {}
for char in pattern:
char_counts[char] = char_counts.get(char, 0) + 1
total_chars = len(pattern)
shannon_info = 0
for count in char_counts.values():
p = count / total_chars
shannon_info -= p * np.log2(p)
# φ-权重信息量
phi_info = 0
for i, char in enumerate(pattern):
if char == '1':
phi_info += 1 / (self.phi ** i)
# 归一化密度
density = (shannon_info + phi_info) / len(pattern)
return density
def verify_density_invariance(self, base_pattern: str, scales: List[int]) -> Dict[str, float]:
"""验证密度在尺度变换下的不变性"""
base_density = self.calculate_information_density(base_pattern)
results = {}
for scale in scales:
scaled_pattern = self._scale_pattern(base_pattern, scale)
scaled_density = self.calculate_information_density(scaled_pattern)
# 密度比率（应该接近1表示不变性）
if base_density > 0:
density_ratio = scaled_density / base_density
else:
density_ratio = 1 if scaled_density == 0 else float('inf')
results[f"scale_{scale}"] = density_ratio
return results
def _scale_pattern(self, pattern: str, scale: int) -> str:
"""缩放模式同时保持约束"""
if scale <= 0:
return pattern
# 智能缩放：保持φ-结构
scaled = ""
for i, char in enumerate(pattern):
if char == '1':
# 1的缩放：考虑φ-比例
scaled += '1' + '0' * (scale - 1)
else:
# 0的缩放：直接重复
scaled += '0' * scale
# 应用no-11约束
return self._ensure_no11_constraint(scaled)
def _ensure_no11_constraint(self, pattern: str) -> str:
"""确保no-11约束满足"""
result = ""
prev_char = ""
for char in pattern:
if prev_char == '1' and char == '1':
# 插入分隔符
if len(result) > 0:
result = result[:-1] + '10'
char = '0' # 当前字符改为0
result += char
prev_char = char
return result
4. 分形维数计算
4.1 维数计算器
class FractalDimensionCalculator:
"""计算φ-分形的各种维数"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def hausdorff_dimension(self, pattern_sequence: List[str]) -> float:
"""计算Hausdorff维数"""
if len(pattern_sequence) < 2:
return 1.0
# 计算长度序列
lengths = [len(p) for p in pattern_sequence]
# 拟合幂律关系 L_n ~ φ^(d*n)
if len(lengths) >= 3:
# 使用对数回归拟合维数
n_points = np.arange(len(lengths))
log_lengths = np.log(lengths)
# 线性拟合 log(L) = d*log(φ)*n + C
if len(n_points) > 1:
slope, _ = np.polyfit(n_points, log_lengths, 1)
dimension = slope / np.log(self.phi)
else:
dimension = 1.0
else:
# 理论维数
dimension = np.log(self.phi + 1) / np.log(self.phi)
return max(0, dimension)
def box_counting_dimension(self, pattern: str, box_sizes: List[int]) -> float:
"""使用盒计数法计算维数"""
if not pattern or not box_sizes:
return 1.0
counts = []
for box_size in box_sizes:
if box_size >= len(pattern):
counts.append(1)
continue
# 计算需要多少个盒子覆盖模式
boxes_needed = 0
for i in range(0, len(pattern), box_size):
box_content = pattern[i:i+box_size]
if '1' in box_content: # 盒子包含信息
boxes_needed += 1
counts.append(boxes_needed)
# 拟合 N(r) ~ r^(-d)
if len(box_sizes) >= 2 and len(counts) >= 2:
log_sizes = np.log(box_sizes)
log_counts = np.log([max(1, c) for c in counts])
# 线性拟合
slope, _ = np.polyfit(log_sizes, log_counts, 1)
dimension = -slope # 负号因为 N(r) ~ r^(-d)
else:
dimension = 1.0
return max(0, dimension)
def information_dimension(self, pattern: str, scales: List[int]) -> float:
"""计算信息维数"""
if not pattern or not scales:
return 1.0
entropies = []
for scale in scales:
# 在给定尺度下计算信息熵
scaled_entropy = self._calculate_scaled_entropy(pattern, scale)
entropies.append(scaled_entropy)
# 信息维数 D_I = lim_{r->0} I(r) / log(r)
if len(scales) >= 2 and len(entropies) >= 2:
log_scales = np.log(scales)
# 拟合线性关系
slope, _ = np.polyfit(log_scales, entropies, 1)
dimension = slope
else:
dimension = 1.0
return max(0, dimension)
def _calculate_scaled_entropy(self, pattern: str, scale: int) -> float:
"""计算给定尺度下的熵"""
if scale >= len(pattern):
return 0
# 将模式分割成scale大小的块
blocks = {}
for i in range(0, len(pattern) - scale + 1):
block = pattern[i:i+scale]
blocks[block] = blocks.get(block, 0) + 1
# 计算Shannon熵
total_blocks = sum(blocks.values())
entropy = 0
for count in blocks.values():
p = count / total_blocks
entropy -= p * np.log(p)
return entropy
5. 验证系统
5.1 尺度不变性验证器
class ScaleInvarianceVerifier:
"""完整的尺度不变性验证系统"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.fractal_gen = PhiFractal()
self.structure_verifier = StructurePreservationVerifier()
self.density_analyzer = InformationDensityAnalyzer()
self.dimension_calc = FractalDimensionCalculator()
def comprehensive_verification(self, test_patterns: List[str],
scales: List[int]) -> Dict[str, Dict[str, float]]:
"""全面验证尺度不变性"""
results = {}
for i, pattern in enumerate(test_patterns):
pattern_results = {}
# 1. 结构保持性
structure_results = self.structure_verifier.verify_scale_invariance(pattern, scales)
pattern_results['structure_preservation'] = np.mean(list(structure_results.values()))
# 2. 信息密度不变性
density_results = self.density_analyzer.verify_density_invariance(pattern, scales)
density_variations = [abs(1 - ratio) for ratio in density_results.values()]
pattern_results['density_invariance'] = 1 - np.mean(density_variations)
# 3. 分形维数稳定性
pattern_sequence = self.fractal_gen.generate_phi_fractal(len(scales), pattern)
hausdorff_dim = self.dimension_calc.hausdorff_dimension(pattern_sequence)
theoretical_dim = np.log(self.phi + 1) / np.log(self.phi)
pattern_results['dimension_consistency'] = 1 - abs(hausdorff_dim - theoretical_dim) / theoretical_dim
# 4. no-11约束保持
constraint_preserved = all(
self._verify_no11_constraint(self.structure_verifier._apply_scale_transform(pattern, scale))
for scale in scales
)
pattern_results['constraint_preservation'] = 1.0 if constraint_preserved else 0.0
# 5. 自相似性
self_similarity_scores = []
for scale in scales[1:]: # 跳过scale=1
similarity = self.fractal_gen.measure_self_similarity(pattern, scale)
self_similarity_scores.append(similarity)
pattern_results['self_similarity'] = np.mean(self_similarity_scores) if self_similarity_scores else 0.0
results[f'pattern_{i}'] = pattern_results
return results
def _verify_no11_constraint(self, pattern: str) -> bool:
"""验证no-11约束"""
return '11' not in pattern
def generate_test_report(self, verification_results: Dict[str, Dict[str, float]]) -> str:
"""生成验证报告"""
report = "# P6 尺度不变性验证报告\n\n"
# 总体统计
all_scores = []
for pattern_results in verification_results.values():
all_scores.extend(pattern_results.values())
overall_score = np.mean(all_scores)
report += f"## 总体评分: {overall_score:.3f}\n\n"
# 详细结果
report += "## 详细结果\n\n"
for pattern_name, results in verification_results.items():
report += f"### {pattern_name}\n\n"
for metric, score in results.items():
report += f"- {metric}: {score:.3f}\n"
report += "\n"
# 结论
if overall_score > 0.8:
report += "## 结论\n\n尺度不变性命题得到强有力支持。"
elif overall_score > 0.6:
report += "## 结论\n\n尺度不变性命题得到部分支持，需要进一步改进。"
else:
report += "## 结论\n\n尺度不变性命题需要重新审视。"
return report
6. 总结
本形式化框架提供了：
- 完整的φ-表示系统：满足no-11约束的尺度变换
- 分形生成器：自动生成φ-分形结构
- 多维度验证：结构、密度、维数、约束的综合验证
- 量化分析：精确的数值度量和验证报告
这为P6尺度不变性命题提供了严格的数学基础和验证工具。