C11-3 理论不动点 - 形式化规范
模块结构
from dataclasses import dataclass, field
from typing import Set, Dict, List, Tuple, Optional, Callable
from abc import ABC, abstractmethod
import hashlib
from no11_number_system import No11Number
from test_C11_1 import Theory, Formula, ReflectionOperator
from test_C11_2 import EntropyCalculator
核心数据结构
1. 理论不动点表示
@dataclass
class TheoryFixedPoint:
"""理论不动点的表示"""
theory: Theory
reflection_depth: int # 达到不动点的反射深度
structural_entropy: float # 结构熵
is_exact: bool # 是否精确不动点（True）或近似（False）
def verify_fixed_point(self) -> bool:
"""验证不动点性质"""
reflector = ReflectionOperator()
reflected = reflector.reflect(self.theory)
return self.theory.is_isomorphic_to(reflected)
def __post_init__(self):
"""初始化后验证"""
if self.is_exact and not self.verify_fixed_point():
raise ValueError("声称的不动点未通过验证")
2. 不动点检测器
@dataclass
class FixedPointDetector:
"""检测理论反射序列中的不动点"""
max_iterations: int = 100
isomorphism_checker: 'IsomorphismChecker' = field(default_factory=lambda: IsomorphismChecker())
def find_fixed_point(self, initial_theory: Theory) -> Optional[TheoryFixedPoint]:
"""
寻找不动点
返回:
TheoryFixedPoint 如果找到
None 如果在max_iterations内未找到
"""
reflector = ReflectionOperator()
current = initial_theory
history = []
for depth in range(self.max_iterations):
# 检查是否达到精确不动点
reflected = reflector.reflect(current)
if self.isomorphism_checker.are_isomorphic(current, reflected):
entropy = self._compute_structural_entropy(current)
return TheoryFixedPoint(
theory=current,
reflection_depth=depth,
structural_entropy=entropy,
is_exact=True
)
# 检查是否形成循环
for i, past_theory in enumerate(history):
if self.isomorphism_checker.are_isomorphic(current, past_theory):
# 找到循环，构造循环不动点
cycle_length = depth - i
fixed_point = self._construct_cycle_fixed_point(
history[i:], cycle_length
)
return fixed_point
history.append(current)
current = reflected
# 未找到精确不动点，返回近似
return self._find_approximate_fixed_point(history)
def _construct_cycle_fixed_point(self, cycle: List[Theory], length: int) -> TheoryFixedPoint:
"""从循环构造不动点"""
# 循环中的理论共同构成不动点结构
# 这里简化为返回循环中熵最大的理论
max_entropy = 0
best_theory = cycle[0]
for theory in cycle:
entropy = self._compute_structural_entropy(theory)
if entropy > max_entropy:
max_entropy = entropy
best_theory = theory
return TheoryFixedPoint(
theory=best_theory,
reflection_depth=len(cycle),
structural_entropy=max_entropy,
is_exact=False # 循环不动点不是精确的
)
def _find_approximate_fixed_point(self, history: List[Theory]) -> Optional[TheoryFixedPoint]:
"""寻找近似不动点"""
if not history:
return None
# 找到变化最小的理论
min_distance = float('inf')
best_index = -1
reflector = ReflectionOperator()
for i in range(len(history) - 1):
reflected = reflector.reflect(history[i])
distance = self._theory_distance(history[i], reflected)
if distance < min_distance:
min_distance = distance
best_index = i
if best_index >= 0:
entropy = self._compute_structural_entropy(history[best_index])
return TheoryFixedPoint(
theory=history[best_index],
reflection_depth=best_index,
structural_entropy=entropy,
is_exact=False
)
return None
def _compute_structural_entropy(self, theory: Theory) -> float:
"""计算理论的结构熵"""
# 基于理论的各个组成部分
axiom_bits = len(theory.axioms) * 10
symbol_bits = len(theory.language.symbols) * 5
rule_bits = len(theory.inference_rules) * 15
total_bits = axiom_bits + symbol_bits + rule_bits
# 归一化到[0, 1]
return min(total_bits / 1000.0, 1.0)
def _theory_distance(self, t1: Theory, t2: Theory) -> float:
"""计算两个理论之间的距离"""
# 简化的距离度量
axiom_diff = len(t1.axioms.symmetric_difference(t2.axioms))
symbol_diff = len(set(t1.language.symbols.keys()).symmetric_difference(
set(t2.language.symbols.keys())))
return axiom_diff + symbol_diff * 0.5
3. 同构性检测
@dataclass
class IsomorphismChecker:
"""检测两个理论是否同构"""
def are_isomorphic(self, t1: Theory, t2: Theory) -> bool:
"""
检测两个理论是否同构
同构意味着存在双射保持所有结构
"""
# 必要条件：基数相等
if not self._check_cardinality(t1, t2):
return False
# 尝试构造同构映射
iso_map = self._find_isomorphism(t1, t2)
if iso_map is None:
return False
# 验证映射保持所有结构
return self._verify_isomorphism(t1, t2, iso_map)
def _check_cardinality(self, t1: Theory, t2: Theory) -> bool:
"""检查基数是否匹配"""
return (len(t1.axioms) == len(t2.axioms) and
len(t1.language.symbols) == len(t2.language.symbols) and
len(t1.inference_rules) == len(t2.inference_rules))
def _find_isomorphism(self, t1: Theory, t2: Theory) -> Optional[Dict]:
"""尝试找到同构映射"""
# 这是一个NP完全问题，这里使用启发式方法
# 首先匹配符号
symbol_map = self._match_symbols(t1.language.symbols, t2.language.symbols)
if symbol_map is None:
return None
# 基于符号映射检查公理是否可以对应
axiom_map = self._match_axioms(t1.axioms, t2.axioms, symbol_map)
if axiom_map is None:
return None
return {'symbols': symbol_map, 'axioms': axiom_map}
def _match_symbols(self, symbols1: Dict, symbols2: Dict) -> Optional[Dict]:
"""匹配符号"""
if len(symbols1) != len(symbols2):
return None
# 按类型分组
by_type1 = self._group_by_type(symbols1)
by_type2 = self._group_by_type(symbols2)
# 每种类型的符号数必须相等
if by_type1.keys() != by_type2.keys():
return None
mapping = {}
for sym_type in by_type1:
if len(by_type1[sym_type]) != len(by_type2[sym_type]):
return None
# 简单地按顺序匹配（实际应该更智能）
for s1, s2 in zip(sorted(by_type1[sym_type]),
sorted(by_type2[sym_type])):
mapping[s1] = s2
return mapping
def _group_by_type(self, symbols: Dict) -> Dict:
"""按类型分组符号"""
groups = {}
for name, symbol in symbols.items():
if symbol.type not in groups:
groups[symbol.type] = []
groups[symbol.type].append(name)
return groups
def _match_axioms(self, axioms1: Set[Formula], axioms2: Set[Formula],
symbol_map: Dict) -> Optional[Dict]:
"""基于符号映射匹配公理"""
# 简化实现：如果公理数相等就认为可以匹配
if len(axioms1) == len(axioms2):
return {ax1: ax2 for ax1, ax2 in zip(axioms1, axioms2)}
return None
def _verify_isomorphism(self, t1: Theory, t2: Theory, iso_map: Dict) -> bool:
"""验证映射是否真的是同构"""
# 简化验证：检查基本性质
return True # 实际实现应该更严格
4. 熵分离计算器
@dataclass
class EntropySeparator:
"""分离并计算结构熵和过程熵"""
def compute_structural_entropy(self, theory: Theory) -> float:
"""
计算结构熵
基于理论的静态结构：符号、公理、规则
"""
# 符号熵
symbol_entropy = self._symbol_entropy(theory.language.symbols)
# 公理熵
axiom_entropy = self._axiom_entropy(theory.axioms)
# 规则熵
rule_entropy = self._rule_entropy(theory.inference_rules)
# 编码熵
encoding_entropy = self._encoding_entropy(theory)
# 组合
total = symbol_entropy + axiom_entropy + rule_entropy + encoding_entropy
return min(total / 4.0, 1.0) # 归一化
def compute_process_entropy(self, theory: Theory, steps: int = 1000) -> float:
"""
计算过程熵
基于理论的动态行为：证明、计算、推理
"""
# 证明搜索的熵
proof_entropy = self._proof_search_entropy(theory, steps)
# 定理生成的熵
theorem_entropy = self._theorem_generation_entropy(theory, steps)
# 反射计算的熵
reflection_entropy = self._reflection_computation_entropy(theory)
# 组合
total = proof_entropy + theorem_entropy + reflection_entropy
return total / 3.0 # 平均
def _symbol_entropy(self, symbols: Dict) -> float:
"""计算符号系统的熵"""
if not symbols:
return 0.0
# 基于符号类型的分布
type_counts = {}
for symbol in symbols.values():
type_counts[symbol.type] = type_counts.get(symbol.type, 0) + 1
total = len(symbols)
entropy = 0.0
for count in type_counts.values():
p = count / total
if p > 0:
entropy -= p * math.log2(p)
return entropy
def _axiom_entropy(self, axioms: Set[Formula]) -> float:
"""计算公理集的熵"""
if not axioms:
return 0.0
# 基于公理的复杂度分布
complexities = [self._formula_complexity(ax) for ax in axioms]
avg_complexity = sum(complexities) / len(complexities)
# 复杂度的方差作为熵的度量
variance = sum((c - avg_complexity) ** 2 for c in complexities) / len(complexities)
return math.sqrt(variance) / 100.0 # 归一化
def _formula_complexity(self, formula: Formula) -> int:
"""计算公式的复杂度"""
# 简单计数：符号数、嵌套深度等
return len(str(formula)) # 简化实现
def _rule_entropy(self, rules: Set) -> float:
"""计算推理规则的熵"""
return len(rules) * 0.1 # 简化实现
def _encoding_entropy(self, theory: Theory) -> float:
"""计算编码的熵"""
# 基于理论自编码的长度
try:
encoding = theory.encode_self()
return len(encoding.bits) / 1000.0
except:
return 0.5 # 默认值
def _proof_search_entropy(self, theory: Theory, steps: int) -> float:
"""计算证明搜索过程的熵"""
# 模拟证明搜索，测量搜索树的分支因子
return 0.3 # 简化实现
def _theorem_generation_entropy(self, theory: Theory, steps: int) -> float:
"""计算定理生成的熵"""
# 测量新定理的生成率
initial_theorems = len(theory.theorems)
# 模拟推理过程...
final_theorems = initial_theorems + steps // 10 # 简化模型
growth_rate = (final_theorems - initial_theorems) / steps
return growth_rate * 10 # 缩放
def _reflection_computation_entropy(self, theory: Theory) -> float:
"""计算反射计算的熵"""
# 测量反射操作的计算复杂度
reflector = ReflectionOperator()
# 可以通过计时或其他方式测量
return 0.4 # 简化实现
5. 不动点构造器
@dataclass
class FixedPointConstructor:
"""构造理论不动点"""
def construct_minimal_fixed_point(self) -> Theory:
"""
构造最小不动点理论
包含：
- 自引用公理
- 反射规则
- 最小符号集
"""
from test_C10_1 import FormalSystem, Symbol, SymbolType
from test_C11_1 import Theory
# 创建最小语言
system = FormalSystem("MinimalFixedPoint")
# 基本符号
system.add_symbol(Symbol("T", SymbolType.CONSTANT)) # 理论自身
system.add_symbol(Symbol("Reflect", SymbolType.FUNCTION, 1)) # 反射函数
system.add_symbol(Symbol("=", SymbolType.RELATION, 2)) # 等于
# 自引用公理：Reflect(T) = T
t_const = ConstantTerm(system.symbols["T"])
reflect_t = FunctionTerm(
system.symbols["Reflect"],
(t_const,)
)
fixed_point_axiom = AtomicFormula(
system.symbols["="],
(reflect_t, t_const)
)
# 创建理论
theory = Theory(
name="FixedPointTheory",
language=system,
axioms={fixed_point_axiom},
inference_rules=set()
)
return theory
def approach_fixed_point(self, initial: Theory, iterations: int) -> List[Theory]:
"""
通过迭代反射逼近不动点
返回反射序列
"""
reflector = ReflectionOperator()
sequence = [initial]
current = initial
for _ in range(iterations):
current = reflector.reflect(current)
sequence.append(current)
# 检查是否已达到不动点
detector = FixedPointDetector(max_iterations=1)
if detector.find_fixed_point(current) is not None:
break
return sequence
6. 不动点分析器
@dataclass
class FixedPointAnalyzer:
"""分析不动点的性质"""
def analyze_convergence(self, sequence: List[Theory]) -> Dict[str, float]:
"""
分析序列收敛到不动点的过程
返回:
收敛速度、振荡程度等指标
"""
if len(sequence) < 2:
return {'convergence_rate': 0.0, 'oscillation': 0.0}
# 计算相邻理论间的距离
distances = []
for i in range(1, len(sequence)):
dist = self._theory_distance(sequence[i-1], sequence[i])
distances.append(dist)
# 收敛速度：距离的递减率
convergence_rate = 0.0
if len(distances) > 1:
decreases = sum(1 for i in range(1, len(distances))
if distances[i] < distances[i-1])
convergence_rate = decreases / (len(distances) - 1)
# 振荡程度：距离的方差
if distances:
mean_dist = sum(distances) / len(distances)
oscillation = math.sqrt(
sum((d - mean_dist) ** 2 for d in distances) / len(distances)
)
else:
oscillation = 0.0
return {
'convergence_rate': convergence_rate,
'oscillation': oscillation,
'final_distance': distances[-1] if distances else 0.0
}
def verify_attracting_fixed_point(self, fixed_point: TheoryFixedPoint,
test_theories: List[Theory]) -> bool:
"""
验证不动点是否是吸引子
测试多个初始理论是否都收敛到此不动点
"""
detector = FixedPointDetector()
for theory in test_theories:
found = detector.find_fixed_point(theory)
if found is None:
return False
# 检查是否收敛到同一个不动点
checker = IsomorphismChecker()
if not checker.are_isomorphic(found.theory, fixed_point.theory):
return False
return True
def compute_basin_of_attraction(self, fixed_point: TheoryFixedPoint,
sample_size: int = 100) -> float:
"""
估计不动点的吸引域大小
返回：被吸引的理论比例
"""
# 生成随机理论样本
attracted = 0
constructor = FixedPointConstructor()
for _ in range(sample_size):
# 创建随机变异的理论
random_theory = self._generate_random_theory()
sequence = constructor.approach_fixed_point(random_theory, 50)
# 检查是否收敛到给定不动点
if sequence:
final = sequence[-1]
checker = IsomorphismChecker()
if checker.are_isomorphic(final, fixed_point.theory):
attracted += 1
return attracted / sample_size
def _theory_distance(self, t1: Theory, t2: Theory) -> float:
"""计算理论间距离"""
# 复用FixedPointDetector中的实现
detector = FixedPointDetector()
return detector._theory_distance(t1, t2)
def _generate_random_theory(self) -> Theory:
"""生成随机理论用于测试"""
# 简化实现：返回最小不动点的变体
constructor = FixedPointConstructor()
base = constructor.construct_minimal_fixed_point()
# 随机添加一些公理
import random
if random.random() > 0.5:
# 添加一个简单公理
new_axiom = AtomicFormula(
Symbol("P", SymbolType.RELATION, 0),
()
)
base.axioms.add(new_axiom)
return base
集成点
与C11-1的集成
# 使用ReflectionOperator进行反射操作
from test_C11_1 import ReflectionOperator, Theory
# 不动点检测依赖于反射机制
detector = FixedPointDetector()
fixed_point = detector.find_fixed_point(initial_theory)
与C11-2的集成
# 不动点理论仍然满足不完备性
from test_C11_2 import IncompletenessAnalyzer
analyzer = IncompletenessAnalyzer(fixed_point.theory)
assert analyzer.verify_first_incompleteness() # 仍然不完备
熵计算的一致性
# 使用统一的熵计算框架
from test_C11_2 import EntropyCalculator
calc = EntropyCalculator()
entropy = calc.compute_entropy(fixed_point.theory)
验证要求
- 不动点存在性：在有限步内找到不动点或循环
- 同构性正确：正确判定理论同构
- 熵分离：结构熵饱和，过程熵持续增长
- 吸引性：不动点是局部吸引子
- No-11约束：所有编码满足约束
数学保证
- 反射序列在有限理论空间中必然循环或收敛
- 不动点在同构意义下唯一
- 熵增原理在过程维度持续有效
- 构造的最小不动点确实满足不动点条件