Skip to main content

C9-3 自指代数形式化规范

模块依赖​

from typing import List, Callable, Tuple, Optional, Set, Dict
from dataclasses import dataclass
from abc import ABC, abstractmethod
from no11_number_system import No11Number
from test_C9_1 import SelfReferentialArithmetic
from test_C9_2 import RecursiveNumberTheory

核心数据结构​

群结构定义​

@dataclass
class GroupElement:
    """群元素的形式化表示"""
    value: No11Number
    group_id: str  # 所属群的标识
    
    def __eq__(self, other: 'GroupElement') -> bool:
        return self.value == other.value and self.group_id == other.group_id
    
    def __hash__(self) -> int:
        return hash((self.value.value, self.group_id))

class SelfReferentialGroup:
    """自指群的形式化实现"""
    def __init__(self, 
                 elements: Set[No11Number],
                 operation: Callable[[No11Number, No11Number], No11Number],
                 identity: Optional[No11Number] = None):
        self.elements = elements
        self.operation = operation
        self.identity = identity or self._find_identity()
        self._cayley_table: Optional[Dict[Tuple[No11Number, No11Number], No11Number]] = None
        self._verify_closure()
        self._verify_associativity()
        self._verify_identity()
        self._verify_inverses()
    
    def operate(self, a: No11Number, b: No11Number) -> No11Number:
        """群运算，保证结果满足self-collapse性质"""
        if a not in self.elements or b not in self.elements:
            raise ValueError(f"Elements {a}, {b} not in group")
        result = self.operation(a, b)
        if result not in self.elements:
            raise ValueError(f"Operation not closed: {a} * {b} = {result}")
        return result
    
    def inverse(self, a: No11Number) -> No11Number:
        """计算群元素的逆元"""
        for x in self.elements:
            if self.operate(a, x) == self.identity:
                return x
        raise ValueError(f"No inverse found for {a}")
    
    def is_abelian(self) -> bool:
        """检查群是否是阿贝尔群"""
        for a in self.elements:
            for b in self.elements:
                if self.operate(a, b) != self.operate(b, a):
                    return False
        return True
    
    def subgroup(self, subset: Set[No11Number]) -> 'SelfReferentialGroup':
        """生成子群"""
        if not subset.issubset(self.elements):
            raise ValueError("Subset contains elements not in group")
        return SelfReferentialGroup(subset, self.operation)
    
    def cosets(self, subgroup: 'SelfReferentialGroup') -> List[Set[No11Number]]:
        """计算子群的陪集"""
        coset_list = []
        remaining = self.elements.copy()
        
        while remaining:
            a = remaining.pop()
            coset = {self.operate(a, h) for h in subgroup.elements}
            coset_list.append(coset)
            remaining -= coset
        
        return coset_list
    
    def _find_identity(self) -> No11Number:
        """寻找单位元"""
        for e in self.elements:
            if all(self.operation(e, x) == x and self.operation(x, e) == x 
                   for x in self.elements):
                return e
        raise ValueError("No identity element found")
    
    def _verify_closure(self):
        """验证封闭性"""
        for a in self.elements:
            for b in self.elements:
                if self.operation(a, b) not in self.elements:
                    raise ValueError(f"Operation not closed: {a} * {b}")
    
    def _verify_associativity(self):
        """验证结合律"""
        for a in self.elements:
            for b in self.elements:
                for c in self.elements:
                    left = self.operation(self.operation(a, b), c)
                    right = self.operation(a, self.operation(b, c))
                    if left != right:
                        raise ValueError(f"Associativity failed: ({a}*{b})*{c} != {a}*({b}*{c})")
    
    def _verify_identity(self):
        """验证单位元性质"""
        for x in self.elements:
            if self.operation(self.identity, x) != x or self.operation(x, self.identity) != x:
                raise ValueError(f"Identity property failed for {x}")
    
    def _verify_inverses(self):
        """验证每个元素都有逆元"""
        for a in self.elements:
            try:
                self.inverse(a)
            except ValueError:
                raise ValueError(f"Element {a} has no inverse")

环结构定义​

class SelfReferentialRing:
    """自指环的形式化实现"""
    def __init__(self,
                 elements: Set[No11Number],
                 add_op: Callable[[No11Number, No11Number], No11Number],
                 mul_op: Callable[[No11Number, No11Number], No11Number]):
        self.elements = elements
        self.add_op = add_op
        self.mul_op = mul_op
        
        # 验证环公理
        self.additive_group = SelfReferentialGroup(elements, add_op)
        self._verify_multiplicative_closure()
        self._verify_distributivity()
        
        # 找出特殊元素
        self.zero = self.additive_group.identity
        self.one = self._find_multiplicative_identity()
    
    def add(self, a: No11Number, b: No11Number) -> No11Number:
        """环的加法运算"""
        return self.additive_group.operate(a, b)
    
    def multiply(self, a: No11Number, b: No11Number) -> No11Number:
        """环的乘法运算"""
        if a not in self.elements or b not in self.elements:
            raise ValueError(f"Elements {a}, {b} not in ring")
        result = self.mul_op(a, b)
        if result not in self.elements:
            raise ValueError(f"Multiplication not closed")
        return result
    
    def is_commutative(self) -> bool:
        """检查环是否交换"""
        for a in self.elements:
            for b in self.elements:
                if self.multiply(a, b) != self.multiply(b, a):
                    return False
        return True
    
    def is_integral_domain(self) -> bool:
        """检查是否是整环"""
        if not self.is_commutative():
            return False
        
        # 检查无零因子
        for a in self.elements:
            if a == self.zero:
                continue
            for b in self.elements:
                if b == self.zero:
                    continue
                if self.multiply(a, b) == self.zero:
                    return False
        return True
    
    def ideal(self, generators: Set[No11Number]) -> 'Ideal':
        """生成理想"""
        return Ideal(self, generators)
    
    def quotient_ring(self, ideal: 'Ideal') -> 'SelfReferentialRing':
        """构造商环"""
        # 构造等价类
        equivalence_classes = ideal.equivalence_classes()
        
        # 定义商环运算
        def quotient_add(cls1: Set[No11Number], cls2: Set[No11Number]) -> Set[No11Number]:
            rep1 = next(iter(cls1))
            rep2 = next(iter(cls2))
            sum_rep = self.add(rep1, rep2)
            return ideal.equivalence_class(sum_rep)
        
        def quotient_mul(cls1: Set[No11Number], cls2: Set[No11Number]) -> Set[No11Number]:
            rep1 = next(iter(cls1))
            rep2 = next(iter(cls2))
            prod_rep = self.multiply(rep1, rep2)
            return ideal.equivalence_class(prod_rep)
        
        return SelfReferentialRing(equivalence_classes, quotient_add, quotient_mul)
    
    def _find_multiplicative_identity(self) -> Optional[No11Number]:
        """寻找乘法单位元"""
        for e in self.elements:
            if all(self.mul_op(e, x) == x and self.mul_op(x, e) == x 
                   for x in self.elements if x != self.zero):
                return e
        return None
    
    def _verify_multiplicative_closure(self):
        """验证乘法封闭性"""
        for a in self.elements:
            for b in self.elements:
                if self.mul_op(a, b) not in self.elements:
                    raise ValueError(f"Multiplication not closed: {a} * {b}")
    
    def _verify_distributivity(self):
        """验证分配律"""
        for a in self.elements:
            for b in self.elements:
                for c in self.elements:
                    # 左分配律: a*(b+c) = a*b + a*c
                    left = self.multiply(a, self.add(b, c))
                    right = self.add(self.multiply(a, b), self.multiply(a, c))
                    if left != right:
                        raise ValueError(f"Left distributivity failed")
                    
                    # 右分配律: (a+b)*c = a*c + b*c
                    left = self.multiply(self.add(a, b), c)
                    right = self.add(self.multiply(a, c), self.multiply(b, c))
                    if left != right:
                        raise ValueError(f"Right distributivity failed")

class Ideal:
    """理想的形式化表示"""
    def __init__(self, ring: SelfReferentialRing, generators: Set[No11Number]):
        self.ring = ring
        self.generators = generators
        self.elements = self._generate_ideal()
    
    def _generate_ideal(self) -> Set[No11Number]:
        """从生成元生成理想"""
        ideal_elements = {self.ring.zero}
        
        # 添加生成元的所有线性组合
        for g in self.generators:
            for r in self.ring.elements:
                # 左理想: r*g
                ideal_elements.add(self.ring.multiply(r, g))
                # 右理想: g*r
                ideal_elements.add(self.ring.multiply(g, r))
        
        # 闭包under加法
        changed = True
        while changed:
            changed = False
            new_elements = set()
            for a in ideal_elements:
                for b in ideal_elements:
                    sum_elem = self.ring.add(a, b)
                    if sum_elem not in ideal_elements:
                        new_elements.add(sum_elem)
                        changed = True
            ideal_elements.update(new_elements)
        
        return ideal_elements
    
    def contains(self, element: No11Number) -> bool:
        """检查元素是否在理想中"""
        return element in self.elements
    
    def equivalence_class(self, element: No11Number) -> Set[No11Number]:
        """计算元素的等价类"""
        return {self.ring.add(element, i) for i in self.elements}
    
    def equivalence_classes(self) -> List[Set[No11Number]]:
        """计算所有等价类"""
        classes = []
        remaining = self.ring.elements.copy()
        
        while remaining:
            rep = remaining.pop()
            eq_class = self.equivalence_class(rep)
            classes.append(eq_class)
            remaining -= eq_class
        
        return classes

域结构定义​

class SelfReferentialField:
    """自指域的形式化实现"""
    def __init__(self,
                 elements: Set[No11Number],
                 add_op: Callable[[No11Number, No11Number], No11Number],
                 mul_op: Callable[[No11Number, No11Number], No11Number]):
        # 首先验证是环
        self.ring = SelfReferentialRing(elements, add_op, mul_op)
        
        # 验证域特有性质
        self._verify_field_axioms()
        
        self.zero = self.ring.zero
        self.one = self.ring.one
    
    def add(self, a: No11Number, b: No11Number) -> No11Number:
        """域的加法"""
        return self.ring.add(a, b)
    
    def multiply(self, a: No11Number, b: No11Number) -> No11Number:
        """域的乘法"""
        return self.ring.multiply(a, b)
    
    def multiplicative_inverse(self, a: No11Number) -> No11Number:
        """计算乘法逆元"""
        if a == self.zero:
            raise ValueError("Zero has no multiplicative inverse")
        
        # 构造非零元素的乘法群
        nonzero = self.ring.elements - {self.zero}
        mul_group = SelfReferentialGroup(nonzero, self.ring.mul_op)
        return mul_group.inverse(a)
    
    def divide(self, a: No11Number, b: No11Number) -> No11Number:
        """域的除法 a/b = a * b^(-1)"""
        if b == self.zero:
            raise ValueError("Division by zero")
        return self.multiply(a, self.multiplicative_inverse(b))
    
    def characteristic(self) -> int:
        """计算域的特征"""
        if self.one is None:
            return 0
        
        char = 1
        sum_val = self.one
        while sum_val != self.zero:
            sum_val = self.add(sum_val, self.one)
            char += 1
            if char > len(self.ring.elements):
                return 0  # 特征为0
        return char
    
    def _verify_field_axioms(self):
        """验证域公理"""
        # 验证交换环
        if not self.ring.is_commutative():
            raise ValueError("Field must be commutative")
        
        # 验证有单位元
        if self.ring.one is None:
            raise ValueError("Field must have multiplicative identity")
        
        # 验证每个非零元素都有乘法逆元
        nonzero = self.ring.elements - {self.ring.zero}
        try:
            mul_group = SelfReferentialGroup(nonzero, self.ring.mul_op)
        except ValueError as e:
            raise ValueError(f"Non-zero elements don't form multiplicative group: {e}")

同态映射定义​

class AlgebraicHomomorphism(ABC):
    """代数同态的抽象基类"""
    @abstractmethod
    def map(self, element: No11Number) -> No11Number:
        """同态映射"""
        pass
    
    @abstractmethod
    def verify_homomorphism_property(self) -> bool:
        """验证同态性质"""
        pass

class GroupHomomorphism(AlgebraicHomomorphism):
    """群同态"""
    def __init__(self,
                 source: SelfReferentialGroup,
                 target: SelfReferentialGroup,
                 mapping: Callable[[No11Number], No11Number]):
        self.source = source
        self.target = target
        self.mapping = mapping
        
        if not self.verify_homomorphism_property():
            raise ValueError("Mapping is not a group homomorphism")
    
    def map(self, element: No11Number) -> No11Number:
        """应用同态映射"""
        if element not in self.source.elements:
            raise ValueError(f"Element {element} not in source group")
        result = self.mapping(element)
        if result not in self.target.elements:
            raise ValueError(f"Mapping result {result} not in target group")
        return result
    
    def kernel(self) -> SelfReferentialGroup:
        """计算同态的核"""
        ker_elements = {e for e in self.source.elements 
                       if self.map(e) == self.target.identity}
        return self.source.subgroup(ker_elements)
    
    def image(self) -> SelfReferentialGroup:
        """计算同态的像"""
        img_elements = {self.map(e) for e in self.source.elements}
        return self.target.subgroup(img_elements)
    
    def is_isomorphism(self) -> bool:
        """检查是否是同构"""
        # 单射：核只包含单位元
        if len(self.kernel().elements) > 1:
            return False
        # 满射：像等于目标群
        if self.image().elements != self.target.elements:
            return False
        return True
    
    def verify_homomorphism_property(self) -> bool:
        """验证群同态性质: φ(a*b) = φ(a)*φ(b)"""
        for a in self.source.elements:
            for b in self.source.elements:
                left = self.map(self.source.operate(a, b))
                right = self.target.operate(self.map(a), self.map(b))
                if left != right:
                    return False
        return True

class RingHomomorphism(AlgebraicHomomorphism):
    """环同态"""
    def __init__(self,
                 source: SelfReferentialRing,
                 target: SelfReferentialRing,
                 mapping: Callable[[No11Number], No11Number]):
        self.source = source
        self.target = target
        self.mapping = mapping
        
        if not self.verify_homomorphism_property():
            raise ValueError("Mapping is not a ring homomorphism")
    
    def map(self, element: No11Number) -> No11Number:
        """应用同态映射"""
        if element not in self.source.elements:
            raise ValueError(f"Element {element} not in source ring")
        result = self.mapping(element)
        if result not in self.target.elements:
            raise ValueError(f"Mapping result {result} not in target ring")
        return result
    
    def verify_homomorphism_property(self) -> bool:
        """验证环同态性质"""
        # 验证加法同态: φ(a+b) = φ(a)+φ(b)
        for a in self.source.elements:
            for b in self.source.elements:
                left = self.map(self.source.add(a, b))
                right = self.target.add(self.map(a), self.map(b))
                if left != right:
                    return False
        
        # 验证乘法同态: φ(a*b) = φ(a)*φ(b)
        for a in self.source.elements:
            for b in self.source.elements:
                left = self.map(self.source.multiply(a, b))
                right = self.target.multiply(self.map(a), self.map(b))
                if left != right:
                    return False
        
        # 验证单位元映射（如果存在）
        if self.source.one is not None:
            if self.map(self.source.one) != self.target.one:
                return False
        
        return True

接口规范​

群论接口​

class GroupTheoryInterface:
    """群论操作的标准接口"""
    def create_cyclic_group(self, n: int) -> SelfReferentialGroup:
        """创建n阶循环群"""
        pass
    
    def create_symmetric_group(self, n: int) -> SelfReferentialGroup:
        """创建n次对称群（考虑No-11约束）"""
        pass
    
    def direct_product(self, G1: SelfReferentialGroup, 
                      G2: SelfReferentialGroup) -> SelfReferentialGroup:
        """计算群的直积"""
        pass
    
    def find_subgroups(self, G: SelfReferentialGroup) -> List[SelfReferentialGroup]:
        """找出所有子群"""
        pass
    
    def is_simple_group(self, G: SelfReferentialGroup) -> bool:
        """检查是否是单群"""
        pass

环论接口​

class RingTheoryInterface:
    """环论操作的标准接口"""
    def create_polynomial_ring(self, base_ring: SelfReferentialRing, 
                              variable: str) -> SelfReferentialRing:
        """创建多项式环"""
        pass
    
    def create_matrix_ring(self, base_ring: SelfReferentialRing, 
                          n: int) -> SelfReferentialRing:
        """创建n×n矩阵环"""
        pass
    
    def find_prime_ideals(self, R: SelfReferentialRing) -> List[Ideal]:
        """找出所有素理想"""
        pass
    
    def find_maximal_ideals(self, R: SelfReferentialRing) -> List[Ideal]:
        """找出所有极大理想"""
        pass

域论接口​

class FieldTheoryInterface:
    """域论操作的标准接口"""
    def create_prime_field(self, p: int) -> SelfReferentialField:
        """创建素数阶有限域"""
        pass
    
    def field_extension(self, base_field: SelfReferentialField,
                       irreducible_poly: 'Polynomial') -> SelfReferentialField:
        """通过不可约多项式构造域扩张"""
        pass
    
    def splitting_field(self, base_field: SelfReferentialField,
                       polynomial: 'Polynomial') -> SelfReferentialField:
        """构造分裂域"""
        pass
    
    def algebraic_closure(self, field: SelfReferentialField) -> SelfReferentialField:
        """构造代数闭包（在No-11约束下）"""
        pass

验证规范​

自指性验证​

def verify_self_reference(algebraic_structure) -> bool:
    """验证代数结构的自指性质"""
    # 1. 运算表是collapse的不动点
    # 2. 所有运算结果满足No-11约束
    # 3. 结构保持熵增性质
    pass

完备性验证​

def verify_completeness(algebraic_structure) -> bool:
    """验证代数结构的完备性"""
    # 1. 所有必需的运算都已定义
    # 2. 所有公理都得到满足
    # 3. 与C9-1, C9-2的兼容性
    pass

熵增验证​

def verify_entropy_increase(operation, inputs, output) -> bool:
    """验证运算的熵增性质"""
    # 1. 输出的信息量≥输入的信息量
    # 2. 结构复杂度单调增加
    # 3. 不可逆性的度量
    pass

错误处理规范​

所有代数运算必须进行严格的错误检查：

  1. 输入验证: 确保输入元素属于正确的代数结构
  2. 运算封闭性: 确保运算结果仍在结构内
  3. 约束保持: 确保No-11约束始终得到满足
  4. 熵增检查: 确保信息量不减少

性能要求​

  1. 群运算: O(1) 对于预计算的Cayley表
  2. 环运算: O(n) 其中n是环的大小
  3. 域运算: O(log n) 对于有限域的运算
  4. 同态计算: O(n²) 对于验证同态性质

测试规范​

每个代数结构必须通过以下测试：

  1. 公理测试: 验证所有代数公理
  2. 自指测试: 验证self-collapse性质
  3. 一致性测试: 与C9-1, C9-2的兼容性
  4. 熵增测试: 验证信息量增加
  5. 边界测试: 特殊情况和极限情况