C11-1 理论自反射形式化规范

模块依赖

from typing import Set, Dict, List, Tuple, Optional, Callable, Any
from dataclasses import dataclass, field
from abc import ABC, abstractmethod
from enum import Enum
import hashlib
from no11_number_system import No11Number
from test_C10_1 import (
    FormalSystem, Formula, Symbol, SymbolType,
    VariableTerm, ConstantTerm, AtomicFormula,
    Axiom, Proof, ProofStep, Model, Interpretation,
    GödelEncoder
)
from test_C10_2 import (
    CategoryObject, Morphism, IdentityMorphism, ComposedMorphism,
    Category, Functor, NaturalTransformation,
    FormalSystemCategory, CollapseFunctor
)

核心数据结构

理论表示

@dataclass
class Theory:
    """完整理论的表示"""
    name: str
    language: FormalSystem
    axioms: Set[Formula]
    inference_rules: Set['InferenceRule']
    theorems: Set[Formula] = field(default_factory=set)
    proofs: Dict[Formula, Proof] = field(default_factory=dict)
    
    def encode(self) -> No11Number:
        """理论的No-11编码"""
        encoder = GödelEncoder(self.language)
        # 编码语言、公理、规则
        lang_code = self.language.encode_self()
        axiom_codes = [encoder.encode_formula(ax) for ax in self.axioms]
        rule_codes = [rule.encode() for rule in self.inference_rules]
        
        # 组合编码
        combined = lang_code.value
        for code in axiom_codes + rule_codes:
            combined = (combined * 100 + code.value) % 1000000
        
        return No11Number(combined)
    
    def add_axiom(self, axiom: Formula):
        """添加公理"""
        self.axioms.add(axiom)
    
    def add_theorem(self, theorem: Formula, proof: Proof):
        """添加定理及其证明"""
        self.theorems.add(theorem)
        self.proofs[theorem] = proof
    
    def includes(self, other: 'Theory') -> bool:
        """检查是否包含另一理论"""
        return (self.axioms >= other.axioms and 
                self.inference_rules >= other.inference_rules)

@dataclass
class InferenceRule:
    """推理规则"""
    name: str
    premises: List[Formula]  # 模式
    conclusion: Formula      # 模式
    
    def encode(self) -> No11Number:
        """规则的No-11编码"""
        name_hash = sum(ord(c) for c in self.name) % 1000
        return No11Number(name_hash)
    
    def apply(self, formulas: List[Formula]) -> Optional[Formula]:
        """应用规则到具体公式"""
        # 模式匹配和替换
        pass

反射机制

class ReflectionError(Exception):
    """反射错误基类"""
    pass

@dataclass
class TheoryEncoding:
    """理论在其内部的编码表示"""
    theory: Theory
    encoding_formula: Formula  # 表示编码的公式
    
    def verify_correctness(self) -> bool:
        """验证编码的正确性"""
        # 检查encoding_formula确实编码了theory
        return True

class ReflectionOperator:
    """反射操作符"""
    
    def __init__(self, max_depth: int = 10):
        self.max_depth = max_depth
        self.encoder = None  # 将根据具体理论设置
    
    def reflect(self, theory: Theory) -> Theory:
        """计算理论的反射"""
        # 创建新理论
        reflected = Theory(
            name=f"Reflect({theory.name})",
            language=self._extend_language(theory.language),
            axioms=theory.axioms.copy(),
            inference_rules=theory.inference_rules.copy()
        )
        
        # 添加原理论的所有内容
        for theorem in theory.theorems:
            reflected.theorems.add(theorem)
        
        # 添加反射公理
        self._add_reflection_axioms(theory, reflected)
        
        # 添加证明反射
        self._add_proof_reflection(theory, reflected)
        
        return reflected
    
    def _extend_language(self, language: FormalSystem) -> FormalSystem:
        """扩展语言以支持反射"""
        extended = FormalSystem(f"{language.name}_extended")
        
        # 复制原有符号
        for symbol in language.symbols.values():
            extended.add_symbol(symbol)
        
        # 添加反射符号
        extended.add_symbol(Symbol("Theory", SymbolType.PREDICATE, 1))
        extended.add_symbol(Symbol("Proves", SymbolType.PREDICATE, 2))
        extended.add_symbol(Symbol("Encode", SymbolType.FUNCTION, 1))
        
        return extended
    
    def _add_reflection_axioms(self, original: Theory, reflected: Theory):
        """添加反射公理"""
        # 编码器
        self.encoder = GödelEncoder(reflected.language)
        
        # 对每个原公理添加反射
        for axiom in original.axioms:
            # 构造 "axiom ∈ theory" 的公式
            axiom_code = self.encoder.encode_formula(axiom)
            theory_code = original.encode()
            
            # 创建反射公式
            reflection = self._create_membership_formula(
                axiom_code, theory_code, reflected.language
            )
            
            reflected.add_axiom(reflection)
    
    def _add_proof_reflection(self, original: Theory, reflected: Theory):
        """添加证明的反射"""
        for theorem, proof in original.proofs.items():
            # 构造 "exists proof of theorem in theory"
            theorem_code = self.encoder.encode_formula(theorem)
            proof_code = self._encode_proof(proof)
            
            # 创建存在性公式
            proof_exists = self._create_proof_exists_formula(
                theorem_code, proof_code, reflected.language
            )
            
            # 添加为定理（需要构造证明）
            meta_proof = self._construct_meta_proof(proof_exists)
            reflected.add_theorem(proof_exists, meta_proof)
    
    def _create_membership_formula(self, element_code: No11Number, 
                                 set_code: No11Number,
                                 language: FormalSystem) -> Formula:
        """创建成员关系公式"""
        # element ∈ set 的形式化
        element_term = ConstantTerm(str(element_code.value))
        set_term = ConstantTerm(str(set_code.value))
        
        theory_pred = language.symbols["Theory"]
        
        return AtomicFormula(
            predicate=theory_pred,
            terms=[element_term]
        )
    
    def _create_proof_exists_formula(self, theorem_code: No11Number,
                                   proof_code: No11Number,
                                   language: FormalSystem) -> Formula:
        """创建证明存在性公式"""
        # ∃p: Proves(p, theorem)
        proves_pred = language.symbols["Proves"]
        
        theorem_term = ConstantTerm(str(theorem_code.value))
        proof_term = ConstantTerm(str(proof_code.value))
        
        return AtomicFormula(
            predicate=proves_pred,
            terms=[proof_term, theorem_term]
        )
    
    def _encode_proof(self, proof: Proof) -> No11Number:
        """编码证明"""
        # 简化实现
        proof_hash = hash(tuple(step.formula for step in proof.steps))
        return No11Number(abs(proof_hash) % 100000)
    
    def _construct_meta_proof(self, formula: Formula) -> Proof:
        """构造元证明"""
        # 简化实现
        return Proof([
            ProofStep(formula, "reflection", [])
        ])

理论塔构造

@dataclass
class TheoryTower:
    """理论塔的表示"""
    base: Theory
    levels: List[Theory] = field(default_factory=list)
    reflection_operator: ReflectionOperator = field(default_factory=ReflectionOperator)
    
    def __post_init__(self):
        if not self.levels:
            self.levels = [self.base]
    
    def build_to_level(self, n: int):
        """构建到第n层"""
        while len(self.levels) <= n:
            current = self.levels[-1]
            next_level = self.reflection_operator.reflect(current)
            
            # 检查是否达到不动点
            if self._is_fixpoint(current, next_level):
                print(f"Reached fixpoint at level {len(self.levels)-1}")
                break
            
            self.levels.append(next_level)
    
    def _is_fixpoint(self, theory1: Theory, theory2: Theory) -> bool:
        """检查是否是不动点"""
        # 简化：比较公理数量
        return len(theory1.axioms) == len(theory2.axioms)
    
    def get_level(self, n: int) -> Optional[Theory]:
        """获取第n层理论"""
        if n < len(self.levels):
            return self.levels[n]
        return None
    
    def compute_entropy_growth(self) -> List[float]:
        """计算熵增曲线"""
        entropies = []
        for theory in self.levels:
            # 简化：使用公理数量作为熵的代理
            entropy = len(theory.axioms) + len(theory.theorems)
            entropies.append(float(entropy))
        return entropies

class TheoryFixpoint:
    """理论不动点的特殊处理"""
    
    @staticmethod
    def find_fixpoint(base: Theory, max_iterations: int = 100) -> Optional[Theory]:
        """寻找反射不动点"""
        tower = TheoryTower(base)
        
        for i in range(max_iterations):
            tower.build_to_level(i + 1)
            if len(tower.levels) <= i + 1:
                # 已经找到不动点
                return tower.levels[-1]
        
        return None
    
    @staticmethod
    def verify_fixpoint(theory: Theory) -> bool:
        """验证理论是否是不动点"""
        reflector = ReflectionOperator()
        reflected = reflector.reflect(theory)
        
        # 比较公理集
        return theory.axioms == reflected.axioms

证明谓词

class ProofPredicate:
    """证明谓词的实现"""
    
    def __init__(self, theory: Theory):
        self.theory = theory
        self.encoder = GödelEncoder(theory.language)
    
    def provable(self, formula: Formula) -> Formula:
        """构造Prov_T(φ)谓词"""
        # ∃p: Proof_T(p, φ)
        phi_code = self.encoder.encode_formula(formula)
        
        # 创建存在性公式
        # 简化实现
        proves_symbol = self.theory.language.symbols.get("Proves")
        if not proves_symbol:
            raise ReflectionError("Proves predicate not in language")
        
        return AtomicFormula(
            predicate=proves_symbol,
            terms=[VariableTerm("p"), ConstantTerm(str(phi_code.value))]
        )
    
    def proof_of(self, proof: Proof, formula: Formula) -> Formula:
        """构造Proof_T(p, φ)谓词"""
        p_code = self._encode_proof(proof)
        phi_code = self.encoder.encode_formula(formula)
        
        proves_symbol = self.theory.language.symbols["Proves"]
        
        return AtomicFormula(
            predicate=proves_symbol,
            terms=[ConstantTerm(str(p_code.value)), 
                  ConstantTerm(str(phi_code.value))]
        )
    
    def _encode_proof(self, proof: Proof) -> No11Number:
        """编码证明序列"""
        # 编码每个证明步骤
        step_codes = []
        for step in proof.steps:
            formula_code = self.encoder.encode_formula(step.formula)
            step_codes.append(formula_code.value)
        
        # 组合编码
        combined = 0
        for code in step_codes:
            combined = (combined * 100 + code) % 1000000
        
        return No11Number(combined)

范畴论视角

class ReflectionFunctor(Functor):
    """反射作为函子"""
    
    def __init__(self):
        # 理论范畴到自身
        theory_cat = Category("Theory")
        super().__init__("Reflection", theory_cat, theory_cat)
        self.reflection_op = ReflectionOperator()
    
    def setup_mapping(self, theories: List[Theory]):
        """设置函子映射"""
        for theory in theories:
            # 对象映射
            theory_obj = CategoryObject(theory.name, theory)
            reflected_theory = self.reflection_op.reflect(theory)
            reflected_obj = CategoryObject(reflected_theory.name, reflected_theory)
            
            self.object_map[theory_obj] = reflected_obj
            
            # 态射映射（理论态射）
            # 简化：只考虑恒等态射
            id_mor = IdentityMorphism(theory_obj)
            reflected_id = IdentityMorphism(reflected_obj)
            self.morphism_map[id_mor] = reflected_id

class ReflectionMonad:
    """反射作为Monad"""
    
    def __init__(self):
        self.functor = ReflectionFunctor()
        
    def unit(self, theory: Theory) -> Morphism:
        """单位态射 η: T → R(T)"""
        theory_obj = CategoryObject(theory.name, theory)
        reflected_obj = self.functor.map_object(theory_obj)
        
        # 包含映射
        def inclusion_map(t):
            return t  # 理论包含在其反射中
        
        return Morphism(theory_obj, reflected_obj, "unit", inclusion_map)
    
    def multiplication(self, theory: Theory) -> Morphism:
        """乘法 μ: R(R(T)) → R(T)"""
        # R(R(T)) → R(T) 的平坦化
        pass

算法实现

核心反射算法

def reflect_theory(theory: Theory) -> Theory:
    """
    完整的理论反射算法
    """
    reflector = ReflectionOperator()
    return reflector.reflect(theory)

def build_reflection_tower(base: Theory, height: int) -> TheoryTower:
    """
    构建反射塔到指定高度
    """
    tower = TheoryTower(base)
    tower.build_to_level(height)
    return tower

def find_reflection_fixpoint(base: Theory) -> Optional[Theory]:
    """
    寻找反射不动点
    """
    return TheoryFixpoint.find_fixpoint(base)

证明反射算法

def reflect_proof(theory: Theory, proof: Proof) -> Proof:
    """
    反射一个证明
    """
    # 在反射理论中构造关于原证明的证明
    reflected_theory = reflect_theory(theory)
    proof_pred = ProofPredicate(reflected_theory)
    
    # 构造元证明
    meta_steps = []
    for step in proof.steps:
        # 每个步骤都被反射
        reflected_step = ProofStep(
            formula=proof_pred.proof_of(proof, step.formula),
            justification="reflection",
            dependencies=[]
        )
        meta_steps.append(reflected_step)
    
    return Proof(meta_steps)

接口规范

理论反射接口

class TheoryReflectionInterface:
    """理论反射系统的标准接口"""
    
    def create_theory(self, name: str) -> Theory:
        """创建新理论"""
        pass
    
    def reflect(self, theory: Theory) -> Theory:
        """反射理论"""
        pass
    
    def build_tower(self, base: Theory, height: int) -> TheoryTower:
        """构建理论塔"""
        pass
    
    def find_fixpoint(self, base: Theory) -> Optional[Theory]:
        """寻找不动点"""
        pass
    
    def encode_in_theory(self, theory: Theory, object: Any) -> Formula:
        """在理论内编码对象"""
        pass
    
    def verify_self_reference(self, theory: Theory) -> bool:
        """验证理论的自引用能力"""
        pass

验证规范

反射正确性验证

def verify_reflection_correctness(original: Theory, reflected: Theory) -> bool:
    """
    验证反射的正确性
    """
    # 1. 原理论包含在反射中
    if not reflected.includes(original):
        return False
    
    # 2. 反射公理的正确性
    # 检查每个原公理都有对应的反射
    
    # 3. 证明反射的正确性
    # 检查每个原证明都被正确反射
    
    return True

不动点验证

def verify_fixpoint(theory: Theory) -> bool:
    """
    验证理论是否是反射不动点
    """
    return TheoryFixpoint.verify_fixpoint(theory)

熵增验证

def verify_entropy_increase(tower: TheoryTower) -> bool:
    """
    验证理论塔的熵增
    """
    entropies = tower.compute_entropy_growth()
    
    # 检查严格递增（除非达到不动点）
    for i in range(1, len(entropies)):
        if entropies[i] < entropies[i-1]:
            return False
        if entropies[i] == entropies[i-1]:
            # 可能达到不动点
            if i < len(tower.levels) - 1:
                return False
    
    return True

错误处理规范

所有反射操作必须进行严格的错误检查：

1. 编码错误: 对象无法编码为No-11数
2. 语言错误: 缺少必要的反射符号
3. 循环错误: 自引用导致的无限循环
4. 资源错误: 反射超过最大深度
5. 一致性错误: 反射导致不一致

性能要求

1. 编码操作: O(n log n) 其中n是理论大小
2. 反射操作: O(n²) 基本情况
3. 不动点查找: O(φⁿ) 指数级
4. 理论塔构建: O(h · n²) 其中h是高度
5. 熵计算: O(n) 线性复杂度

测试规范

每个反射组件必须通过以下测试：

1. 基础测试: 理论创建和编码
2. 反射测试: 单次反射的正确性
3. 塔测试: 理论塔的构建和属性
4. 不动点测试: 不动点的存在性和唯一性
5. 范畴测试: 函子和Monad结构
6. 熵增测试: 验证熵增原理