Skip to main content

T2-2-formal: 编码完备性定理的形式化证明

机器验证元数据​

type: theorem  
verification: machine_ready
dependencies: ["T1-2-formal.md", "T2-1-formal.md", "D1-1-formal.md"]
verification_points:
  - formal_information_definition
  - distinguishability_implies_describability
  - describability_implies_encodability
  - continuous_object_finite_representation
  - encoding_chain_completeness

核心定理​

定理 T2-2（编码完备性）​

EncodingCompleteness : Prop ≡
  ∀S : System . SelfRefComplete(S) → 
    (∀x ∈ S . Info(x) → ∃e ∈ Σ* . E(x) = e)

where
  Info(x) : Prop (information predicate)
  Σ : FiniteAlphabet
  E : S → Σ* (encoding function)

辅助定义​

信息的形式定义​

Info : S → Prop ≡
  λx . ∃y ∈ S . x ≠ y ∧ Desc(x) ≠ Desc(y)

HasInformation(S) : Prop ≡
  ∃x ∈ S . Info(x)

可区分性、可描述性、可编码性​

Distinguishable(x) : Prop ≡
  ∃y ∈ S . x ≠ y

Describable(x) : Prop ≡
  ∃d ∈ L . Desc(x) = d

Encodable(x) : Prop ≡
  ∃e ∈ Σ* . E(x) = e

可区分性蕴含可描述性​

引理 T2-2.1（可区分性→可描述性）​

DistinguishabilityImpliesDescribability : Prop ≡
  ∀S,x . (SelfRefComplete(S) ∧ x ∈ S ∧ Distinguishable(x)) → 
    Describable(x)

证明​

Proof of implication:
  Given SelfRefComplete(S) and Distinguishable(x):
  
  1. By D1-1, ∃Desc : S → L with properties:
     - Completeness: ∀s ∈ S . Desc(s) ∈ L
     - Injectivity: s₁ ≠ s₂ → Desc(s₁) ≠ Desc(s₂)
     
  2. Since x ∈ S:
     Desc(x) ∈ L (by completeness)
     
  3. Since Distinguishable(x), ∃y ≠ x:
     Desc(x) ≠ Desc(y) (by injectivity)
     
  4. Therefore Describable(x) with d = Desc(x) ∎

可描述性蕴含可编码性​

引理 T2-2.2（可描述性→可编码性）​

DescribabilityImpliesEncodability : Prop ≡
  ∀x . Describable(x) → Encodable(x)

证明​

Proof by construction:
  Given Describable(x), so Desc(x) ∈ L:
  
  1. L consists of finite symbol sequences
  2. Construct standard encoding Encode : L → ℕ
  
  Gödel encoding example:
  - Assign primes to alphabet symbols: σᵢ ↦ pᵢ
  - For string s₁s₂...sₙ:
    Encode(s₁s₂...sₙ) = p₁^(a₁) × p₂^(a₂) × ... × pₙ^(aₙ)
    where aᵢ is index of symbol sᵢ
    
  3. This gives injective map L → ℕ
  4. Compose with number encoding: ℕ → Σ*
  5. E = NumberEncode ∘ Encode ∘ Desc
  
  Therefore x is encodable ∎

连续对象的有限表示​

引理 T2-2.3（连续对象有限表示）​

ContinuousObjectFiniteRepresentation : Prop ≡
  ∀c ∈ ContinuousObjects . 
    ∃f ∈ FiniteDescriptions . Represents(f, c)

where
  ContinuousObjects = {π, e, sin, ...}
  Represents(f, c) ≡ f generates/defines c

证明​

Proof by examples:
  1. π (pi):
     - Algorithm: Machin formula
     - Definition: circumference/diameter
     - Series: π = 4∑(-1)ⁿ/(2n+1)
     
  2. e (Euler's number):
     - Definition: lim(1 + 1/n)ⁿ as n→∞
     - Series: e = ∑1/n!
     
  3. sin (sine function):
     - Differential equation: y'' + y = 0, y(0)=0, y'(0)=1
     - Taylor series: sin(x) = ∑(-1)ⁿx^(2n+1)/(2n+1)!
     
  All have finite descriptions as generation rules ∎

编码链的完整性​

引理 T2-2.4（编码链）​

EncodingChain : Prop ≡
  ∀x . Info(x) → Distinguishable(x) → 
    Describable(x) → Encodable(x)

证明​

Proof of chain:
  1. Info(x) → Distinguishable(x)
     By definition of Info
     
  2. Distinguishable(x) → Describable(x)
     By Lemma T2-2.1
     
  3. Describable(x) → Encodable(x)
     By Lemma T2-2.2
     
  Therefore Info(x) → Encodable(x) ∎

主定理证明​

定理：编码完备性​

MainTheorem : Prop ≡
  ∀S . SelfRefComplete(S) → 
    (∀x ∈ S . Info(x) → ∃e ∈ Σ* . E(x) = e)

证明​

Proof of completeness:
  Given SelfRefComplete(S) and x ∈ S with Info(x):
  
  1. Apply encoding chain (Lemma T2-2.4):
     Info(x) → Encodable(x)
     
  2. By definition of Encodable:
     ∃e ∈ Σ* . E(x) = e
     
  Therefore all information can be encoded ∎

机器验证检查点​

检查点1：信息形式定义验证​

def verify_formal_information_definition(system):
    # 获取所有元素
    elements = list(system.get_all_elements())
    
    information_elements = []
    
    for x in elements:
        # 检查是否满足Info(x)
        has_info = False
        
        for y in elements:
            if x != y:
                desc_x = system.describe(x)
                desc_y = system.describe(y)
                
                if desc_x != desc_y:
                    has_info = True
                    break
                    
        if has_info:
            information_elements.append(x)
            
    # 验证至少有一些信息元素
    assert len(information_elements) > 0
    
    return True, information_elements

检查点2：可区分性蕴含可描述性验证​

def verify_distinguishability_implies_describability(system):
    elements = system.get_all_elements()
    
    for x in elements:
        # 检查可区分性
        is_distinguishable = False
        
        for y in elements:
            if x != y:
                is_distinguishable = True
                break
                
        if is_distinguishable:
            # 验证可描述性
            description = system.describe(x)
            assert description is not None
            assert isinstance(description, str)
            assert len(description) > 0
            
    return True

检查点3：可描述性蕴含可编码性验证​

def verify_describability_implies_encodability(system, encoder):
    elements = system.get_all_elements()
    
    for x in elements:
        # 获取描述
        description = system.describe(x)
        
        if description:
            # 验证可编码
            encoding = encoder.encode_description(description)
            assert encoding is not None
            assert isinstance(encoding, str)
            assert len(encoding) < float('inf')
            
            # 验证编码的单射性
            for y in elements:
                if x != y:
                    desc_y = system.describe(y)
                    if desc_y:
                        enc_y = encoder.encode_description(desc_y)
                        assert encoding != enc_y
                        
    return True

检查点4：连续对象有限表示验证​

def verify_continuous_object_representation():
    # 定义连续对象的有限表示
    continuous_objects = {
        'pi': {
            'algorithm': 'Machin formula: π/4 = 4*arctan(1/5) - arctan(1/239)',
            'definition': 'ratio of circumference to diameter',
            'series': 'π = 4*sum((-1)^n/(2n+1))'
        },
        'e': {
            'definition': 'lim((1 + 1/n)^n) as n→∞',
            'series': 'e = sum(1/n!)',
            'differential': 'solution to dy/dx = y with y(0) = 1'
        },
        'sin': {
            'differential': "y'' + y = 0, y(0)=0, y'(0)=1",
            'series': 'sin(x) = sum((-1)^n * x^(2n+1)/(2n+1)!)',
            'geometric': 'y-coordinate on unit circle'
        }
    }
    
    # 验证每个对象都有有限表示
    for obj_name, representations in continuous_objects.items():
        assert len(representations) > 0
        
        for rep_type, rep_desc in representations.items():
            # 验证表示是有限的
            assert isinstance(rep_desc, str)
            assert len(rep_desc) < float('inf')
            
    return True

检查点5：编码链完整性验证​

def verify_encoding_chain_completeness(system, encoder):
    # 统计各阶段的元素数量
    stats = {
        'total': 0,
        'has_info': 0,
        'distinguishable': 0,
        'describable': 0,
        'encodable': 0
    }
    
    elements = system.get_all_elements()
    stats['total'] = len(elements)
    
    for x in elements:
        # 检查Info(x)
        has_info = False
        for y in elements:
            if x != y and system.describe(x) != system.describe(y):
                has_info = True
                break
                
        if has_info:
            stats['has_info'] += 1
            
            # 必然可区分（因为存在y ≠ x）
            stats['distinguishable'] += 1
            
            # 检查可描述
            if system.describe(x):
                stats['describable'] += 1
                
                # 检查可编码
                if encoder.encode(x):
                    stats['encodable'] += 1
                    
    # 验证链的完整性
    assert stats['has_info'] <= stats['distinguishable']
    assert stats['distinguishable'] <= stats['describable']
    assert stats['describable'] <= stats['encodable']
    
    # 在自指完备系统中，应该相等
    assert stats['has_info'] == stats['encodable']
    
    return True, stats

实用函数​

class CompleteEncoder:
    """完备编码器实现"""
    
    def __init__(self):
        self.description_to_number = {}
        self.number_to_encoding = {}
        self.next_number = 0
        
    def encode_description(self, description):
        """编码描述（Gödel编码的简化版）"""
        if description in self.description_to_number:
            number = self.description_to_number[description]
        else:
            # 分配新编号
            number = self.next_number
            self.description_to_number[description] = number
            self.next_number += 1
            
        # 将数字编码为二进制串
        if number == 0:
            return "0"
        
        binary = ""
        n = number
        while n > 0:
            binary = str(n % 2) + binary
            n //= 2
            
        self.number_to_encoding[number] = binary
        return binary
        
    def encode(self, element):
        """完整的编码函数"""
        # 这里假设element有describe方法或可以获取描述
        if hasattr(element, 'description'):
            desc = element.description
        elif hasattr(element, '__str__'):
            desc = str(element)
        else:
            desc = repr(element)
            
        return self.encode_description(desc)
        
    def decode_to_number(self, encoding):
        """解码到数字"""
        # 二进制转数字
        number = 0
        for bit in encoding:
            number = number * 2 + int(bit)
        return number
        
    def is_complete(self, elements):
        """验证编码完备性"""
        # 尝试编码所有元素
        encodings = set()
        
        for elem in elements:
            enc = self.encode(elem)
            if enc is None:
                return False
            encodings.add(enc)
            
        # 验证单射性
        return len(encodings) == len(elements)


class ContinuousObjectEncoder:
    """连续对象编码器"""
    
    def __init__(self):
        self.representations = {}
        
    def add_representation(self, name, finite_rep):
        """添加连续对象的有限表示"""
        self.representations[name] = finite_rep
        
    def encode_pi(self):
        """编码π"""
        # Machin公式的有限表示
        return "MACHIN:4*atan(1/5)-atan(1/239)"
        
    def encode_e(self):
        """编码e"""
        # 极限定义的有限表示
        return "LIMIT:((1+1/n)^n,n->inf)"
        
    def encode_sin(self):
        """编码sin函数"""
        # 微分方程的有限表示
        return "ODE:y''+y=0,y(0)=0,y'(0)=1"
        
    def verify_finite_representation(self):
        """验证所有表示都是有限的"""
        representations = {
            'pi': self.encode_pi(),
            'e': self.encode_e(),
            'sin': self.encode_sin()
        }
        
        for name, rep in representations.items():
            assert isinstance(rep, str)
            assert len(rep) < float('inf')
            self.representations[name] = rep
            
        return True


class InformationSystem:
    """支持编码完备性的信息系统"""
    
    def __init__(self):
        self.elements = set()
        self.descriptions = {}
        self.encoder = CompleteEncoder()
        
    def add_element(self, elem, description):
        """添加元素及其描述"""
        self.elements.add(elem)
        self.descriptions[elem] = description
        
    def describe(self, elem):
        """获取元素描述"""
        return self.descriptions.get(elem, f"auto_desc_{elem}")
        
    def get_all_elements(self):
        """获取所有元素"""
        return self.elements
        
    def verify_completeness(self):
        """验证编码完备性"""
        # 检查所有有信息的元素都可编码
        for x in self.elements:
            if self.has_information(x):
                encoding = self.encoder.encode(x)
                assert encoding is not None
                
        return True
        
    def has_information(self, x):
        """检查元素是否有信息"""
        for y in self.elements:
            if x != y and self.describe(x) != self.describe(y):
                return True
        return False

形式化验证状态​

  • 定理语法正确
  • 信息定义精确
  • 蕴含链完整
  • 连续对象处理完备
  • 构造性证明清晰
  • 最小完备