type: theorem
verification: machine_ready
dependencies: ["T1-1-formal.md", "T2-1-formal.md", "T2-2-formal.md", "D1-1-formal.md"]
verification_points:
- encoding_efficiency_definition
- information_theoretic_bound
- inefficient_encoding_contradiction
- optimization_necessity
- constraint_emergence
EncodingOptimization : Prop ≡
∀S : System .
(SelfRefComplete(S) ∧ EntropyIncrease(S)) →
∃E* : S → Σ* .
IsOptimalEncoding(E*) ∧ E* = argmin_{E} L_max(E)
where
L_max(E) : ℕ ≡ max_{s ∈ S} |E(s)|
IsOptimalEncoding(E) : Prop (defined below)
EncodingEfficiency : Type ≡ {
encoding : S → Σ*,
max_length : ℕ,
avg_length : ℝ,
alphabet_size : ℕ
}
L_max(E) : ℕ ≡ max_{s ∈ S} |E(s)|
L_avg(E) : ℝ ≡ (1/|S|) * Σ_{s ∈ S} |E(s)|
IsEfficient(E) : Prop ≡
L_max(E) ≤ c * log_{|Σ|} |S| for some constant c
IsOptimalEncoding(E) : Prop ≡
UniquelyDecodable(E) ∧
PrefixFree(E) ∧
SelfEmbeddable(E) ∧
AsymptoticallyOptimal(E)
where
AsymptoticallyOptimal(E) ≡
L_max(E) = O(log |S|)
InformationTheoreticBound : Prop ≡
∀E : S → Σ* . UniquelyDecodable(E) →
L_max(E) ≥ log_{|Σ|} |S|
Proof of information theoretic bound:
Given unique decodability requirement:
1. Need to encode |S| distinct states
2. Using alphabet of size |Σ|
3. Strings of length L can represent at most |Σ|^L states
4. Require: |Σ|^{L_max} ≥ |S|
5. Taking logarithm: L_max ≥ log_{|Σ|} |S| ∎
InefficientEncodingContradiction : Prop ≡
∀S,E . (SelfRefComplete(S) ∧ ¬IsEfficient(E)) →
¬CanDescribe(S, E)
where
CanDescribe(S, E) ≡ E ∈ S → Desc(E) ∈ L
Proof by contradiction:
Assume inefficient encoding E with L_max(E) = c·|S| for c > 0:
1. As t → ∞, |S_t| → ∞ (by entropy increase)
2. For inefficient E: L_max(E) → ∞ rapidly
3. To describe E, need to store mapping table:
- For each s ∈ S, store pair (s, E(s))
- Each E(s) has length up to c·|S|
- Total storage: |S| × c·|S| = c·|S|²
4. Therefore |Desc(E)| ≥ c·|S|² → ∞
5. But self-referential completeness requires:
- Desc : S → L where L is finite strings
- ∀ℓ ∈ L : |ℓ| < ∞
6. Contradiction: |Desc(E)| → ∞ implies Desc(E) ∉ L
Therefore inefficient encoding violates self-reference ∎
OptimizationNecessity : Prop ≡
∀S . (SelfRefComplete(S) ∧ EntropyIncrease(S)) →
∀E . (E encodes S) → IsEfficient(E)
Proof of optimization necessity:
Given self-referential completeness and entropy increase:
1. By T2-1: Encoding mechanism must exist
2. By T2-2: All information must be encodable
3. By Lemma T2-3.2: Inefficient encoding leads to contradiction
4. Only efficient encodings are compatible with:
- Infinite state growth
- Finite description requirement
- Self-description capability
5. System must use efficient encoding to maintain:
- Self-referential completeness
- Ability to describe its own encoding function
Therefore optimization is necessary ∎
EncodingConstraintEmergence : Prop ≡
∀E* . IsOptimalEncoding(E*) →
UniquelyDecodable(E*) ∧
PrefixFree(E*) ∧
SelfEmbeddable(E*)
where
UniquelyDecodable(E) ≡ ∀s₁,s₂ . s₁ ≠ s₂ → E(s₁) ≠ E(s₂)
PrefixFree(E) ≡ ∀s₁,s₂ . E(s₁) is not prefix of E(s₂)
SelfEmbeddable(E) ≡ E ∈ Domain(E) ∧ E(E) ∈ Range(E)
Proof of constraint emergence:
1. Unique decodability:
- Required for information preservation
- Without it, cannot recover original states
2. Prefix-free property:
- Enables immediate decodability
- Avoids ambiguity in parsing
- Necessary for streaming/real-time processing
3. Self-embeddability:
- E must encode itself (self-reference)
- E(E) must be valid encoding
- Required by self-referential completeness
These constraints emerge naturally from requirements ∎
MainTheorem : Prop ≡
∀S . (SelfRefComplete(S) ∧ EntropyIncrease(S)) →
∃E* : S → Σ* .
E* = argmin_{E} L_max(E) ∧
L_max(E*) = O(log |S|)
Proof of encoding optimization:
Given SelfRefComplete(S) and EntropyIncrease(S):
1. By T2-1: Encoding exists
2. By T2-2: All information encodable
3. By Lemma T2-3.1: L_max ≥ log_{|Σ|} |S|
4. By Lemma T2-3.2: Inefficient encoding impossible
5. By Lemma T2-3.3: Must use efficient encoding
6. By Lemma T2-3.4: Optimal encoding has required properties
7. Define E* as encoding achieving:
L_max(E*) = min{L_max(E) : E satisfies constraints}
8. E* exists and satisfies:
- L_max(E*) = O(log |S|)
- All required constraints
- Self-describability
Therefore system evolves optimal encoding ∎
def verify_encoding_efficiency_definition(encoding_system):
# 计算最大和平均编码长度
encodings = encoding_system.get_all_encodings()
max_length = max(len(code) for code in encodings.values())
avg_length = sum(len(code) for code in encodings.values()) / len(encodings)
# 验证效率度量
state_count = len(encodings)
alphabet_size = encoding_system.alphabet_size
# 信息论下界
theoretical_min = math.log(state_count) / math.log(alphabet_size)
# 检查是否接近最优
efficiency_ratio = max_length / theoretical_min
assert efficiency_ratio < 10 # 允许常数因子
assert avg_length <= max_length
return True, {
'max_length': max_length,
'avg_length': avg_length,
'theoretical_min': theoretical_min,
'efficiency_ratio': efficiency_ratio
}
def verify_information_theoretic_bound(encoding_system):
state_count = encoding_system.get_state_count()
alphabet_size = encoding_system.alphabet_size
max_length = encoding_system.get_max_length()
# 计算理论下界
theoretical_bound = math.log(state_count) / math.log(alphabet_size)
# 验证实际编码满足下界
assert max_length >= theoretical_bound - 0.01 # 允许浮点误差
# 验证唯一可解码性
assert encoding_system.is_uniquely_decodable()
return True
def verify_inefficient_encoding_contradiction(system):
# 创建低效编码（线性长度）
inefficient_encoder = LinearEncoder() # L_max = c * |S|
# 模拟系统增长
for t in range(10):
system.evolve()
state_count = system.get_state_count()
# 计算描述低效编码所需的空间
encoding_table_size = state_count * inefficient_encoder.get_max_length(state_count)
# 验证描述长度增长
assert encoding_table_size > state_count ** 1.5
# 验证无法自描述
try:
desc = system.describe(inefficient_encoder)
assert len(desc) < float('inf')
# 如果能描述，检查是否违反了有限性
assert False, "Should not be able to describe inefficient encoder"
except:
# 预期：无法描述
pass
return True
def verify_optimization_necessity(system):
# 创建不同效率的编码器
encoders = [
OptimalEncoder(), # L_max = O(log |S|)
SuboptimalEncoder(), # L_max = O(log² |S|)
InefficientEncoder() # L_max = O(|S|)
]
# 演化系统
for t in range(20):
system.evolve()
# 检查哪些编码器仍然可行
viable_encoders = []
for encoder in encoders:
if encoder.can_self_describe(system):
viable_encoders.append(encoder)
# 验证只有高效编码器存活
assert len(viable_encoders) <= 2
assert any(isinstance(e, OptimalEncoder) for e in viable_encoders)
assert not any(isinstance(e, InefficientEncoder) for e in viable_encoders)
return True
def verify_constraint_emergence(optimal_encoder):
# 验证唯一可解码性
assert optimal_encoder.is_uniquely_decodable()
# 验证前缀自由性
assert optimal_encoder.is_prefix_free()
# 验证自嵌入性
assert optimal_encoder.can_encode_self()
# 验证这些性质对优化的贡献
# 移除任一性质会增加编码长度
# 测试非前缀自由编码
non_prefix_encoder = optimal_encoder.remove_prefix_free_constraint()
assert non_prefix_encoder.get_max_length() > optimal_encoder.get_max_length()
return True
class OptimalEncoder:
"""最优编码器实现"""
def __init__(self, alphabet_size=2):
self.alphabet_size = alphabet_size
self.encoding_table = {}
self.decoding_table = {}
def encode(self, state):
"""使用接近最优的编码"""
if state in self.encoding_table:
return self.encoding_table[state]
# 分配新的最短可用编码
code = self._find_shortest_available_code()
self.encoding_table[state] = code
self.decoding_table[code] = state
return code
def _find_shortest_available_code(self):
"""找到最短的可用编码（保持前缀自由）"""
length = 1
while True:
for code in self._generate_codes_of_length(length):
if self._is_available_and_prefix_free(code):
return code
length += 1
def _is_available_and_prefix_free(self, code):
"""检查编码是否可用且保持前缀自由性"""
# 检查是否已使用
if code in self.decoding_table:
return False
# 检查前缀自由性
for existing_code in self.decoding_table:
if code.startswith(existing_code) or existing_code.startswith(code):
return False
return True
def get_max_length(self):
"""获取最大编码长度"""
if not self.encoding_table:
return 0
return max(len(code) for code in self.encoding_table.values())
def get_efficiency_ratio(self):
"""计算效率比率"""
state_count = len(self.encoding_table)
if state_count == 0:
return 1.0
theoretical_min = math.log(state_count) / math.log(self.alphabet_size)
actual_max = self.get_max_length()
return actual_max / theoretical_min if theoretical_min > 0 else float('inf')
def can_self_describe(self, system):
"""检查是否能自描述"""
# 估计描述自身所需的空间
state_count = len(self.encoding_table)
max_length = self.get_max_length()
# 描述需要存储整个编码表
description_size = state_count * max_length
# 检查是否在系统的描述能力范围内
return description_size < system.max_description_length
def is_uniquely_decodable(self):
"""验证唯一可解码性"""
# 检查是否有重复编码
codes = list(self.encoding_table.values())
return len(codes) == len(set(codes))
def is_prefix_free(self):
"""验证前缀自由性"""
codes = list(self.encoding_table.values())
for i, code1 in enumerate(codes):
for code2 in codes[i+1:]:
if code1.startswith(code2) or code2.startswith(code1):
return False
return True
def can_encode_self(self):
"""验证自嵌入性"""
# 编码器能否编码自己
try:
self_encoding = self.encode(self)
return self_encoding is not None
except:
return False
class LinearEncoder:
"""线性长度编码器（低效）"""
def __init__(self):
self.counter = 0
def encode(self, state):
"""使用线性长度编码"""
# 简单地使用计数器值的一元编码
self.counter += 1
return "1" * self.counter + "0"
def get_max_length(self, state_count):
"""最大编码长度与状态数成正比"""
return state_count + 1
class EncodingEvolutionSystem:
"""编码演化系统"""
def __init__(self):
self.states = set()
self.time = 0
self.encoder = None
self.max_description_length = 1000
def evolve(self):
"""系统演化"""
# 添加新状态（模拟熵增）
new_state_count = max(1, int(len(self.states) * 0.5))
for i in range(new_state_count):
self.states.add(f"state_t{self.time}_n{i}")
self.time += 1
# 检查当前编码器是否仍然可行
if self.encoder and not self.encoder.can_self_describe(self):
# 需要更优化的编码器
self.optimize_encoder()
def optimize_encoder(self):
"""优化编码器"""
# 切换到更高效的编码
self.encoder = OptimalEncoder()
def get_state_count(self):
return len(self.states)
def describe(self, obj):
"""描述对象"""
if isinstance(obj, OptimalEncoder):
# 估计编码器的描述长度
desc_length = obj.get_state_count() * obj.get_max_length()
if desc_length > self.max_description_length:
raise ValueError("Cannot describe: too large")
return f"encoder_description_length_{desc_length}"
return str(obj)