Neural Theorem Proving（神经定理证明）
概述
神经定理证明（Neural Theorem Proving）是将神经网络与形式化定理证明相结合的研究领域，旨在利用深度学习的模式识别能力辅助或自动化数学推理过程。
形式化定理证明的核心工具包括：
- Lean：微软研究院开发的定理证明器，核心引擎基于依赖类型理论
- Isabelle/HOL：高阶逻辑定理证明器
- Coq：基于归纳构造演算的证明助手
神经定理证明的目标：
- 证明自动化：自动发现证明策略
- 证明搜索指导：使用神经网络预测下一步证明动作
- 证明修复：自动修复不完整的证明
- 反例检测：使用神经网络发现证明中的错误
形式化定理证明基础
Lean定理证明器
Lean使用依赖类型理论（Dependent Type Theory），核心概念包括：
-- 自然数的类型
#check Nat -- Nat : Type
-- 命题即类型
#check 2 + 2 = 4 -- Eq (2 + 2) 4 : Prop
-- 证明即程序
theorem add_comm (n m : Nat) : n + m = m + n := Nat.add_comm n m
-- 函数定义
def factorial : Nat → Nat
| 0 => 1
| (n + 1) => (n + 1) * factorial n
-- 递归函数
def fib : Nat → Nat
| 0 => 0
| 1 => 1
| (n + 2) => fib n + fib (n + 1)证明策略（Tactics）
Lean中的**策略（tactic）**是证明搜索的基本动作：
-- 基础策略示例
theorem example1 (p q : Prop) : p → p :=
fun hp : p => hp -- λ抽象，hp是p的证明
theorem example2 (p q : Prop) : p → q → p :=
fun hp : p => fun hq : q => hp -- 多个假设
theorem example3 (p q : Prop) : p ∧ q → p :=
fun h : p ∧ q => And.left h -- 合取消去
theorem example4 (p q : Prop) : p ∨ q → q ∨ p :=
fun h : p ∨ q =>
Or.elim h
(fun hp : p => Or.inr hp) -- 分类讨论
(fun hq : q => Or.inl hq)数学库：Mathlib
Mathlib是Lean的最大数学库，包含：
- 实分析、复分析
- 线性代数、抽象代数
- 拓扑学、几何学
- 数论、组合数学
神经网络辅助定理证明
证明状态表示
证明状态（Proof State）包含：
- 上下文（Context）：当前可用的假设
- 目标（Goal）：待证明的命题
from dataclasses import dataclass
from typing import List, Dict
@dataclass
class ProofState:
"""证明状态"""
context: List[str] # 假设列表
goals: List[str] # 目标列表
location: str # 当前位置
def to_text(self) -> str:
"""转换为文本表示"""
lines = []
if self.context:
lines.append("Context:")
for i, hyp in enumerate(self.context):
lines.append(f" {i}: {hyp}")
lines.append("Goal:")
for i, goal in enumerate(self.goals):
lines.append(f" {i}: {goal}")
return "\n".join(lines)策略预测网络
使用序列到序列模型预测下一步策略：
import torch
import torch.nn as nn
from transformers import AutoTokenizer, AutoModel
class TacticPredictor(nn.Module):
"""
策略预测器：给定证明状态，预测下一个策略
"""
def __init__(self, model_name: str = "microsoft/codebert-base", hidden_dim: int = 256):
super().__init__()
self.encoder = AutoModel.from_pretrained(model_name)
self.hidden_dim = self.encoder.config.hidden_size
# 证明状态编码
self.state_projection = nn.Linear(self.hidden_dim, hidden_dim)
# 策略解码器
self.tactic_decoder = nn.GRU(
hidden_dim,
hidden_dim,
num_layers=2,
batch_first=True
)
# 策略词汇表投影
self.tactic_head = nn.Linear(hidden_dim, 50000) # 策略词汇表大小
# 参数绑定预测
self.bindings_head = nn.Linear(hidden_dim, 128) # 假设/变量绑定
def encode_proof_state(self, proof_state_text: str) -> torch.Tensor:
"""编码证明状态"""
tokens = self.tokenizer(
proof_state_text,
return_tensors="pt",
padding=True,
truncation=True,
max_length=512
)
outputs = self.encoder(**tokens)
# 使用[CLS] token表示
return outputs.last_hidden_state[:, 0, :]
def forward(self, proof_state_text: str, tactic_prefix: List[str] = None):
"""
前向传播
Args:
proof_state_text: 证明状态的文本表示
tactic_prefix: 已生成的部分策略（用于自回归生成）
Returns:
tactic_logits: 策略的logits
binding_logits: 参数绑定的logits
"""
# 编码证明状态
state_encoding = self.encode_proof_state(proof_state_text)
state_proj = self.state_projection(state_encoding)
# 解码策略
if tactic_prefix:
tactic_tokens = self.tokenizer(
tactic_prefix,
return_tensors="pt",
padding=True,
add_special_tokens=False
)
tactic_emb = self.encoder.embeddings(tactic_tokens.input_ids)
outputs, _ = self.tactic_decoder(tactic_emb)
tactic_logits = self.tactic_head(outputs)
else:
tactic_logits = None
return tactic_logits, state_proj
class ProofSearch:
"""证明搜索：结合神经网络和搜索算法"""
def __init__(self, predictor: TacticPredictor, beam_width: int = 5):
self.predictor = predictor
self.beam_width = beam_width
def search(self, initial_state: ProofState, max_steps: int = 50) -> List[ProofState]:
"""
束搜索寻找证明
Args:
initial_state: 初始证明状态
max_steps: 最大搜索步数
Returns:
证明路径或空列表
"""
beam = [(initial_state, [], 0.0)] # (状态, 策略序列, 对数概率)
for step in range(max_steps):
candidates = []
for state, tactics, log_prob in beam:
# 检查是否完成
if not state.goals:
return self.reconstruct_proof(state, tactics)
# 预测策略
state_text = state.to_text()
tactic_logits, _ = self.predictor(state_text)
# 获取top-k策略
top_k = torch.topk(tactic_logits[0], k=self.beam_width)
for idx, score in zip(top_k.indices, top_k.values):
tactic_text = self.idx_to_tactic(idx.item())
new_state = self.apply_tactic(state, tactic_text)
if new_state is not None: # 策略有效
candidates.append((
new_state,
tactics + [tactic_text],
log_prob + torch.log_softmax(tactic_logits[0], dim=0)[idx].item()
))
# 保留top-k
beam = sorted(candidates, key=lambda x: x[2], reverse=True)[:self.beam_width]
if not beam:
break
return []
def apply_tactic(self, state: ProofState, tactic: str) -> ProofState:
"""应用策略到证明状态"""
# 这里需要调用Lean的证明引擎
# 简化实现
pass大语言模型定理证明
LLM for Theorem Proving
from transformers import AutoModelForCausalLM, AutoTokenizer
class LLMTheoremProver:
"""基于LLM的定理证明器"""
SYSTEM_PROMPT = """You are an expert Lean theorem prover.
Given a proof state, propose the next tactic to make progress.
Use the following tactics: intro, apply, exact, rw, simp, cases, induction, have, let.
Format your response as:
TACTIC: <tactic>
REASONING: <why this tactic helps>"""
def __init__(self, model_name: str = "meta-llama/Llama-3-8B-Instruct"):
self.tokenizer = AutoTokenizer.from_pretrained(model_name)
self.model = AutoModelForCausalLM.from_pretrained(
model_name,
device_map="auto",
torch_dtype=torch.bfloat16
)
def format_proof_state(self, proof_state: ProofState) -> str:
"""格式化证明状态"""
return proof_state.to_text()
def propose_tactic(self, proof_state: ProofState) -> str:
"""提出策略建议"""
prompt = f"{self.SYSTEM_PROMPT}\n\nProof State:\n{self.format_proof_state(proof_state)}"
messages = [{"role": "user", "content": prompt}]
inputs = self.tokenizer.apply_chat_template(
messages,
return_tensors="pt"
).to(self.model.device)
outputs = self.model.generate(
inputs,
max_new_tokens=256,
temperature=0.7,
do_sample=True
)
response = self.tokenizer.decode(outputs[0][len(inputs[0]):], skip_special_tokens=True)
return self.extract_tactic(response)
def extract_tactic(self, response: str) -> str:
"""从响应中提取策略"""
# 简单解析
for line in response.split('\n'):
if line.startswith('TACTIC:'):
return line.replace('TACTIC:', '').strip()
return ""Reinforcement Learning for Proof Search
class RLTheoremProver:
"""强化学习定理证明器"""
def __init__(self, llm_prover: LLMTheoremProver):
self.prover = llm_prover
self.value_network = ValueNetwork()
def collect_proofs(self, theorems: List[str], n_rollouts: int = 8) -> List[ProofTrajectory]:
"""
收集证明轨迹用于训练
Args:
theorems: 待证明定理列表
n_rollouts: 每个定理的rollout次数
Returns:
证明轨迹列表
"""
trajectories = []
for theorem in theorems:
initial_state = self.parse_theorem(theorem)
for _ in range(n_rollouts):
trajectory = self.rollout(initial_state)
trajectories.append(trajectory)
return trajectories
def rollout(self, initial_state: ProofState) -> ProofTrajectory:
"""单个证明rollout"""
trajectory = ProofTrajectory()
state = initial_state
while len(trajectory.states) < 50:
# 获取策略
tactic = self.prover.propose_tactic(state)
# 获取价值估计
state_encoding = self.prover.format_proof_state(state)
value = self.value_network(state_encoding)
# 应用策略
new_state = self.apply_tactic(state, tactic)
trajectory.add(state, tactic, value)
if new_state is None:
trajectory.reward = -1 # 失败
break
if not new_state.goals:
trajectory.reward = 1 # 成功
break
state = new_state
return trajectory
def compute_advantages(self, trajectories: List[ProofTrajectory]) -> Dict:
"""计算GAE优势"""
advantages = []
returns = []
for traj in trajectories:
# 计算回报
G = 0
for i, step in enumerate(reversed(traj.steps)):
G = step['reward'] + 0.99 * G
returns.insert(0, G)
# GAE计算
if i == 0:
adv = G - step['value']
else:
adv = step['reward'] + 0.99 * step['value'] - prev_value
advantages.insert(0, adv)
prev_value = step['value']
return {
'advantages': advantages,
'returns': returns
}MinWikiVerif证明验证框架
验证条件生成
class VerificationConditionGenerator:
"""从形式化证明生成验证条件"""
def __init__(self):
self.lean_engine = LeanEngine()
def generate_vc(self, theorem: str) -> List[VerificationCondition]:
"""生成验证条件"""
# 解析定理
ast = self.lean_engine.parse(theorem)
# 生成验证条件
vcs = []
for lemma in self.extract_lemmas(ast):
vc = self.lemma_to_vc(lemma)
vcs.append(vc)
return vcs
def lemma_to_vc(self, lemma) -> VerificationCondition:
"""将引理转换为验证条件"""
# 提取前置条件
preconditions = self.extract_preconditions(lemma)
# 提取后置条件
postconditions = self.extract_postconditions(lemma)
# 生成VC: preconditions => postconditions
vc = VerificationCondition(
name=lemma.name,
preconditions=preconditions,
postconditions=postconditions,
proof=lemma.proof
)
return vc
class NeuralVCVerifier:
"""使用神经网络验证验证条件"""
def __init__(self):
self.encoder = VCEncoder()
self.verifier = VerificationMLP()
def verify(self, vc: VerificationCondition, model_weights) -> bool:
"""
验证VC是否成立
Args:
vc: 验证条件
model_weights: 模型权重（用于提取约束）
Returns:
验证结果
"""
# 编码验证条件
vc_encoding = self.encoder(vc)
# 提取模型约束
model_constraints = self.extract_constraints(model_weights)
# 构建验证问题
verification_input = torch.cat([
vc_encoding,
model_constraints
], dim=-1)
# 预测验证结果
with torch.no_grad():
score = self.verifier(verification_input)
return score > 0.5LeanDojo：神经定理证明平台
平台架构
class LeanDojoInterface:
"""LeanDojo接口"""
def __init__(self, lean_version: str = "3.4.2"):
self.lean_version = lean_version
self.docker_container = None
def start(self):
"""启动Lean环境"""
# 启动Docker容器
pass
def interact(self, theorem: str, tactic: str) -> ProofState:
"""
与Lean交互
Args:
theorem: 定理声明
tactic: 策略
Returns:
新的证明状态
"""
# 发送交互请求
request = {
'theorem': theorem,
'tactic': tactic
}
response = self.send_to_lean(request)
return ProofState.from_json(response)
def evaluate_tactic(self, state: ProofState, tactic: str) -> Dict:
"""评估策略"""
result = self.interact(state.theorem, tactic)
return {
'success': len(result.goals) < len(state.goals),
'new_state': result,
'error': result.error if hasattr(result, 'error') else None
}证明修复（Proof Repair）
class ProofRepairer:
"""自动修复失败的证明"""
def __init__(self, prover: LLMTheoremProver):
self.prover = prover
def repair_proof(self, failed_proof: FailedProof) -> FixedProof:
"""
修复失败的证明
Args:
failed_proof: 失败的证明，包含错误位置
Returns:
修复后的证明
"""
# 定位错误位置
error_location = self.locate_error(failed_proof)
# 分析错误类型
error_type = self.classify_error(failed_proof.error)
# 修复策略
if error_type == "tactic_failed":
return self.repair_tactic(failed_proof, error_location)
elif error_type == "induction_needed":
return self.add_induction(failed_proof, error_location)
elif error_type == "lemma_missing":
return self.add_lemma(failed_proof, error_location)
else:
return self.regenerate_subproof(failed_proof, error_location)
def repair_tactic(self, proof: FailedProof, location: int) -> FixedProof:
"""修复失败的策略"""
# 获取失败状态
failed_state = proof.states[location]
# 生成替代策略
alternatives = []
for _ in range(5):
tactic = self.prover.propose_tactic(failed_state)
alternatives.append(tactic)
# 尝试每个替代策略
for alt_tactic in alternatives:
new_proof = proof.copy()
new_proof.tactics[location] = alt_tactic
if self.verify_proof(new_proof):
return new_proof
return None
def add_induction(self, proof: FailedProof, location: int) -> FixedProof:
"""添加归纳步骤"""
# 分析目标类型
inductive_type = self.extract_inductive_type(proof.goals[location])
# 生成归纳策略
induction_tactic = f"induction {inductive_type}"
new_proof = proof.copy()
new_proof.tactics.insert(location, induction_tactic)
return new_proof数据集与基准
常用数据集
| 数据集 | 描述 | 规模 |
|---|---|---|
| MiniF2F | 高中数学竞赛题目 | 488定理 |
| ProofNet | 大学数学证明 | 10,000+定理 |
| NatProver | 自然数证明 | 5,000+定理 |
| LeanDojo Benchmark | Mathlib子集 | 30,000+定理 |
评估指标
def evaluate_prover(prover, test_set: List[Theorem]) -> Dict:
"""评估证明器性能"""
results = {
'total': len(test_set),
'proved': 0,
'failed': 0,
'avg_steps': [],
'avg_time': []
}
for theorem in test_set:
start_time = time.time()
proof = prover.prove(theorem)
elapsed = time.time() - start_time
if proof is not None:
results['proved'] += 1
results['avg_steps'].append(len(proof.tactics))
else:
results['failed'] += 1
results['avg_time'].append(elapsed)
results['success_rate'] = results['proved'] / results['total']
results['avg_steps'] = np.mean(results['avg_steps'])
results['avg_time'] = np.mean(results['avg_time'])
return results与神经网络验证的结合
神经定理证明可以与形式化验证结合：
class VerifiedNeuralTheoremProver:
"""可验证的神经定理证明器"""
def __init__(self, prover: LLMTheoremProver, verifier: NeuralVCVerifier):
self.prover = prover
self.verifier = verifier
def prove_with_verification(self, theorem: str) -> Tuple[Proof, bool]:
"""
证明并验证
Returns:
(证明, 验证结果)
"""
# 生成候选证明
proof = self.prover.prove(theorem)
if proof is None:
return None, False
# 生成验证条件
vc_generator = VerificationConditionGenerator()
vcs = vc_generator.generate_vc(theorem)
# 验证每个VC
all_verified = True
for vc in vcs:
if not self.verifier.verify(vc, proof.model_weights):
all_verified = False
break
return proof, all_verified挑战与未来方向
当前挑战
- 长程依赖：数学证明往往需要数百步
- 符号操作：需要精确的符号推理能力
- 创造性问题：某些证明需要”灵光一现”
- 数据稀缺：形式化证明数据有限
未来方向
- 工具集成：更好的LLM与形式化工具集成
- 多模态推理：结合自然语言和形式化语言
- 元学习：学习如何学习证明
- 人机协作：人类专家与AI协同证明
总结
神经定理证明代表了人工智能与数学交叉的前沿研究方向。通过结合：
- 神经网络的模式识别能力：预测证明策略
- 形式化验证的严格性：保证证明正确性
- 强化学习的探索能力：发现新颖证明路径
神经定理证明正在逐步接近解决复杂数学问题的目标，未来有望在数学研究、科学发现等领域发挥重要作用。