AI辅助形式化验证

#!/usr/bin/env python3"""演示一个极简的AI辅助证明搜索：用策略函数选择推理规则，在Lean风格证明树中寻找可达目标的路径。""" import numpy as npfrom collections import deque  class ProofState(object):    """表示当前证明状态：已推导事实集合和目标公式。"""     def __init__(self, facts, goal):        self.facts = facts        self.goal = goal        self._hash = None     def __eq__(self, other):        return self.facts == other.facts and self.goal == other.goal     def __hash__(self):        if self._hash is None:            self._hash = hash((frozenset(self.facts), self.goal))        return self._hash  class RuleSet(object):    """简单的符号推理规则集。每条规则返回新加入的事实或None。"""     def __init__(self):        self.rules = [            ("mp", self._modus_ponens),            ("and_intro", self._and_intro),            ("contra", self._contraposition),        ]     def _modus_ponens(self, facts):        for f in list(facts):            if isinstance(f, tuple) and f[0] == "imp":                _, a, b = f                if a in facts:                    return b        return None     def _and_intro(self, facts):        for a in facts:            for b in facts:                if (a, b) not in facts:                    return ("and", a, b)        return None     def _contraposition(self, facts):        for f in list(facts):            if isinstance(f, tuple) and f[0] == "imp":                _, a, b = f                nb = ("not", b)                na = ("not", a)                if nb in facts and na not in facts:                    return na        return None     def applicable(self, state):        """返回当前可推出的新事实集合。"""        cands = []        for _, rule in self.rules:            new = rule(state.facts)            if new is not None and new not in state.facts:                cands.append(new)        return cands  class PolicyNetwork(object):    """用随机权重模拟AI策略网络：对候选动作打分。"""     def __init__(self, seed=42):        self.rng = np.random.RandomState(seed)        self.emb = {None: self.rng.randn(4)}     def score(self, action):        key = action[0] if isinstance(action, tuple) else str(action)        if key not in self.emb:            self.emb[key] = self.rng.randn(4)        return float(np.dot(self.emb[key], self.rng.randn(4)))  class AIProver(object):    """基于策略优先和队列搜索的极简证明器。"""     def __init__(self):        self.rules = RuleSet()        self.policy = PolicyNetwork()     def prove(self, init_facts, goal, max_steps=500):        start = ProofState(frozenset(init_facts), goal)        queue = deque([start])        visited = set([start])        parent = {start: (None, None)}         steps = 0        while queue and steps < max_steps:            state = queue.popleft()            if state.goal in state.facts:                return self._reconstruct(parent, state)            cands = self.rules.applicable(state)            cands.sort(key=self.policy.score, reverse=True)            for c in cands:                new_facts = set(state.facts)                new_facts.add(c)                ns = ProofState(frozenset(new_facts), state.goal)                if ns not in visited:                    visited.add(ns)                    parent[ns] = (state, c)                    queue.append(ns)                    steps += 1        return None     def _reconstruct(self, parent, node):        path = []        while parent[node][0] is not None:            prev, action = parent[node]            path.append(action)            node = prev        path.reverse()        return path  def main():    # 公理：P -> Q, Q -> R, P    facts = [        ("imp", "P", "Q"),        ("imp", "Q", "R"),        "P",    ]    prover = AIProver()    proof = prover.prove(facts, "R")    print("证明步骤:", proof)    if proof is not None:        assert ("imp", "P", "Q") in facts        print("证明成功，R已被推出")    else:        print("未找到证明")  if __name__ == "__main__":    main()