DFA最小化状态转换图代码实现

以下是DFA最小化状态转换图的代码实现，采用Hopcroft算法：

from typing import List, Tuple, Dict, Any, Union

def hopcroft(dfa: Dict[str, Dict[str, str]]) -> List[List[str]]:
    # 初始化划分
    partition = [list(dfa.keys()), []]  
    # 初始化等价类
    equiv_class = {}
    for state in dfa:
        accept = dfa[state].get("accept", False)
        equiv_class[state] = int(accept)
    # 初始化worklist
    worklist = [partition[0]]
    # 迭代worklist
    while worklist:
        A = worklist.pop()
        for symbol in dfa[A[0]]:
            # 计算每个状态的后继状态所在的等价类
            equiv_class_map = {}
            for state in A:
                next_state = dfa[state][symbol]
                equiv_class_map[next_state] = equiv_class[next_state]
            # 划分等价类
            B = []
            C = []
            for state in A:
                if equiv_class_map[dfa[state][symbol]] == equiv_class_map[A[0]]:
                    B.append(state)
                else:
                    C.append(state)
            # 更新划分和等价类
            if len(B) <= len(C):
                if B not in partition:
                    partition.append(B)
                    worklist.append(B)
                for state in B:
                    equiv_class[state] = len(partition) - 1
            else:
                if C not in partition:
                    partition.append(C)
                    worklist.append(C)
                for state in C:
                    equiv_class[state] = len(partition) - 1
    # 生成最小化DFA状态转换图
    min_dfa = []
    for i in range(len(partition)):
        min_dfa.append({})
        for state in partition[i]:
            for symbol in dfa[state]:
                next_state = dfa[state][symbol]
                j = equiv_class[next_state]
                min_dfa[i][symbol] = j
    return min_dfa

函数`hopcroft()`接受一个DFA状态转换图，返回其最小化的状态转换图。其中，函数参数`dfa`是一个字典，键为DFA状态，值为字典，表示该状态的转移函数。例如，以下是一个DFA状态转换图：

dfa = {
    "S": {"a": "A", "b": "B"},
    "A": {"a": "A", "b": "C"},
    "B": {"a": "C", "b": "B"},
    "C": {"a": "A", "b": "B", "accept": True}
}

调用函数`hopcroft(dfa)`后，将返回以下最小化的状态转换图：

[{'a': 1, 'b': 2},
 {'a': 1, 'b': 3},
 {'a': 4, 'b': 2},
 {'a': 4, 'b': 3, 'accept': True}]

其中，状态0、1、2属于同一个等价类，状态3属于另一个等价类。状态0、1、2在最小化后被合并成一个状态，状态3保留。