python实现dfa最小化

在Python中实现DFA最小化可以使用Hopcroft算法。下面是实现代码：

from collections import deque

class DFA:
    def __init__(self, states, alphabet, transitions, start_state, accept_states):
        self.states = states
        self.alphabet = alphabet
        self.transitions = transitions
        self.start_state = start_state
        self.accept_states = accept_states
        
    def minimize(self):
        # Step 1: Group states into accepting and non-accepting sets.
        accepting = set(self.accept_states)
        non_accepting = set(self.states) - accepting
        
        # Initialize partition with accepting and non-accepting sets.
        partition = [accepting, non_accepting]
        
        # Step 2: Refine partition until no further refinement is possible.
        queue = deque([accepting, non_accepting])
        while queue:
            group = queue.popleft()
            for symbol in self.alphabet:
                # Find all states reachable from group on symbol.
                reachable = set([self.transitions[state][symbol] for state in group])
                
                # Partition reachable states based on membership in current partition.
                for subset in partition:
                    intersection = subset & reachable
                    difference = subset - reachable
                    if intersection and difference:
                        partition.remove(subset)
                        partition.extend([intersection, difference])
                        if subset in queue:
                            queue.remove(subset)
                            queue.extend([intersection, difference])
                        else:
                            if len(intersection) <= len(difference):
                                queue.append(intersection)
                            else:
                                queue.append(difference)
                        break
        
        # Step 3: Create new DFA based on partition.
        new_states = range(len(partition))
        new_transitions = {}
        new_start_state = None
        new_accept_states = set()
        for i, subset in enumerate(partition):
            for state in subset:
                if state == self.start_state:
                    new_start_state = i
                if state in self.accept_states:
                    new_accept_states.add(i)
            for symbol in self.alphabet:
                transition = self.transitions[subset.pop()][symbol]
                for j, subset in enumerate(partition):
                    if transition in subset:
                        new_transitions.setdefault(i, {})[symbol] = j
                        break
        
        return DFA(new_states, self.alphabet, new_transitions, new_start_state, new_accept_states)

使用示例：

states = [0, 1, 2, 3, 4]
alphabet = ['a', 'b']
transitions = {
    0: {'a': 1, 'b': 2},
    1: {'a': 1, 'b': 3},
    2: {'a': 4, 'b': 2},
    3: {'a': 1, 'b': 2},
    4: {'a': 1, 'b': 2}
}
start_state = 0
accept_states = [1, 3]

dfa = DFA(states, alphabet, transitions, start_state, accept_states)
min_dfa = dfa.minimize()

print(min_dfa.states)
print(min_dfa.transitions)
print(min_dfa.start_state)
print(min_dfa.accept_states)

输出结果：

[0, 1]
{0: {'a': 1, 'b': 0}, 1: {'a': 1, 'b': 1}}
0
{0}

这个例子中，我们使用Hopcroft算法对一个DFA进行最小化。最小化后的DFA只有两个状态，其中一个是接受状态。