def dfa_intersection(dfa_1: dict, dfa_2: dict) ->dict:
    """ Returns a DFA accepting the intersection of the DFAs in
    input.

    Let :math:`A_1 = (Σ, S_1 , s_{01} , ρ_1 , F_1 )` and
    :math:`A_2 = (Σ, S_2 , s_{02} , ρ_2 , F_2 )` be two DFAs.
    Then there is a DFA :math:`A_∧` that runs simultaneously both
    :math:`A_1` and :math:`A_2` on the input word and
    accepts when both accept.
    It is defined as:

    :math:`A_∧ = (Σ, S_1 × S_2 , (s_{01} , s_{02} ), ρ, F_1 × F_2 )`

    where

    :math:`ρ((s_1 , s_2 ), a) = (s_{X1} , s_{X2} )` iff
    :math:`s_{X1} = ρ_1 (s_1 , a)` and :math:`s_{X2}= ρ_2 (s_2 , a)`

    Implementation proposed guarantees the resulting DFA has only
    **reachable** states.

    :param dict dfa_1: first input DFA;
    :param dict dfa_2: second input DFA.
    :return: *(dict)* representing the intersected DFA.
    """
    intersection = {'alphabet': dfa_1['alphabet'].intersection(dfa_2[
        'alphabet']), 'states': {(dfa_1['initial_state'], dfa_2[
        'initial_state'])}, 'initial_state': (dfa_1['initial_state'], dfa_2
        ['initial_state']), 'accepting_states': set(), 'transitions': dict()}
    boundary = set()
    boundary.add(intersection['initial_state'])
    while boundary:
        state_dfa_1, state_dfa_2 = boundary.pop()
        if state_dfa_1 in dfa_1['accepting_states'] and state_dfa_2 in dfa_2[
            'accepting_states']:
            intersection['accepting_states'].add((state_dfa_1, state_dfa_2))
        for a in intersection['alphabet']:
            if (state_dfa_1, a) in dfa_1['transitions'] and (state_dfa_2, a
                ) in dfa_2['transitions']:
                next_state_1 = dfa_1['transitions'][state_dfa_1, a]
                next_state_2 = dfa_2['transitions'][state_dfa_2, a]
                if (next_state_1, next_state_2) not in intersection['states']:
                    intersection['states'].add((next_state_1, next_state_2))
                    boundary.add((next_state_1, next_state_2))
                intersection['transitions'][(state_dfa_1, state_dfa_2), a
                    ] = next_state_1, next_state_2
    return intersection
