You can certainly add as many disconnected states as you
want. However, I think that the definition of DFA or NFA stipulates
that the number of states is finite (that is what the F stands
for). So it would not be a DFA. However, even with an infinity of
states, it can certainly recognize a regular language, if
appropriately built (as you suggest, for example).
The reason why the definition stipulates a finite number of states is
that an infinite number of states will allow you to recognize about
anything. After all, a Turing Machine is an automaton just like the
DFA, but with an infinite number of states, that you can actually
enumerate, as you can enumerate the transitions (also infinite in
I have not looked at it previously, but I believe that you can
actually do much better than Turing Machine with a DIA (Deterministic
Infinite Automaton). You have only countably many TM, but you have
continuously many DIAs, as many as you have real numbers. So I should
expect that they would do things one does not even expect from TM.
On the other hand, anything they do, they will do in finite number of
steps, so that remains a limitation (as compared to Zeno machines, or machines using looping time-lines). I
did not see anything in a fast look on the hypercomputation page of wikipedia.
I am sure someone has worked on continuous machines (I do not dare
call them real machines, real numbers are enough of an abusive claim). Maybe looking at work on computable reals,
and related literature can put some light on this.
They can solve the halting problem for ordinary TM. You just follow
the usual diagonalization construction that describe a solution for
the halting problem. Since you do not require that construction to be
describable by an ordinary Turing Machine, there is no
contradiction. Isn't this a happy world?
The only difficulty, albeit a minor one, is that we do not know how to
implement them. Miracles are not of this world.. . yet.