I don’t think computability is the right framework to think about this question. Mostly this is because, for somewhat silly reasons, there exists a Turing machine that answers any fixed finite sequence of questions. It’s just the Turing machine which prints out the answers to those questions. “But which Turing machine is that?” Dunno, this is an existence proof. But the point is that you can’t just say “you can’t build a device that solves the halting problem” because, for any fixed finite set of Turing machines, you can trivially build such a device, you just can’t recognize it once you’ve built it...
It follows that in the framework of computability, to distinguish a “real halting oracle” from a rock with stuff scribbled on it, you need to ask infinitely many questions. And that’s pretty hard to do.
I think Toby Ord actually has a good response to this objection:
It is worth responding, at this point, to a persistent argument that is made against the coherence of physical hypercomputation. It states that a machine must perform an infinite number of computations for it to count as hypercomputational. After all, for any function f on the integers, there is a Turing machine that computes f(n) for an arbitrarily large finite number of values of n. It is thus claimed either that we could not know that a prospective machine was a hypermachine after witnessing finitely many computations or that it simply would not be a hypermachine since its behaviour could be simulated by a Turing machine. Hypercomputation is thus claimed to be on shaky ground. However, it is easy to see that this argument cannot achieve what it hopes to since it does not just collapse hypercomputation to classical computation, but instead it collapses all computation on the integers down to that of finite state machines. This is because for any function f on the integers, there is also a finite state machine that computes f(n) on an arbitrarily large finite domain. Thus, as explained in Copeland [10], whatever force this argument is supposed to have with respect to hypercomputation, it must also have with respect to refuting the existence of classical (general recursive) computation. Since the latter is agreed to be on firm ground, it is difficult to see why this argument should count against the former.
Recall that a physical computer is technically a finite state machine (it doesn’t have infinite memory). This is another reason I don’t think Turing machines are a good formalism for talking about computation in the real world.
It is thus claimed either that we could not know that a prospective machine was a hypermachine after witnessing finitely many computations or that it simply would not be a hypermachine since its behaviour could be simulated by a Turing machine. Hypercomputation is thus claimed to be on shaky ground.
The former suggestion seems like the more important point here. While true that the hypercomputer’s behavior can be simulated on a Turing machine, this is only true of the single answer given and received, not of the abstractly defined problem being solved. The hypercomputer still cannot be proven by a Turing machine to have knowledge of a fact that a Turing machine could not.
And so the words “shaky ground” are used loosely here. The argument doesn’t refute the theoretic “existence” of recursive computation any more than it refutes the existence of theoretic hypercomputation. That finite state machines are the only realizable form of Turing machine is hardly a point in their generalized disfavor.
I think Toby Ord actually has a good response to this objection:
Recall that a physical computer is technically a finite state machine (it doesn’t have infinite memory). This is another reason I don’t think Turing machines are a good formalism for talking about computation in the real world.
The former suggestion seems like the more important point here. While true that the hypercomputer’s behavior can be simulated on a Turing machine, this is only true of the single answer given and received, not of the abstractly defined problem being solved. The hypercomputer still cannot be proven by a Turing machine to have knowledge of a fact that a Turing machine could not.
And so the words “shaky ground” are used loosely here. The argument doesn’t refute the theoretic “existence” of recursive computation any more than it refutes the existence of theoretic hypercomputation. That finite state machines are the only realizable form of Turing machine is hardly a point in their generalized disfavor.