Well, an infinite memory store or an infinite energy source would have infinite mass. So it would either take up the entire universe and have nowhere external to send its results to or, if it had finite size, it would be inside its own Schwarzchild radius, and there would be no way to send a signal out through its event horizon.
So yeah, I'd call infinite storage or power sources (as politely as possible) "unphysical".
And I don't see why you think the halting problem goes away just because you can't put infinite tape in your Turing machine, or because you use finite state automata instead. You still can't set an upper bound on the size of computation needed to determine whether any algorithm in general will terminate, and I kind of thought the point of the halting problem was that it doesn't only apply to actual Turing machines.
So it would either take up the entire universe and have nowhere external to send its results to or, if it had finite size, it would be inside its own Schwarzchild radius, and there would be no way to send a signal out through its event horizon.
These are not the only options. Infinite sets have infinite proper subsets, so an object in a spatially infinite universe could have infinite size without taking up the entire universe. In a universe with an infinite amount of matter, a computational process could requisition an infinite proper subset of that mat...
"Hypercomputation" is a term coined by two philosophers, Jack Copeland and Dianne Proudfoot, to refer to allegedly computational processes that do things Turing machines are in principle incapable of doing. I'm somewhat dubious of whether any of the proposals for "hypercomputation" are really accurately described as computation, but here, I'm more interested in another question: is there any chance it's possible to build a physical device that answers questions a Turing machine cannot answer?
I've read a number of Copeland and Proudfoot's articles promoting hypercomputation, and they claim this is an open question. I have, however, seen some indications that they're wrong about this, but my knowledge of physics and computability theory isn't enough to answer this question with confidence.
Some of the ways to convince yourself that "hypercomputation" might be physically possible seem like obvious confusions, for example if you convince yourself that some physical quality is allowed to be any real number, and then notice that because some reals are non-computable, you say to yourself that if only we could measure such a non-computable quantity then we could answer questions no Turing machine could answer. Of course, the idea of doing such a measurement is physically implausible even if you could find a non-computable physical quantity in the first place. And that mistake can be sexed up in various ways, for example by talking about "analog computers" and assuming "analog" means it has components that can take any real-numbered value.
Points similar to the one I've just made exist in the literature on hypercomputation (see here and here, for example). But the critiques of hypercomputation I've found tend to focus on specific proposals. It's less clear whether there are any good general arguments in the literature that hypercomputation is physically impossible, because it would require infinite-precision measurements or something equally unlikely. It seems like it might be possible to make such an argument; I've read that the laws of physics are consiered to be computable, but I don't have a good enough understanding of what that means to tell if it entails that hypercomputation is physically impossible.
Can anyone help me out here?