So if you had a Turing machine traveling along this wordline, it could send a signal to p if and onliy if it halted, and the observer at p is guaranteed to receive the signal if the machine ever halts. No infinite-precision measurements are involved (unless perhaps you believe that a Turing machine operating reliably for an indefinite period of time is tantamount to an infinite-precision measurement).
Even if such spacetimes were possible, in order to exploit them for hypercomptation you would require a true Turing machine with infinite tape, infinite energy supply (unless it was perfectly reversible) and enough durability to run for a literally infinite amount of proper time without breaking. Such requirements seem inconsistent with the known laws of physics.
Why do you believe infinite memory is incompatible with the fundamental laws of physics? In any case, if it is, then Turing machines are physically impossible as well, the only physically possible computational systems are finite state automata, and there is no halting problem. You are right that in this case the infinite run-time provided by M-H spacetimes will not give any computational advantage, since every finite state machine either halts or loops anyway. But in discussions of computability theory I always assume that we are idealizing away memory co...
"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?