A possibly-relevant consideration in the analogy to computation is that the threshold of Turing completeness is in some sense extremely low (see one-instruction set computer, Turing tarpits, Rule 110), and is the final threshold.
Nitpick, but it actually isn’t the final threshold of computation, though the things that would allow you to compute beyond a Turing Machine are basically cases where we are majorly wrong on the physical laws of the universe, or we somehow have a way to control the fundamental physical constants and/or laws of the universe, and the computers that can legitimately claim to go beyond Turing Machines with known physics aren’t useful computers due to the No Free Lunch theorems.
The point is that a random Turing Machine’s output is technically uncomputable, which is nice, but it’s entirely useless because it uses an entirely flat prior, because it entirely picks randomly from all possible universes, and a No Free Lunch argument can be deployed to show why this isn’t useful, because it picks at random from all possible universes/functions.
This, incidentally resolves gedymin’s question on the difference between a random hypercomputer and a useful hypercomputer: A useful hypercomputer trades off performance for certain functions/universes in order to do better in other functions/universes, while a random hypercomputer doesn’t do that and thus is useless.
The point is that a random Turing Machine’s output is technically uncomputable
What do you mean? The output of any Turing machine is computable by definition. Do you mean solving the halting problem for a random Turing machine? Or a random oracle?
Fair. I think this is indeed a nitpick. 😊 In case it wasn’t clear, the point remains something like: When we observe/build computational systems in our world that are “better” along some axis than other systems, that “betterness” is not generally derived from having gone over a new threshold of “even more general” computation (they definitely aren’t deriving it from hypercomputation, and often aren’t even deriving it from universal Turing computation), but through being better suited to the capability in question.
Nitpick, but it actually isn’t the final threshold of computation, though the things that would allow you to compute beyond a Turing Machine are basically cases where we are majorly wrong on the physical laws of the universe, or we somehow have a way to control the fundamental physical constants and/or laws of the universe, and the computers that can legitimately claim to go beyond Turing Machines with known physics aren’t useful computers due to the No Free Lunch theorems.
Just worth keeping that in mind.
Non-sequitur, the no-free-lunch theorems don’t have anything to do with the physical realizability of hypercomputers.
The point is that a random Turing Machine’s output is technically uncomputable, which is nice, but it’s entirely useless because it uses an entirely flat prior, because it entirely picks randomly from all possible universes, and a No Free Lunch argument can be deployed to show why this isn’t useful, because it picks at random from all possible universes/functions.
This, incidentally resolves gedymin’s question on the difference between a random hypercomputer and a useful hypercomputer: A useful hypercomputer trades off performance for certain functions/universes in order to do better in other functions/universes, while a random hypercomputer doesn’t do that and thus is useless.
What do you mean? The output of any Turing machine is computable by definition. Do you mean solving the halting problem for a random Turing machine? Or a random oracle?
Fair. I think this is indeed a nitpick. 😊 In case it wasn’t clear, the point remains something like: When we observe/build computational systems in our world that are “better” along some axis than other systems, that “betterness” is not generally derived from having gone over a new threshold of “even more general” computation (they definitely aren’t deriving it from hypercomputation, and often aren’t even deriving it from universal Turing computation), but through being better suited to the capability in question.