alt Hacker NewsTuring machines can't operate on arbitrary real-numbered values because they require infinite precision.
If the universe indeed uses real values, it's possible that it's more powerful than Turing machines.
Turing machines can't operate on arbitrary real-numbered values because they require infinite precision.
If the universe indeed uses real values, it's possible that it's more powerful than Turing machines.
The problem with arbitrary real numbers is not precision, it is computability. Most real numbers we actually work with in the sciences are computable - pi, e, 2, 0.(2), and so on - they are all computable by Turing Machines. However, Turing himself showed that not all real numbers are - you can construct real numbers like Chaitin's constant that are not actually computable by a TM. In fact, almost all real numbers are non-computable (the set of computable numbers is countable, so it has the same "size" as the Naturals, while the Reals are not countable - so they are much larger).
However, this doesn't mean that TMs can't operate on such numbers. You can compute the fact that pi + pi = 2 * pi even if you can't enumerate all of pi's decimals.