Talk:Blum–Shub–Smale machine

Representing rational numbers in a single tape symbol

The following claim seems dubious to me:

A Turing machine can be empowered to store arbitrary rational numbers in a single tape symbol by making that finite alphabet arbitrarily large (in terms of a physical machine using transistor-based memory, building its memory locations out of enough transistors to store the desired number), but this does not extend to the uncountable real numbers (for example, no number of transistors can accurately represent Pi). Farmdudler (talk) 07:28, 6 July 2023 (UTC)

@Farmdudler: Perfectly reasonable. A rational number is a ratio of integers, so as long as both the top and bottom integers can be represented, the number can be represented. This logically corresponds with adding more symbols: 1/2, 1/3, 1/4, etc. with the top and bottom numbers ranging over the range the hardware provides for. Add more components to extend the range and the number of valid symbolic representations also increases.

Of course, the real problem with this article is the lack of footnotes. Skyerise (talk) 11:32, 6 July 2023 (UTC)

@Skyerise yes, natural numbers can be represented and therefore rational numbers also. but they cannot (all) be represented in a single tape symbol for any given alphabet, because the alphabet still has to be finite. Farmdudler (talk) 11:43, 6 July 2023 (UTC)

@Farmdudler: And yet Achilles still beats the tortoise. It says "arbitrarily large", not "infinitely". Skyerise (talk) 11:50, 6 July 2023 (UTC)

Take any Turing machine. The size of it's alphabet may be arbitrarily large, but it is finite, like you said. Therefore, a single tape cell can represent only a finite number of different things. Since there are infinitely many rational numbers, it can therefore not represent arbitrary rational numbers. (That would mean: if you give me any rational number, whatever number you choose, I can represent it in one tape cell.) So the paragraph in the article is incorrect. Farmdudler (talk) 12:17, 6 July 2023 (UTC)

"AN arbitrary rational number". You pick the specific number that you need to represent. Then you construct the Turing machine in such a way as to be able to represent it. It does NOT say "ALL rational numbers". You seem to misunderstand how "arbitrary" is being used here. I've modified the paragraph to reduce such confusion. Skyerise (talk) 13:42, 6 July 2023 (UTC)