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> And would such a theory have any practical use? Yes, sureley. In Computablilty Theory you have the famous Ackermann-function[1]. It is actually an operator-e
by maze-le 11y ago
> And would such a theory have any practical use?
Yes, sureley. In Computablilty Theory you have the famous Ackermann-function[1]. It is actually an operator-extension, like you just described. It is important, because it grows overexponentially, but is still computable (unlike e.g. the busy-beaver-function).
[1]: https://en.wikipedia.org/wiki/Ackermann_function https://en.wikipedia.org/wiki/Ackermann_function
- infogulch 11y agoGiven that Graham's Number is an (over-)extension of the Ackermann-function that means that it's still computable, correct? Given the series leading up to Graham's Number, G = g_64 (g_1, g_2, ...), BB(n) will outgrow g_n, right? If so what's the smallest n such that BB(n) > g_n?
- infogulch 11y agoAfter a bit of research I found that there is a proof [0][1] that BB(12) > g_1. But that's still a ways from g_2, let alone g_64. [0]: "A Lower Bound on Rado's Sigma Function for Binary Turing Machines" by Milton Green (1964) [1]: https://en.wikipedia.org/wiki/Busy_beaver#Known_values_for_.CE.A3_and_S https://en.wikipedia.org/wiki/Busy_beaver#Known_values_for_....