Hyper operators in computability theory

[tommy1729]

Recently I asked this question on MO http://mathoverflow.net/questions/261538...n-analysis

And I'm curious if anyone has encountered anything similar. As in, the main value of the hyper-operators defined for natural numbers is its computational aspect. Is there a similar idea in analysis? Can anyone give me any ideas of where to talk about these things. About how to phrase the fact that the computational complexity of \( \sqrt{2}\uparrow^n x \) grows hyper operationally with \( n \).

In the most General case computational complexity is a very hard and unsolved area of research.
For instance take euler's gamma : if irrational , the complexity is finite. But we do not know.
The fastest algorithm or even a quadratic speed algorithm for its digits is unknown.

As for your case : 

I hope you meant superexponentially INSTEAD of hyper operationally.

Second , it seems you want a fastcut for functions like exp exp and exp exp exp.

Well if the stirling Numbers or its generalisations will not help , I assume it can not be done.

Reminds me of Stephen Wolfram's irreducible complexity.

Besides the acceleration by 2 or 3 iterates at once and the alike , one could try a nonconstant iteration speedup, but that would require a superfunction or Abel function AND a fast method for THAT Abel or super.

Combinatorical methods probably reduce to the above.

Number theory seems unrelated in a noncombinatorical sense.

Fake function theory can be fast but not precise.

Contour integrals ??

Im not optimistic since i just Summarized imho the most realistic ideas.


Sorry


Regards

Tommy1729