Functional super-iteration and hierarchy of functional hyper-iterations

[bo198214]

bo, you needed to see my explanation.

1. Evaluate g(x) at c first.
2. Hyper-n-iterate f to the OUTPUT of Step 1.
3. Evaluate the resulting function at c.

For n=1 (iteration), to evaluate this expression at any given natural x=c:
1. Evaluate g(x) at c first.
2. Iterate f to the OUTPUT of Step 1.
3. Evaluate the resulting function at c.

The set of all ordered pairs resulting from this evaluation is {(x, [f It_1 g(x)] (x))}.

I know, but I dont know what you want to say with that *questioningly look*.
We agree on the meaning, the question is more how to write it down properly.

Edit: Oh ok I see you extended your article. Yes and that computer program can be expressed with the 3 lines (1),(2),(3) that I gave.
Perhaps the difficulty of mutual understanding results from that you have no experience in functional programming.

[Base-Acid Tetration]

bo, you needed to see my explanation.

1. Evaluate g(x) at c first.
2. Hyper-n-iterate f to the OUTPUT of Step 1.
3. Evaluate the resulting function at c.

For n=1 (iteration), to evaluate this expression at any given natural x=c:
1. Evaluate g(x) at c first.
2. Iterate f to the OUTPUT of Step 1.
3. Evaluate the resulting function at c.

The set of all ordered pairs resulting from this evaluation is {(x, [f It_1 g(x)] (x))}.

I know, but I dont know what you want to say with that *questioningly look*.
We agree on the meaning, the question is more how to write it down properly.

Ok, I now see that you were only trying to clarify my notation. I would agree that your notation emphasizes that hyper-iterations are actually: function * OUTPUT of a function -> function.

tongue in cheek edit: all this editing posts only to see it was too late biz, i know... wastes such a h**k of time...

[bo198214]

tongue in cheek edit: all this editing posts only to see it was too late biz, i know... wastes such a h**k of time...

Haha, see I edited my previous too, after you replied to it

[Base-Acid Tetration]

To test the usefulness of this concept, I am now trying to define the hyper-operations in terms of hyper-iteration of the successor operation x+1. to try to see if the levels of hyper operations correspond to the hyper-iterations. if S(a) := a+1, than iteration of this function is addition: \( a+b = [S \operatorname{It}_1 b] (a). \) Problem is, \( [S \operatorname{It}_2 b] (a) \) produces ab+1 instead of ab, so it needs to be written as \( [S \operatorname{It}_2 (b)] (a)-1 \). So to define exponentiation etc... How do I put it in It_3? Don't know how to do this! Dammit!

[bo198214]

So to define exponentiation etc... How do I put it in It_3? Don't know how to do this! Dammit!

I dont think they are compatible:
Roughly your hyper-iteration ladder does:
\( f_{n+1} (x) = f^{[x]}(x) \)
while the hyper-operations ladder does:
\( f_{n+1}(x) = f^{[x]}( c) \)
where \( c=0 \) or \( c=1 \).