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Functional super-iteration and hierarchy of functional hyper-iterations
#21
(05/04/2009, 08:57 PM)Tetratophile Wrote: 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.
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#22
(05/04/2009, 09:01 PM)bo198214 Wrote:
(05/04/2009, 08:57 PM)Tetratophile Wrote: 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...
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#23
(05/04/2009, 09:06 PM)Tetratophile Wrote: 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 Tongue
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#24
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: Problem is, produces ab+1 instead of ab, so it needs to be written as . So to define exponentiation etc... How do I put it in It_3? Don't know how to do this! Dammit!
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#25
(05/12/2009, 02:29 AM)Tetratophile Wrote: 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:

while the hyper-operations ladder does:

where or .
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