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+--- Thread: Functional super-iteration and hierarchy of functional hyper-iterations (/showthread.php?tid=284)
RE: Functional super-iteration and hierarchy of functional hyper-iterations - bo198214 - 05/04/2009
(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.
RE: Functional super-iteration and hierarchy of functional hyper-iterations - Base-Acid Tetration - 05/04/2009
(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...
RE: Functional super-iteration and hierarchy of functional hyper-iterations - bo198214 - 05/04/2009
(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
RE: Functional super-iteration and hierarchy of functional hyper-iterations - Base-Acid Tetration - 05/12/2009
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!
RE: Functional super-iteration and hierarchy of functional hyper-iterations - bo198214 - 05/12/2009
(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:
\( f_{n+1} (x) = f^{[x]}(x) \)
while the hyper-operations ladder does:
\( f_{n+1}(x) = f^{[x]}( c) \)
where \( c=0 \) or \( c=1 \).