05/02/2009, 02:23 AM
(This post was last modified: 05/12/2009, 02:12 AM by Base-Acid Tetration.)
We all know the functional "exponentiation" or "power" for functional iteration, \( f^{\circ n} (x). \)
(I use F^circle n, f^n (x), or f^circle n (x) if clarity is needed, for iteration)
That means we can do this for some function:
\( f^{g(x)}(x), \) where the function f is iterated g(x) times at x.
We could investigate the properties and results of functional "tetration", or super-iteration. The "upper-left exponent" notation can be abuse, or use the circle notation like \( {}^{\circ n} f(x) \)
Initial Definition: (might have gotten my variables wrong)
Given a \( f : S \rightarrow S \) and a \( n \in \mathbb{N}, \)
a functional super-iteration is defined by the recurrence relation:
\( (\forall n \, \forall x) \mbox{ so that the result of the iteration is defined, } {}^{\circ n+1} f(x) =[ f^ {{}^{\circ{n}} f(x)}(x) ] , \)
\( {}^{\circ 1} f(x) = f(x). \)
Already a hierarchy of operations on functions that is much like the arithmetic hyper-operations is evident:
Function composition: \( g \circ f (x) = g(f(x)),
\, g(x) = c \rightarrow g(f(x)) = c. \)
Function iteration: \( f^{n+1}(x) = f (f^n (x)),
\, f^0(x) = x. \) A function can be iterated a constant times, or by g(x) times.
Functional superiteration: \( {}^{\circ n+1} f(x) = \left [ {}^{\circ n} f^{\circ f(x)} \right ] (x)}, {}^{\circ 1} f(x) = f(x). \)
(I use F^circle n, f^n (x), or f^circle n (x) if clarity is needed, for iteration)
That means we can do this for some function:
\( f^{g(x)}(x), \) where the function f is iterated g(x) times at x.
We could investigate the properties and results of functional "tetration", or super-iteration. The "upper-left exponent" notation can be abuse, or use the circle notation like \( {}^{\circ n} f(x) \)
Initial Definition: (might have gotten my variables wrong)
Given a \( f : S \rightarrow S \) and a \( n \in \mathbb{N}, \)
a functional super-iteration is defined by the recurrence relation:
\( (\forall n \, \forall x) \mbox{ so that the result of the iteration is defined, } {}^{\circ n+1} f(x) =[ f^ {{}^{\circ{n}} f(x)}(x) ] , \)
\( {}^{\circ 1} f(x) = f(x). \)
Already a hierarchy of operations on functions that is much like the arithmetic hyper-operations is evident:
Function composition: \( g \circ f (x) = g(f(x)),
\, g(x) = c \rightarrow g(f(x)) = c. \)
Function iteration: \( f^{n+1}(x) = f (f^n (x)),
\, f^0(x) = x. \) A function can be iterated a constant times, or by g(x) times.
Functional superiteration: \( {}^{\circ n+1} f(x) = \left [ {}^{\circ n} f^{\circ f(x)} \right ] (x)}, {}^{\circ 1} f(x) = f(x). \)
