Approaches to Tetration

[Gottfried]

Code:

- Approaches to tetration

  We may classify the approaches to tetration in two classes: a "binary operator
  approach" and an "iterative series approach"

  -- the operator approach ----------------------------------

  Here one tries to extend the hierarchy of binary operator
   a+b : addition      a*b : multiplication    a^b : exponentiation    a^^b : tetration
  in a meaningful way
  
  Extensions of this are made in two ways
  assigning an index to the operation and to evolve this
  to higher or lesser indices
   a[0]b : "zeration"       a[1]b : addition     a[2]b : multiplication
   a[3]b : exponentiation   a[4]b : tetration    a[5]b : pentation
    ...
[see links]

  * Ackermann-function
  Closely related is the concept of the Ackermann-function
  A(a,index,iterate)
  [see: Ackermann-function, <literature>]
  
  * Reihenalgebra

  Even an approach to extend this operator-hierarchy to fractional indices is known
  where the index was adapted:
  
                  a[1]x : "do nothing"
   a[2]x : add one to a     a[1/2]b : subtract one // unary operator
   a[3]b : addition         a[1/3]b : subtraction
   a[4]b : multiplication   a[1/4]b : division
   a[5]b : exponentiation   a[1/5,subscript]b : inverses root or logarithm
   a[6]b : tetration        a[1/6,subscript]b : inverses depending on evaluation-precedence
   ...

  The author tried then to find meaningful interpolations for this indexing scheme.
  [markus Müller, Reihenalgebra]


  -- the iterative series approach -----------------------------------------
  
  Here operations/functions are expressed as series; for instance the operation
  of exponentiation as powerseries, such that
  
   exp(x) = 1 + x + x^2/2 + ...

  and then iteration

   exp°2(x) = exp(exp(x)) = 1 + exp(x) + exp(x)^2/2! + ...
                          = K + Ax + Bx^2 + Cx^3 + ...

   --- powerseries
  
   The most common approach, see for instance []
    b^x     = 1 + log(b)x + log(b)^2 x^2/2! +...
    (b,x)°1 = b^x
    (b,x)°2 = b^(b^x) = 1 + log(b)b^x + log(b)^2 b^2x/2! + ...
    (b,x)°h = (b,b^x)°(h-1)
    
    The needed manipulations on powerseries are sometimes explicitely
    expressed in matrix-notation [see links]

   ---functions defined by general series ( dirichlet-like series,..)

    The needed calculations are performed using derivatives of the functions
    in the places where for powerseries we use their coefficients

       [see Bell-matrix, Carleman-matrix]
      
   --- series on integer iterates of the function itself

     (b,x)°h = A x + B*(b,x)°1 + C*(b,x)°2 + ...
     [for instance see : binomial expansion]
    
  The iterative series approach implies also a different view: one assumes
  an "initial-value" x, to which the operation is applied iteration-times
  which then gives a final value.
  
  So in
    (b,x)°h we have a ternary operation where we apply the exponentiation
    using base b h-times to the initial-value x.

  This paradigm is followed in [see: dynamical systems, iteration theory, ... ]
  
  The reverse-engineering of this paradigm leads then to the redefinition of
  the common "binary operators" using the iteration-form (we use a local,
  one-way notation here)
  
                b occurs h times
  ---------------------------------  -------------------
  (+,b,x)°h  :  x + b + b + ... b  : iterated addition
  (+,b,x)°-h :  x - b - b - ... b  : iterated subtraction

  (*,b,x)°h :  x * b * b * ... b   : iterated multiplication
  (*,b,x)°-h : x / b / b / ... b   : iterated division

  (^,b,x)°h :  b^ b ^ ... b^x      : iterated exponentiation, right associative
  (^,b,x)°-h : Log_b(Log_b(...Log_b(x)))

  Though for iterated addition and multiplication no powerseries is required,
  they may be consistently expressed the same way using matrix-operators on
  formal powerseries, which perform addition, multiplication and exponentiation
  in the ring of formal powerseries according to this scheme [see link]
  
  The latter form of notation (or some convenient adaptions [see "notations"])
  like exp_b°h(x) for (^,b,x)°h allows then to write the iteration-scheme
  for any function, for instance
  
  (dxp,b,x)°h = dxp_b°h(x) :  where dxp_b(x):= b^x - 1
  (sin,b,x)°h = sin_b°h(x) :  where sin_b(x):= i*(b^(ix) - b^(-ix))/2

  --- Connection between series-iteration and operator paradigms ------------

  The common binary operators can then be seen as reduced forms
  of the series-iteration approach
  
  a + b        == (+,b,a)°1
  a + b + b
     = a + 2*b == (+,b,a)°2
  a - b        == (+,b,a)°-1
  b - b  = 0   == (+,b,b)°-1
  b + b  = 2*b == (+,b,b)°1 = (+,b,(+,b,b)°-1))°2 = (+,b,0)°2
               == (*,b,2)°1
  
  
  a * b      == (*,b,a)°1
  a * b * b
     = a*b^2 == (+,b,a)°2
  a / b      == (*,b,a)°-1
  b / b = 1  == (*,b,b)°-1
  b * b      == (*,b,b)°1 = (*,b,(*,b,b)°-1))°2 = (*,b,1)°2
             == (^,b,2)°1



  b ^ a        == (^,b,a)°1
  b ^(b ^ a)   == (^,b,a)°2
  log_b(a)     == (^,b,a)°-1
  log_b(b) = 1 == (^,b,b)°-1
  b ^ b        == (^,b,b)°1 = (^,b,(^,b,b)°-1))°2 = (^,b,1)°2

[bo198214]

Code:

- Approaches to tetration

  We may classify the approaches to tetration in two classes: a "binary operator
  approach" and an "iterative series approach"

I dont get this classification.
The main goal is to find a[4]x or generally a[n+1]x.
As a tool to do so we use non-integer iteration.
As a tool for non-integer iteration we use series expansions or limit formulas (like for example the formula for the schroeder function \( \sigma(x)=\lim_{n\to\infty} f^{\circ n}(x)/f'(0)^n \)).

And if we are in a good mood we even consider a[q]b for non-integer q.

[Gottfried]

Code:

- Approaches to tetration

  We may classify the approaches to tetration in two classes: a "binary operator
  approach" and an "iterative series approach"

I dont get this classification.
The main goal is to find a[4]x or generally a[n+1]x.
As a tool to do so we use non-integer iteration.
As a tool for non-integer iteration we use series expansions or limit formulas (like for example the formula for the schroeder function \( \sigma(x)=\lim_{n\to\infty} f^{\circ n}(x)/f'(0)^n \)).

And if we are in a good mood we even consider a[q]b for non-integer q.

Hmm - I'm focusing different paradigms here; the goal to extend the operator hierarchy is not always driven by a functional representation. One may call it a "naive" approach - but this sounds somehow pejorative in some ears. Anyway - there is a strong effort to extend the (binary) operator hierarchy solely based on the properties of the usual operators.
I think, we should refer to this as well (it is often the first approach to tetration, btw).

[bo198214]

Hmm - I'm focusing different paradigms here; the goal to extend the operator hierarchy is not always driven by a functional representation. One may call it a "naive" approach - but this sounds somehow pejorative in some ears. Anyway - there is a strong effort to extend the (binary) operator hierarchy solely based on the properties of the usual operators.

I dont know what you mean. The only ways I know of to extend tetration is via series and limits. What binary operator properties lead to a definition of tetration?

[Gottfried]

Hmm - I'm focusing different paradigms here; the goal to extend the operator hierarchy is not always driven by a functional representation. One may call it a "naive" approach - but this sounds somehow pejorative in some ears. Anyway - there is a strong effort to extend the (binary) operator hierarchy solely based on the properties of the usual operators.

I dont know what you mean. The only ways I know of to extend tetration is via series and limits. What binary operator properties lead to a definition of tetration?

Hmm - for me, that is sort of the "obvious". But unfortunately the "obvious" is often the most difficult to explain...

What actually don't you get?

That there *are* different paradigms, with wich someone might approach a certain problem (here: the extension of the collection/hierarchy of common binary operators)?
Or that such different paradigms actually were/are *present* in this case?

The latter may be discussed, and empirically be confirmed or disproved - just by asking the people/researchers. I think, empirically they are present ("obviously" - for me); just compare the discussion about zeration and that about the "technical" solutions for series-interpolation in our forum.

The first - I don't know, whether this can be discussed... In all historical review (also in mathematical) the reflection shows, that there were different paradigms which may have converged (or not) and may have evolved the theory via the discourse.
One example, where the difference of paradigms was explicitely discussed, but still persists as duality (and re-occurs with each individual developing to a mathematician or mathamathical thinking subject) is the duality between discrete and continuous number-theory approaches. An example of this discussion which I just came across today is in a historical treatise about Dedekind and his specific contribution to numbertheory. I'll cite it here, although I haven't thought much about it, but the article was fun to read. So below of this it goes...

Did I get the catch?

Gottfried

---

An important part of the dichotomy, as traditionally understood, was that magnitudes and ratios of them were not systematically thought of as numerical entities (with arithmetic operations defined on them), but in a more concrete geometric way (as lengths, areas, volumes, angles, etc. and as relations between them). More particularly, while Eudoxos' theory provides a contextual criterion for the equality of ratios, it does not provide for a definition of the ratios themselves, so that they are not conceived of as independent entities (Stein 1990, Cooke 2005). Such features do little harm with respect to empirical applications of the theory. They lead to inner-mathematical tensions, however, when considering solutions to certain kinds of algebraic equations (some of which could be represented numerically, but others only geometrically). This tension came increasingly to the fore in the mathematics of the early modern period, especially after Descartes' integration of algebra and geometry. What was called for, in response, was a unified treatment of discrete and continuous quantities.

More directly, Dedekind's essay was tied to the arithmetization of analysis in the nineteenth century—pursued by Cauchy, Bolzano, Weierstrass, and others—which in turn was a reaction to tensions within the differential and integral calculus, introduced earlier by Newton, Leibniz, and their followers (Jahnke 2003, chs. 3–6). As is well known, the inventors of the calculus relied on appeals to “infinitesimal” quantities, typically backed up by geometric or even mechanical considerations. This came to be seen as problematic. The early “arithmetizers” found a way to avoid infinitesimals (in terms of the epsilon-delta characterization of limit processes familiar from current introductions to the calculus). But this again, or even more, led to the need for a rigorous and comprehensive characterization of various quantities conceived of as numerical entities, including a unified treatment of rational and irrational numbers.

(from: http://plato.stanford.edu/entries/dedekind-foundations/)

---

[bo198214]

... the goal to extend the operator hierarchy is not always driven by a functional representation ... there is a strong effort to extend the (binary) operator hierarchy solely based on the properties of the usual operators.

What means "driven by functional representation"?

Of course there is this area of the operator hierarchy and then there is this area of non-integer iteration. But these are not different approaches but the non-integer iteration is a way to get the operations demanded in the operator hierarchy.

From the operator hierarchy we have the law:
\( b[4]0=1 \)
\( b[4](x+1)=b[3](b[4]x) \)

So we need an operation [4] (with real right operand) which satisfies these conditions. That means for each \( b \) we need a function
\( f (x)=b[4]x \) (with certain smoothness) which satisfies:
\( f(0)=1 \)
\( f(x+1)=b^{f(x)} \).

We can obtain such a function by non-integer iteration of \( \exp_b \), (perhaps there are also other possibilities):

\( f(x)=\exp_b^{\circ x}(1) \).

And we can perform non-integer iteration via those several approaches we discussed here (natural, diagonalization, regular). And those approaches then may have variants for which we need a series expansion or for which we dont need a series expansion.

So the really concurrent approaches are not operator-hierarchy and series-expansion. But our 3 different approaches to function iteration.

Operator hierarchy and series expansion are just parts of the whole area, where operator hierarchy is a framework into which you can plugin different approaches for non-integer iteration, including those performed by series expansion.