G. F. Romerio
Le Cannet  France ; Saluzzo  Italy
Registered: Sep 2004
Posts: 15 
Hyperoperations
NKS Bibliography paper on:
“Ackermann's Function and New Arithmetical Operation”.
C. A. Rubtsov and G. F. Romerio.
http://www.rotarysaluzzo.it/Iperoperazioni.htm
http://www.rotarysaluzzo.it/filePDF...zioni%20(1).pdf
Mathematicians and AI experts are aware of the infinite hierarchy of arithmetical operations (the Grzegorczyk hierarchy), implied by Ackermann’s function. The small hierarchical levels point to standard operations such as addition (s=1), multiplication (s=2), power/exponentiation (s=3), as well as to a very compact unusual operation called “tower” or “tetration” (s=4), i.e. iterated exponentiation, about which some documentation starts to be available. Actually, this hyperoperations hierarchy is unlimited.
Moreover, in 1989, Constantin Rubtsov has shown that Ackermann’s function itself can be used to define a new operation with a hierarchical level less than that of addition, i.e. for s=0 [Rubtsov, C. A.  Algorithms ingredients in a set of algebraic operations  Cybernetics  Kiev  1989  N° 3, p. 111112 (In Russian). http://numbers.newmail.ru/english/01.htm ]. This new operation has been called “zeration”. The inverse operation of zeration (commutative) generates a new class of numbers (the Rubtsov’s “delta” numbers) that can be put in bijection with the set of the logarithms of negative numbers.
“Tetration” is easily analysed for all natural levels s>0. Nevertheless, its study for negative arguments implies again the logarithms of negative numbers and, therefore, is connected with the analysis of the inverse of “zeration”.
The objectives of the study of the hyperoperations hierarchy can be placed in various domains of the NKS thinking:
a) The hierarchy is infinite. This could be put in evidence in a new presentation of the theory of numbers. In fact, emergence of new classes of numbers should be expected, in performing the inverse operations of any of the hyperoperations with ranks other than 1, 2, and 3.
b) The new zeration operation can be used to systematically describe discontinuities such those normally defined by the step or Dirac’s function.
c) The study of the new delta numbers, obtained via the inverse of zeration and corresponding to complex multivalue numbers, might have an important theoretical impact.
d) Tetration can be used to represent very large numbers. Representation of large numbers via iterated exponentiations, could be used, in high level computer programs like Mathematica, to show “tetration orders of magnitudes”, instead of simply “overflow”. Appropriate Mathematica hyperoperators could be also systematically defined.
Tetration and higherlevel hyperoperations give an idea of “immensity” and could be a tool for a new approach, for instance, to the theory of infinite ordinals
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GFR
Last edited by G. F. Romerio on 10032004 at 09:05 PM
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