G. F. Romerio
Le Cannet - France ; Saluzzo - Italy
Registered: Sep 2004
NKS Bibliography paper on:
“Ackermann's Function and New Arithmetical Operation”.
C. A. Rubtsov and G. F. Romerio.
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 hyper-operations 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. 111-112 (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 bi-jection 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 hyper-operations 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 hyper-operations 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 multi-value 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 hyper-operators could be also systematically defined.
Tetration and higher-level hyper-operations give an idea of “immensity” and could be a tool for a new approach, for instance, to the theory of infinite ordinals
Last edited by G. F. Romerio on 10-03-2004 at 09:05 PM
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