[Hyper-operations. Progress report. Tetration.] - A New Kind of Science: The NKS Forum

A New Kind of Science: The NKS Forum


Hyper-operations. Progress report. Tetration.

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Posted by: G. F. Romerio

NKS Bibliography paper on: “Ackermann's Function and New Arithmetical Operation”.
Progress report.
C. A. Rubtsov and G. F. Romerio.

On 3rd October 2004, we posted a thread in the NKS Forum (New and Announcements) concerning a subject identified with the title “Hyper-operations”, with reference to a paper written by us and included in the NKS Bibliography (C. A. Rubtsov and G. F. Romerio - Ackermann’s Function and New Arithmetical Operations).

See: http://www.rotarysaluzzo.it/filePDF...zioni%20(1).pdf

After that date, we received some direct messages requesting clarifications and information about possible available progress in this interesting subject, particularly concerning “tetration” (hyper-operation of rank 4, power-tower or tower).

A sort of progress report has been prepared for this purpose (see annex) for putting in evidence the following points:

a) A proposal for standardizing the notations of the each particular hyper-operation, for any rank s≥0, as well as of their two respective inverse operations (of the “root” and of the “log” type);

b) Description of some recursive and operations’ invariant properties of the direct and inverse hyper-operations, particularly of those of rank s=4 (the super-root and the super-logarithm of a number);

c) The known relationship between the square super-root and the Lambert’s Function (mentioned within “Mathematica” as ProductLog) and the appearance of interesting simple formulas, also concerning the “infinite towers”;

d) The use of a parameter called “hyper-exponent extension” for defining a new real numbers’ notation similar to the classical floating point notation, but using a higher hyper-operation level, as a tool for representing very large numbers (the RRH ©);

e) The proposal of an acceptable approximation concerning the extension to the real numbers of the tetration operation, with the natural base “e”, for defining approximated tetrational and super-logarithm (superlog) functions, to the base “e”, indispensable for an acceptable tetrational notation of very large numbers;

f) A proposal of using normalized terminology and symbols for facilitating co-operation between specialists and interested researchers in this innovative and fascinating domain.

Among the subjects deserving particular attentions for future analysis we can mention:

I. Study for achieving a complete analytical extension to the real numbers (and not only by simple approximations) of hyper-operations such as tetration (tower) and superlog, by adopting the strategy proposed by Prof S. C. Woon (Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK) for the analytical continuation of operators.

See: http://arxiv.org/PS_cache/hep-th/pdf/9707/9707206.pdf

II. Progress, so far, in the study of the Rubtsov’s “delta numbers”, which are obtained as a result of applying the unique inverse operation of zeration (hyper-operation of rank s=0, commutative). [In preparation].

III. Use of the results obtained in such research fields within Mathematica and, more generally, in the NKS environment.

Posted by: G. F. Romerio

The following software prototypes have been implemented by Constantin Rubtsov and are available for downloading and testing at -


To get the files, follow the above link, then right-click and "save target as" for each file.

- A hyper-calculator prototype showing the possibility of executing hyper-operations of ranks s=0 (zerations), s=1 (additions), s=2 (multiplications), s=3 (exponentiations), s=4 (tetrations), by using both keystrokes and string commands. The hyper-calculator allows to obtain the natural (base e) super-logarithm and the natural tower (tetration, base e) of a number, as well as the square tower and the square super-root of a number. It also includes a set of additional features, which are at a development stage and that require a detailed instruction guide, in the file Calc_KAR_1_00_beta3c.EXE

- Two versions of a real number notation converter, in the files Snumbers.exe and Snumbers2.exe These allow the conversion of numbers notation from the standard scientific floating point notation (base 10) to a proposed new tetrational format (e.g. using the tetration operation with bases 2, e, 10, instead of exponentiation) and vice-versa. This new notation seems very suitable for the representation of very large numbers.

- A prototype tool for the calculation of the tetration and of the super-logarithm of real numbers can be found in the file slog_spe.exe

These prototypes implement the RRH © hyper-format for real number notation (Rubtsov-Romerio Hyper-format; SIAE deposit, Rome, no. 0501658; © C. A. Rubtsov and G. F. Romerio, 2002-2010) They are implemented by means of very simple and “portable” programs, which don’t always allow the full execution of all the operations and often produce an “Error” message . The inclusion of some of their features into a powerful software package like Mathematica is under study.

Giovanni F. Romerio

Posted by: G. F. Romerio

Please find a corrected page 1 of :
Attachment: nks forum ii.pdf

Thank you. GFR

Posted by: G. F. Romerio

ICM-06. Hyper-operations as a Tool for Science and Engineering.
(C. A. Rubtsov – G. F. Romerio)

Sorry to come back on this subject for providing more information and adding some clarifications.

A short communication concerning the proposed practical use of zeration and tetration has been submitted to the International Congress of Mathematicians (ICM-06, Madrid, 22-30 August 2006), as a “Poster”, with the title: “Hyper-operations as a Tool for Science and Engineering”. The poster has been accepted by the Conference Organizers with no. 0480, within the “Algebra” section, and it is now available on line at: http://icm2006.org/AbsDef/Posters/abs_0480.pdf The Congress Web site is at: http://www.icm2006.org/?nav_id=111

It should be recalled that zeration is a new binary operation that belongs to the Grzegorczyk hierarchy and that can be used for systematically describing discontinuous signals, as well as signals with tangent discontinuities.

Please also note that tetration is used in the proposed RRH© number notation hyper-format (s=4) and that any positive real number can be represented as p*(b#n), with this format. Number p (with 0 < p < b), the super-exponent extension, can be found by iteratively applying n times the log b operator, until p < b. In other words, it is not indispensable to solve the problem of extending tetration to the reals (which has a great theoretical importance), before using the RRH© number notation format.

The extension of tetration to the reals is tightly connected with the analytic continuation of operators, as performed, for instance, within the “Fractional Calculus”. A solution is probably … around the corner.

[By the way, K. A. Rubtsov and C. A. Rubtsov are the same person; K or C stay for Konstantin or Constantin, according to two different transliterations from the Cyrillic alphabet].


Posted by: Ingolf Dahl

I have proposed a way to extend tetration to the range of real numbers, see
http://forums.wolfram.com/mathgroup...c/msg00207.html . I have also made a Mathematica notebook, with an extended PowerTower function, available for download on my web page http://web.telia.com/~u31815170/Mathematica/ Thus I might have been able to go around the corner.
I am grateful for any comments on this.

Thus Pi to the hyperpower e (or "Pi tetrated to e") is

Ingolf Dahl

I attach the file here also, but check my web page for the latest version.

Posted by: Ingolf Dahl

I have uploaded a new version of my notebook PowerTower.nb.

Since the last version, I have changed "horse function" to (Exp[x] - 1), to avoid oscillating functions when x->+Infinity This means slightly changed values of the constants I have calculated. I have also included some theory and some graphs in this file, and have structured it better.
Thus the new value of Pi to the hyperpower e (or "Pi tetrated to e") is

Best regards

Ingolf Dahl


Posted by: G. F. Romerio

Hello Mr Dahl !

Thank you for your interest in the problem of extending the validity the tetration operation to real super-exponents. I am writing down these provisional comments during a pause between two trips (Germany and Italy). Despite the fact that, unfortunately, I didn’t have enough time to examine your hypotheses and, obviously, your codes, I should like to congratulate you for the work you have done so far. Some active interest is beginning to focus on this matter, from several horizons.

In Wikipedia, for instance (please see: http://en.wikipedia.org/wiki/Talk:Tetration), Daniel Geisler (the author of the tetration.org Web site) mentioned the postings of the present thread, as well as the Rubtsov/Romerio tetration and slog software, saying (in a rather amusing way) that he can't support the authors’ conclusions but, since he couldn’t really name anyone who support his own conclusions, so it's all good like that (!!!). Moreover, he thinks that the calculator that we proposed does produce a much larger value for e # Pi that he can justify, but that that value is more in line with Andrew Robins' estimate. The solution proposed by Andrew Robins is similar to our simulations and even more accurate.

Indeed, concerning the E-tetra-Pi operation, we have:
for Rubtsov/Romerio: e-tetra-pi = 1.9337456547928 . 10^10
for Ingolf Dahl: e-tetra-pi = 3.71504639065471391171. 10^10
As a provisional conclusion, so to say, our (Rubtsov/Romerio) result is too large according to Daniel Geisler, but near what was obtained by Andrew Robin. Your (Ingor Dahl) value is even larger but, after all, … “only” almost the double of it. Taking into account the order of magnitudes involved in such results [10 ^ 10], this is not a catastrophe. Nevertheless, mathematically speaking, since everybody cannot be right at the same time, probably … somebody (or … everybody) is wrong. I would be happy to tell you that you succeeded and perhaps it is really so. I shall come back to this as soon as possible, also concerning your approximations of Pi-tetra-E.

Actually, the problem of extending tetration (in reality the “tetrational function”) to the real super-exponents x is similar to finding a continuous and derivable function that will coincide will the known values obtained for n = integer(x). Other similar problems are, for example:
(a)– to give a meaning to an expression such as y = b ^ x, for x real, once known
its meaning for x integer (problem solved in the framework of elementary Algebra);
(b)– to give a meaning to y = x ! (the factorial of x, with x real), once known its meaning for x natural (problem solved in the framework of advanced Algebra, with the definition of the Gamma function).

Our general problem here is to give a meaning to an expression such as:
y = b-tetra-x = b # x (with b > 0 and x real > -2)

Unfortunately, we only know how to do it, partially and in some particular cases, by more or less accurate approximations. The general approach should involve the solution of the problem of the analytical continuation of iterated operators. Perhaps fractional calculus and/or the study of functional equations (as you are trying to do, if I am not mistaken) could bring all of us fully “around the corner”. I attach some qualitative simulations obtained by using Mathematica, for various values of the base b. These plots show that, under some conditions, tetration must oscillate.

Giovanni F. Romerio (Gianfranco)

Posted by: G. F. Romerio

Sorry! Here is the attachment foreseen for the previous posting.


Posted by: G. F. Romerio

There is a new value of E-tetra-Pi, calculated by using the Robbins cubical approximation of the slog critical path, rather similar (!) to the value found by Ingolf Dahl. This value is:

y = e [4] π = 3.66901 x 10^10 = 36’690’101’018.425... , to be compared with:
y = e [4] π = 37’150’463’906.547... (estimated by Ingolf Dahl);

Please see also: http://forum.wolframscience.com/sho...s=&threadid=956

Don't give up, we shall succeed.

K. A. Rubtsov - G. F. Romerio

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