Introduction to NEPT and NOPT structures
Introduction to NEPT and NOPT structures
Username: MikeSmith

Hi everybody at tetration forum. Thanks to Bo for all the administration help.
I hope that there are some new ways of looking at familiar things, and some of the material below can be understood and is reasonably clear.
Warning: The material is jam-packed with ideas, but quite difficult to check, so please take your time reading it, and if you’re reading this and new to hyperoperations you should try to work out 3^^3, 3^^^3, 3^^^^3 to get some intuition about the patterns I’ll be discussing below. Also, if you understand Graham’s number construction that can help as well.
Post or email any comments or queries as you wish.

Nept and Nopt structures (Part 1).
A NEPT structure is the representation of a large number as multi-layered Nested Exponentional Power Towers.
A NOPT structure is a generalisation of multi-layered nested power towers. Usually we think of power towers where the operation is exponentiation, but the concept of nested power towers could be used with other operations from the hyperoperation hierarchy as well as any others where height of power tower is well-defined and each power tower in the expression produces a natural number and the operation is a strictly increasing function on the natural numbers.
Other operations are possible for putting into nested power towers for example, such as (+, iterated addition) and (*, iterated multiplication) and (^^, tetration, iterated tetration) and so on. Considering very large numbers, other applications of NOPT structures are possible by using powerful operations such as the gi-sequence from Graham’s number construction where g1=(3)^4(3) (3 hexated to 3) and g64=Graham’s number. Even more powerful operations can be considered such as g-subscript towers where each “subscript power tower” has a well-defined height (of nested g-subscript symbols, ending with g1).
The canonical NOPT structure is the NEPT structure where the operation is exponentiation. The canonical relationship between NEPT and NOPT structures is that an OrderType of a NOPT structure can be well-defined by comparing with the NEPT structure of same order number. In other words, the OrderType of a NOPT structure borrows from the canonical NEPT structure related to some hyperoperation.
So the relationship can be seen like this:
Hyperoperation (n>=4) <-> NEPT (n>=4) <-> NOPT (n>=4)
For example, NOPT structures with OrderType 4 or 5 correspond to the appearance of tetration and pentation numbers when they are written into NEPT from (the form of Nested Exponential Power Towers).
NOPT structures with OrderType 4 or 5 are (the only) Linear NOPT structures.
Hexation NOPT structure has OrderType 6 and has 2-dimensional array structure.
Hepation NOPT structure has OrderType 7 and has (2,1)-dimensional array structure.
Octation NOPT structure has OrderType 8 and has (2,2)-dimensional array structure.
Nonation NOPT structure has OrderType 9 and has (2,2,1)-dimensional array structure. By zigzagging from bottom right corner, in left-up-left-up-left-up-… fashion we can build up NOPT structures with higher dimensions. The computation pathway proceeds from bottom right corner in left-up-left-up-left-up-… fashion until the top left corner.
Linear NOPT structures have Linear Ellipsis Structure.
Non-linear NOPT structures (OrderType>=6) have Multi-layered Ellipsis Structure.
The Ellipsis corresponding to the final top-left corner component of the NOPT structure is where the final computation is achieved. Superficially, it looks like a Linear Nopt structure, but looking at it carefully, notice that the Ellipsis length (of linearly arranged nested power towers) is given by a small number in Tetration and Pentation NOPT structures, but in larger NOPT structures is given by a Multi-layered Ellipsis expression, that is the combined effort of the rest of the interlinked components of the NOPT structure.
The induced Multi-layered Ellipsis Structure from a NOPT structure has a very similar structure to the well-known H-Fractal.

A NOPT-6 structure (hexation) has a 2-dimensional array structure and the induced Multi-layered Ellipsis Structure corresponds with simple H-Fractal at the first level.
A NOPT-7 structure (heptation) has a (2,1)-dimensional array structure and the induced Multi-layered Ellipsis Structure corresponds with the H-Fractal at the second level of resolution.
A NOPT-8 structure (octation) has a (2,2)-dimensional array structure and the induced Multi-layered Ellipsis Structure corresponds with the H-Fractal at the third level of resolution.
A NOPT-9 structure (nonation) has a (2,2,1)-dimensional array structure and the induced Multi-layered Ellipsis Structure corresponds with the H-Fractal at the fourth level of resolution.
You can observe that in terms of written symbolic requirements, going from NOPT-(I) to NOPT-(I+1) structure requires twice as many symbol expressions, due to the symbol folding that proceeds zigzagging L-U-L-U-L-U-L-U from bottom right corner to top right corner.
NOPT structures require a fixed SEEDVALUE, that has the roles of (1) either initiating a Linear NOPT Component OR (2) Controlling Ellipsis Length of a Component.

Can you explain what you mean by OrderType?

Also, what does NOPT stand for?
(11/19/2011, 10:37 PM)andydude Wrote: Can you explain what you mean by OrderType?

Also, what does NOPT stand for?

ok maybe i didn't explicitly say that
NOPT =df Nested Operational Power Towers
NOPT structures are defined for any "suitable" operation.

First consider how to
write 3^^3, 3^^^3, 3^^^^3, 3^^^^^3 and so on
in terms of exponentiation power towers
*not* tetration powers etc
the higher the hyperoperation the more layers of nesting are required.
With 3^^3 , only a single exponential power tower is created
tetration is hyperoperation with n=4 of course
but we say (or define) the first nontrivial NOPT structure
to have OrderType=4 to make a suitable correspondence
with tetration (where tetration is expressed in NEPT form)

You could say that the "trivial" NOPT structure is just "n".
Here n is a small number expressed with a linear sequence of digits with standard interpretation, using SPN (Standard Positional Notation)
And SPN implicitly contains the first 3 operations (+, *, exp)

The first "nontrivial" NOPT structure is just "theta ) n"
"theta" by itself is a "formal power tower", not a number
as there is no height information.
"theta ) n" obviously is a power tower, as the height of the power tower is given by n.

Ah, OK. Now I think I understand what you're talking about. There is a lot of terminology for this kind of thing, and I think it would be good to review it. In the case of addition, it's called N-ary summation, and in the case of multiplication, it's called N-ary products. In computer science, there are also names for NOPT, such as: apply, fold, map, reduce, zip, and many other terms. The most notable have special terms for left-associative nesting and right-associative nesting: foldl and foldr respectively. However, there are many other possibilities, some of which Henryk (BO) discussed in his Ph.D. thesis:

Part of the issue in relating NEPT and hyperoperations is that you have to fix the hyperexponent so you can compare a^^6 and a^(a^a)^(a^a^a) for example. Even then, I don't think that any arithmetic shortcuts will come from this, I think the best that we can hope for in this case will be order relations and inequalities.

Andrew Robbins
Further thoughts
I'm sorry if the writing that follows is sometimes clumsy, but maybe you'll get something out of it. It's a kind of philospophy of natural numbers in the context of hyperoperators and large numbers.

Philosophical considerations
The following is the standard definition of N=Natural Numbers.
“Peano's successor function S(n) = n+1 uniquely covers all numbers 1.. starting from n=0 by iteration of S, and thereby defines the set of natural numbers.”
To understand N better and more accurately consider
N = {N[T, SPN]} U {N[UDC]}
T = Tally; SPN = Standard Positional Notation
UDC = Unbounded Descriptive Capability
This mysterious formula acknowledges the three viewpoints about N
N[T] -> As a tally without bound
(Self-referential concept about the action of counting and size representation)
N[T, SPN] -> As a tally without bound OR As an SPN digit sequence without bound
(This is what people think of about N, in normal, practical situations.)
N[UDC] -> Counting numbers with Unbounded Descriptive Capability
(For example, Scientific Notation, power towers etc are an extended conception of N.)
Counting numbers, in a basic and fundamental way, serve the purpose of indexing and sequencing items, entities etc and one possible way to provide lexicographic ordering.
The tally system allows the fundamental process of pointing to an item, incrementing a counter (recording the presence of the item) changing status from unread to read or removing same item from a collection set.
In this way items in the collection can be counted.
The Thoroughness Property is evident, we believe that incrementing is the unambiguous, systematic method that counts things one-by-one forever.
The reality is that “forever” should be relativised to mean towards a “horizon”,
that is not well-defined but accurately represents an intuition about self-reference regarding “quantity” (the transitions betweeen initial tallying, SPN, and digits-in-sequence tally), and intuition concerning “large enough”.
N[T, SPN] gives the unestimable advantage of allowing a sensible method of Information Condensation while retaining Thoroughness Property.
With SPN, “large enough” can be made small by use of “base” and in so doing frees- up “large enough” to be controlled by other aspects of the description.
And so “large enough” is now the consideration of number of digits in the SPN sequential presentation.
With N[UDC] we have an Emerging Trade-Off between Thoroughness Property and Unbounded Descriptive Capability.
When considering the various big numbers more information is directed towards magnitude and less towards fine details. This is a trade-off between Descriptive Capability and Thoroughness Property. It is an emerging trade-off because there are phase transitions in the trade-off.
For example: A googol is both SPN-describable (a digit sequence of 1 followed by 100 zeros) and UDC-possible (10^100). A googolplex is not SPN-describable but is UDC-possible (10^(10^100)). For numbers between googol and googolplex it is hard to maintain Thoroughness Property. Introducing treelike structures such as HBN (Hereditary Base N) is an attempt to recapture Thoroughness Property but at the expense of greater structural pattern complexity.
Similar phenomena can be observed in the discussions concerning the infinite ordinals.

Considering large numbers and fast growing functions gives a dual reality:
A) A tangible magnitude-into-pattern transformation
B) Traditional perspective of ever-increasing patterns of magnitude

FIFF paradigm
Fuzzy Infinite Fuzzy Finite paradigm
Finite, infinite dichotomy, that is: {1,…,n} versus {1,…}
The appearance seems clear and unambiguous
But this viewpoint is biased by the dominant SPN perspective
And the evidence from considering Nopt structures shows it is a false dichotomy.
Some of the transitions
SPN shows exponentiation as an incremental add-one-digit way. Knuth arrow notation shows Ackermann function as an incremental add-one-to-tally way.
NOPT structures show that Ackermann numbers increase exponentially with respect to Minimal Symbolic Notational requirements on a level playing field (the benchmark or yardstick of using multi-level nested layers with a fixed operation, and power towers)
Nopt structures use a sensible minimal symbolic representation.
Next stage is “zooming in on” HEFTY Nopt structure with a microscope!
Can then introduce another level of chunkiness by storing High Resolution HEFTY Nopts into little boxes… And start the process again…
And so on into the ethereal realms of incomprehensible vastness…

The Inevitable chunkiness of large numbers
In the consideration of fast growing functions there is an inevitable chunkiness that comes about due to natural limit of descriptive capabilities.
You can start out slow with 1, 2, 3 and successor function or fast with Graham’s number and g-subscript power towers but the contemplation of pushing out further into the endless unboundedness of infinity calls upon chunkiness.
NOPT structures are dimensionless until an operation is specified, but even though they are dimensionless, structure can be identified and codified.
Reaching out further and more information hiding is natural and unavoidable.
What about the huge wealth of numbers between g1 and g2 from the gi-sequence leading to Graham’s number? We could traverse the intermediary space by applying the standard math integer functions to the Knuth arrows. And to do this, all the structure leading up to “3 hexated to 3’ could be replicated but this time applying to number of Knuth arrows, padding out the hyperoperator hierarchy to dizzying realms.
The beauty of SPN (standard positional notation) numbers is they preserve the initial successor function, increment natural and successive orders of magnitude while retaining condensation property for as long as a string of digits can go.
A number of visualisations from Wolfram demonstrations show cellular automata applied to binary numbers or other base numbers; we see the condensation property that is systematic reuse and exhaustion of previous orders of magnitude.
An understanding of hyperoperators coded into NOPT structures also has systematic reuse of orders of magnitude but the condensation or thoroughness property from initial successor function is necessarily relaxed. By using exponentiation and the power towers thereof inside a NOPT structure we have the NEPT structures, and now the notion of “successor” is transformed or transmuted into “adjacent power tower”.
The normal successor function we are so familiar with, that is counting distinct symbols is retained and distinctly present in the NEPT structure but we are counting
Power towers symbolically adjacent to one another.
(From some perspectives, in some ways, the traditional finite, infinite separation in maths is flawed, we should think in terms of required chunkiness and layers of nestation.)
A heptation NOPT structure also contains NOPT structures of all previous orders, that is to say, hexation, pentation, tetration, exponentiation are also present and part and parcel of heptation structure. A number such as 53,672 is a 10^4 order number, and also contains information about previous orders of magnitude. A nonation number requires octation, heptation, hexation, pentation, tetration, exponentiation.
Once we get to the gi-sequence it is like a hyperdrive of magnitude that shows the transition between magnitude and pattern.

It seems as if you are interested in large number representations. It also seems as if you are interested in alternative systems with more rapid rates of growth, but one property that many systems do not have is that of uniqueness. Even SPN (as you call it) is not unique for all rational or real numbers, because 1 can be represented as 0.999999... Which is also true of scientific notation:

however, setting the 0.999... issue aside, there is usually only one logical (using common sense) way to write any rational number, uniquely in the form:

\( (-1)^s m \times 10^{e} \) where s=0,1, e is an integer and \( 1 \le m \lt 10 \) which may require a bar for repeating digits. I have also been interested in this enough to talk about it on my blog:

which I would recommend reading. The interesting thing about tetration notation is that this also works, but requires that you change the mantissa range to 0..1. In other words you can write (for base 10, not true of bases less than eta) a representation of a real number (other than -1, 0, 1), uniquely in the form:

\( (-1)^s {\exp_{10}^{h}(m)}^{{(-1)}^r} \) where r,s=0,1, h is an positive integer, and \( 0 \lt m \le 1 \) provided that m is a rational number. While this uniqueness may seem silly at first, it's can be very important. Another property of both scientific notation and tetration notation is the ability to compare without evaluating. For example, if the exponents (e, h) are the same, then we can compare mantissas (m) which can be much, much easier than trying to evaluate a big number.

Andrew Robbins

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