The Ladder Hyper-Operating Transformer function

Part of Counting Really,Really,Really High


Copyright 2020 by Louis Epstein,All Rights Reserved
The Ladder Hyper-Operating Transformer function,or l,raises a base n to a series of hyper-operated exponents numbering as defined by the bound b.
Its basic inspiration is the Ultrex Function; it builds upon the Hyper-Operating Ultrex Function and Simple Hyper-Operating Transformer Function. The first exponent is always n itself,so n l 1 equals (as with the hyper-operating ultrex and simple hyper-operating transformer) n (n up-arrows) n for any value of n.

Again,I mainly represent hyper-operating up-arrows with carets (^) in this article since not all browsers render the ↑ symbol and I will be using some arrows with special powers that need to be noted.

In the ultrex family of functions,certain considerations remain constant. Without the base you have nothing,yet in producing larger numbers the base is much less important than the bound,and the bound much less important than the number of iterations.To put a simple example in large letters,

2 uuu 2 >> 2 uu 3 >> 3 uu 2

and that holds for hu,for s,and for l operations as well.

The l operation is a bit more complex than the previous ones and can have a prescribed number of internal iterations,though the default is one.
2 2l22l3 2
means "2 ladder-hyper-operating-transformed twice with 2 internal iterations and then twice with 3 internal iterations,to a bound of 2."

The shorthand apostrophe used for the other operations to make an easy specification as a particle array entry here expands so that
3l'
means "3 ladder-hyper-operating-transformed three times,each with 3 internal iterations,to a bound of 3."

The punctuations of the operator retain their meanings,so that
l means n l b l's
(l) means n (n l b l's) b l's
[l] means n l(n l(...n l b layers of nesting on the bound)...l b l's
{l} means n [l] n layers of square brackets around the l
etc.

Another common element of the family of functions is that each succeeding exponent is the solution of the entire expression below it.In the hyper- operating versions,each is preceded by a like number of up-arrows.
In the SHOT function a tower is built without regard to the special powers brought into play when it is resolved,but in the LHOT function one climbs up and down the ladder with regularity.

I will here essay to explain 3l' as an example.

With ultrex

3 u 2 = 3^3^27 = 3^7625597484987
This 3,638,334,640,025-digit number--the first four digits being 1258--thus serves as the third exponent of 3 u 3.So to get the value of 3 u 3 you raise 27 (second exponent) to it,then 3 (first exponent) to the result,and finally 3 (base) to that.
3 u 3 is the fourth exponent of 3 u 4,etc.

With hyper-operating ultrex

3 hu 2 = 3^^^3^^^^^^^^^^^^^^^^^^^^^^^^^^^27
The exponents are the same but the operations performed on them have changed. That the arrows are actually hyper-operating arrows (of the h↑ variety) is only taken into account as the final value is calculated.
That 3,638,334,640,025-digit number--though now preceded by that many arrows--is retained as third exponent of 3 hu 3.

With simple-hyper-operating transformation

3 s 2 = 3^^^3^^^...3^^^3 (a power tower of 7625597484987 3's) arrows...^^^ (the value of 3^^^that power tower)
The exponents are now defined by the hyper-operations performed on them but that the arrows are actually hyper-operating arrows (of the s↑ variety) is only taken into account as the final value is calculated.
The value of 3 s 2 with the arrows treated as ordinary arrows is what defines the third exponent of 3 s 3 (which is again preceded by a like number of arrows).

Now,with ladder hyper-operating transformation,we DO take the powers of the arrows involved into account as we calculate each rising exponent.

3 l 1 = 3^^^3 = 3^^(3^^3)

The first two arrows,since we are doing an ordinary l operation, are l-arrows,and the last two,as the final pair facing the final exponent, divide with the one on the right becoming an ordinary arrow cloning the final exponent into itself copies of itself separated by single arrows,the one alone facing the final exponent being a normal arrow,and any previous arrows from this spawning becoming s-arrows; and (to avoid infinite recursion) the one on the left becoming an s-arrow (if we were using l it would become a s↑ and if were using parenthesized,bracketed,or braced l it would become a correspondingly parenthesized,bracketed,or braced s-arrow).

3ll↑(3s↑(3s↑(3↑3)))=
3ll↑(3s↑(3s↑(27)))=
3 is now SHOT to a bound of 27 with the 27 exponents calculated and resolved under the rules of the SHOT function.
3ll↑(3s↑(3 s 27)=
3 is then SHOT to a bound of (3 s 27) with the (3 s 27) exponents calculated and resolved under the rules of the SHOT function.

Since the remaining arrows are only a pair,they too get demoted with the one on the right turning the (3 ss 27) into a power tower of 3 ss 27 (3 ss 27)'s,the last arrow only being normal and the rest all being s-arrows,resolving under the rules of the SHOT function layer by layer until one reaches the bottom of the tower,at which point 3 gets SHOT to this number.

We have now calculated the second exponent of 3 l 2, which must be preceded by as many l-arrows.
We then proceed to climb down the ladder to calculate the full value of 3 l 2,in the course of which many large numbers get SHOT.Only when we have the full value do we get the third exponent of 3 l 3,which likewise is preceded by a like number of arrows.

We calculate this value of one climb-down of 3 l 3,resolving from the top,and remember that we're not only doing 3 ladder hyper-operating transformations,we have specified 3 internal iterations of each of them.

So we LHOT 3 to the bound of this number,for the second iteration,and when we've calculated the full value from the top we do the third iteration to a bound of that value.

Then we repeat this whole process twice more,since
3 l' = 3 lll 3 = 3 l (3 l (3 l 3)).

As noted,if we had used 3 l 3 each time we SHOT we would have done so not once but the preceding-number-SHOT-bounded-to-the-following-number times.
Using 3 (l) 3, 3 [l] 3,or 3 {l} 3 would correspondingly amplify the number of SHOT operations performed.

Of course,using small numbers for initial bases OR bounds is just a teaching tool and the numbers at http://put.com/A/alnumb.html are more productive in creating even bigger ones.


And what about arrays?

Array-making is,as I have said in the previous articles, not something I have done much of,but imagine if you would a linear array specified by a comma thusly:

E(3)l',

It is not practical to lay out the numerous comma-separated entries, but they are the base E(3)l' and every exponent it would take to simple-hyper-operating transform itself to a bound of itself,with each one repeated in ascending order as many times as itself(the number it would have of arrows in hyper-operated ultrexing or SHOT). As with my prior example for the hyper-operating ultrex,imagine the base as performing on stage at a talent show,demonstrating its skill at self-hyperinflation.By the stage,the second entry is the ticker, the third entry is the timekeeper,and the entries after that are arbiters,ranging from the first arbiter(the fourth entry,still very early among the repetitions of E(3)s') to the final arbiter (the very last repetition of the highest exponent).

For its first act,the base (which is itself E(3) ladder-hyper-operating-transformed itself-ladder-hyper-operating-transformed times to itself to itself) ladder-hyper-operating-transforms itself itself-ladder-hyperoperating-transformed-to-itself times to a bound of itself.
The ticker goes down by one,and the base repeats its performance based on its new size.
This repeats until the ticker would reach zero,at which point the timekeeper resets the ticker to the size the base has now reached,and itself goes down by one.


As this performer is skilled with a ladder,unlike the previous examples it is blessed with a sympathetic audience.
Every initial entry except the ticker is part of the sympathetic audience:
every time the base raises itself to an exponent,all the entries that have never yet been reduced or replaced each raise themselves to the same exponent.

This replacement of the ticker while decrementing the timekeeper repeats until the timekeeper would reach zero,at which point the first arbiter brings in a new timekeeper of the size the base has now reached, which resets the ticker to the size the base has now reached,while the first arbiter goes down by one.
As each arbiter reaches zero it is replaced by the next arbiter further from the base,which decreases by one,with a new one the current size of the ever-growing base,as are all other entries between that arbiter and the base.
When the final arbiter reaches zero,it ceases to exist as all entries between it and the base are repopulated with ones equal to the then current size of the base.The arbiter before the last becomes the new final arbiter.
As the arbiters dwindle,the replacement cycles continue,until finally the last arbiter is gone,then the timekeeper,then the ticker.

Calculation halts.

At this point,the critics in the audience who were hoping to catcall "doesn't halt" and throw croutons at the stage file out of the auditorium,muttering "naive extension" and squabbling about how many scroobols of loggion marks are in a ducquaxul,composing letters in their heads about how of course their 2↑↑↑3-dimension arrays,which are TETRATIONAL,have to be bigger than ones with only Skewes' NumberG(centillion)-dimensions that are merely exponential.

The poor expanded base is left on stage wanting to keep going,to conjure up a whole new array of ticker,timekeeper,and arbiters based on its current size.

How can we not have pity on the poor little thing?

A simple additional apostrophe

E(3)l','

denotes the ability to thus regenerate the whole line with updated entries a number of times equal to the starting value.

A subscripted preface to the apostrophe E(3)l',E(5)l''

provides a bound to which that starting value is ladder-hyper-operating-transformed itself times to set the number of times regenerations are permitted.

MORE MAY COME LATER on replenishment,reincarnation,and resurrection.

In the meantime,enjoy this creation and the Big LHOT it has enabled.


First edition April 26,2020.
Revised April 27,2020 to reflect renaming of SHOT and up-powering of the l-arrows in their spawning;typos corrected May 4,16,20,and 21,2020. Internal link added May 23.
{l} generation added June 26.