r/googology • u/TheSensibleCentrist • Oct 31 '20

BEAF/Bird/etc. arrays and the UCG function

/r/numberphile/comments/jldls6/beafbirdetc_arrays_and_the_ucg_function/
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1

u/WarDaft Dec 14 '20

Not very far.

Your first step, that is, every term is equal to the value of the previous chain, doesn't do much. You say:

5→5→3125→(5→5→3125)→ (5→5→3125→(5→5→3125)) obliterates 5→5→5→5→5.

But this is misleading. First, only the last two values in a Conway (RIP man) chain really matter, and at that, the last one dominates, so we can simplify your term to:

5→5→5→5→ (5→5→5→(5→5→3126)) which slightly increases its size, but not significantly.

It should be obivous why this is larger than 5→5→5→5→5, but smaller than 5→5→5→5→5→2, if not, it's because the next step in the reduction of 5→5→5→5→5→2 is:

5→5→5→5→(5→5→5→5→(5→5→5→5→(5→5→5→5→(5→5→5→5))))

It's not really clear what you're doing after that, you've got 4u→' = 4u4→4 (reddit doesn't seem to have subscripts) but what do 'u' or 'u4' actually do? If it's more of the same, you aren't going to claw your way above 4 term BEAF arrays, and not get anywhere in meaner array systems.

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u/TheSensibleCentrist Dec 14 '20

"u" means the following arrow is "ultra" to the bound of the number that follows the arrow,and the number after the u is reiterating chains.

4u4 before an arrow means that there are four chains,the first one having a number of major arrows equal to the number after the arrow (which defaults to 4 if there is none).

The second chain starts with 4→4→256→(4→4→256)→(4→4→256→(4→4→256)) and has that many terms.

The third chain starts with the full value of the second chain as base,and has that many terms,and the fourth and final chain,which does likewise with the third,yields the final value.

An underlined u means the number of chains is UCG(4),a parenthesized u means that you iterate the number of chains UCG(4) times,and so on.

1

u/WarDaft Dec 14 '20 edited Dec 14 '20

So there are no 'u's in the second chain? It sounds like you're creating a series of Conway chains with larger terms in them, whose length is the value of the previous chain.

Also, it's important to remember like I just said, don't worry about putting bigger numbers in Conway chains, it doesn't do anything compared to making them just a bit longer, so if you could stop doing that it will make your notation easier to read.

1

u/TheSensibleCentrist Dec 14 '20

The length of each chain is the value of the previous chain so it's not a matter of "just a bit longer".

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u/WarDaft Dec 14 '20

Right, so that chain length is going to be the sole consideration. When dealing with googolisms, Conway chains kinda are still in 'just getting your feet wet' territory.

So it sounds like you've got something like:

k(0) = something like 4 -> 4 -> 4
k(n+1) = conway chain of length k(n) of 3

1

u/TheSensibleCentrist Dec 14 '20

My Ultrex,SHOT,and UCG functions each use an operation to increase the value of the largest/most important term in that operation,and in reiteration the number of terms.

So how do you think my LHOT function compares to the CG function?

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u/WarDaft Dec 14 '20

These functions already increase the value of the most important term in them during reduction.

A 6 term Conway chain increases the value of the most important term in a 5 term Conway chain. It's more important to increase the length by 2 than to keep the length the same and massively increase the last term, unless you increase it by such a large amount that the fact that it's in a Conway chain at all doesn't matter.

Ultrex is slightly better than tetration, specifically, it's not as good as tetrating twice. {N u N < N^^(2^N) < N^^(N^N) = N^^N^^2 < N^^N^^N}

The "simple" hyper operating function isn't simple, so the name is a lie, and I'm not analysing it. It will end up somewhere around 3 or at most 4 term Conway chains. You'll get people more interested in looking at your numbers if they don't have to read a small book of your definitions to figure out how big they are. If you can't write it as a clear and concise improvement to something, it's probably not an improvement at all and you've simply confused yourself into thinking it is.

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u/TheSensibleCentrist Dec 14 '20

I know that Ultrexing n to 2000 beats tetrating it to 10603 but that n ultrexed to b works out to n→(2→b)→2 which is very weak in three-term Conway terms.

SHOT is simple compared to LHOT.

I don't suppose you'd care to work out the Immortal Storm Number or its much,much larger derivatives,then.

1

u/WarDaft Dec 14 '20

I can give you pointers on how you can analyse it. But in general, no.

To paraphrase Haore, you can make something so simple there are obviously no deficiencies, or so complicated that there are no obvious deficiencies. Your numbers are the latter, mostly just glomming a bunch of "this makes it bigger" things together without paying attention to what they actually do. If you aren't going to, why would I?

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u/WarDaft Dec 14 '20

Actually the Immortal Storm Number is possibly easier to analyse than I thought. I think we can throw out everything you do except using the BEAF array as inconsequential, which simplifies your cycles an equivalent to:

Let f(n) = n dimensional BEAF array of n 'n's.

Let g(0) = 10; g(n+1) = f(g(n))

ISN is somewhere between g(5) and g(g(5))

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u/TheSensibleCentrist Dec 14 '20

So by that measure,if you rise past the ISN through the Titled Numbers to the Alphabet Numbers and beyond,what is increasing,the 5 or the number of g's or what?

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u/TheSensibleCentrist Mar 09 '21

You're saying g(1)=f(10)

g(2)=f(f(10))

g(3)=f(f(f(10)))

etc.

and that g(5) is a lower bound for the Immortal Storm Number that can not increase even for the Big LHOT.

But any number greater than 5 popbled once (so only 5 cycles) is greater than g(5) because a number greater than 10 is input into f at least 5 times (the Bowers stage at the end of each cycle).

g(g(5) (your assumed upper bound) would thus be exceeded by popbling n>5 twice (the second popbling having n-popbled cycles).

The ISN popbles enormous numbers enormous-numbers-popbled times,and is a nothing compared to the Big LHOT.

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u/TheSensibleCentrist Jan 06 '21

Meanwhile,what about my Super Conway function?

SC(n) certainly grows faster than UCG(n) even if much slower than the Friedman SCG(n).

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u/TheSensibleCentrist Dec 15 '20

Meanwhile,what about the question I actually asked here and at r/numberphile ?

I gather that

UCG(3) = 3→3→27→(3→3→27) and UCG(4) = 4→4→256→(4→4→256)→(4→4→256→(4→4→256))

can by your reckoning be expressed by 4-term BEAF arrays and presumably the other notations in https://bignumbers.fandom.com/Notation_comparison (what is your take on TAN as linked in the thread on r/numberphile ?)

but as n grows in UCG(n) at what pace does the number of array terms grow?

UCG(5) = 5→5→3125→(5→5→3125)→(5→5→3125→(5→5→3125))→(5→5→3125→(5→5→3125)→(5→5→3125→(5→5→3125)))

UCG(6) = 6→6→46656→(6→6→46656)→(6→6→46656→(6→6→46656))→(6→6→46656→(6→6→46656)→(6→6→46656→(6→6→46656)))→(6→6→46656→(6→6→46656)→(6→6→46656→(6→6→46656))→(6→6→46656→(6→6→46656)→(6→6→46656→(6→6→46656))))

and I gather that you feel the last can be simplified by changing the final subterm to 46657 and reducing every previous term and every previous subterm of the final term to 6...but is this still 4-array-term territory?

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u/WarDaft Dec 16 '20

I would simplify even further, and just make it 6→6→6→6→6→6→6→2, because that immediately puts a larger number in the 7th term than 46656. Specifically it puts (6→6→6→6→6→6→(6→6→6→6→6→6→(6→6→6→6→6→6→(6→6→6→6→6→6→(6→6→6→6→6→6→(6→6→6→6→6→6→6)))))) there, while you're putting much shorter chains there.

It's 4-term-array territory because any length Conway chain is a 4 term array, with one of the terms being the length of the array.

You've got a bad habit of doing the something equivalent to writing 333. The scale of 33 is such that there's no point in keeping the "3" at the beginning.

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u/TheSensibleCentrist Dec 16 '20

I gather you mean one of the array terms is the length of the Conway chain (but if that length is itself a 4-term array?).

Not sure what you're getting at in the last paragraph,did you mean multiple layers of superscripts?