User blog:GamesFan2000/Chained Array Notation

This is an array notation… the most famous words in my dialect. Oh well. I call this Chained Array Notation.

Digression on Arrays and People Who Deserve a Thanks
First of all, what is array notation? But even before that, what is an array in mathematics? An array is an ordered set of entries that describes a mathematical property, whether it be a function or a pattern or anything in math, really. They are most often expressed as (a1, a2, …an), where every entry is contained with some form of bracket set. Array notation specifically refers to a type of mathematical function that uses arrays to express its results. There are three array notation systems that are commonly in use for googology purposes. First, there’s Jonathan Bowers and his Exploding Array Function. Bowers invented array notation as we know it, and his system, BEAF, is the most commonly used of all the array notations. However, it’s also the most ill-defined of the big three array notations, relying on dimensionality and poorly understood operations. Next up is Chris Bird and his notation. Bird actually made a very important contribution by suggesting that two-entry arrays use exponentiation instead of addition. Bird’s array notation, or BAN, is much better defined, using the concept of nesting arrays and stretching it to the extreme. However, even BAN at its limits is straight-up neutered by the last of the big three. The most well-defined array notation, and beyond a reasonable doubt the absolute strongest computable function ever defined in googology, is strong array notation, created by hyp cos. Hyp cos painstakingly built and analyzed his monster, which goes to the absolute extremes of what googology is capable of producing, and he might still be trying to push it even further. I would like to say that all three of these brave men have my utmost respect. Without their fine work, I wouldn’t be here defining this notation, or any of my array notations, for that matter. Another person who I have much respect for is PsiCubed. He is currently developing a base 10 hierarchy of sorts, comparing many different expressions to each other and levelling them based on size. If you ever need to find out how large your number truly is, then check his Psi Level charts out.

Digression on Chained Arrow Notation and More People to Thank
Even with thanking those who built array notation, I still need to honour three more people who developed functions that are of great use to me. Of all the people to thank, Donald Knuth is one we can all thank for his up-arrow notation, THE gateway notation for googology. If you don’t understand expressions like 3^^^^3, then you’re not going to make it far in this field. Up-arrows are the easiest way to describe hyper-operations, with n^ab meaning n^a-1n…^a-1n, where there is a set of b n’s. But, that’s not of importance to my notation directly. What is of direct importance is the legendary chained arrow notation, developed by John H. Conway and R. K. Guy. Those two especially deserve my thanks, because chained arrows are how I’m going to build this function.

Chained arrow notation is quite a doozy. If we define a function for this, F(n)=n→n→n...→n, with n n’s. Conway’s Tetratri is famous in the googology community for utterly destroying Graham’s number. That’s how powerful this is: even with just four variables, the results are unimaginable. This notation is the base of my arrays.

The Rules: Two Entries
So, with all of that said, we’re ready to begin the process of defining this notation. The first rule is very simple, being that an array with one entry is equal to that entry, and that an array that starts with 1 is equal to 1. The second rule is that (a, b)=a→a→...a, with b+1 a’s, and that (a, 0...)=a.

(2, b, ...)=4

(3, 1)=3→3=27

(4, 1)=256

(5, 1)=3125

(3, 2)=3→3→3=3^^^3

(3, 3)=Conway’s tetratri, 3→3→3→3

The two-entry arrays are very simple to create, but solving them is anything but child’s play. These are impossible to solve within the constraints of our observable universe after a certain point. It has been proven that chained arrows have a limit of Fω^2(n) if compared to the Wainer hierarchy.

Rules: Three Entries
Now we will take the immense size of chained arrow expressions and use them to their fullest potential. Rule 3 states that (a, b, 0, ...)=(a, (a, (a, ...(a, b)...)), ...), with (a, b) recursions on (a, b), and that (a, b, c, ...)=(a, (a, b-1, c, ...), c-1, ...).

(3, 3, 3)=(3, (3, 2, 3), 2)=(3, (3, (3, 1, 3), 2), 2)=(3, (3, (3, 3, 2), 2), 2)=(3, (3, (3, (3, 2, 2), 1), 2), 2) =(3, (3, (3, (3, (3, 1, 2), 1), 1), 2), 2)=(3, (3, (3, (3, (3, 3, 1), 1), 1), 2), 2)=                                           (3, (3, (3, (3, (3, (3, 2, 1), 0), 1), 1), 2), 2)=(3, (3, (3, (3, (3, (3, (3, 3, 0), 0), 0), 1), 1), 2), 2)

Now, as you can see, these do eventually terminate, but it would take longer than the universe will last googolplexes of times over. With that in mind, the numbers are completely gargantuan.

Universal Rule for Chained Array Notation Beyond Three Entries
Now that we’ve defined the three entry arrays, it’s time to define the final rule of the base notation. Rule 4, the universal rule for solving beyond three entries, states that (a, b, c, ...z, 0, n, ...)=(a, b, c, ...z, (a, b, c, ...z, (...(a, b, c, ...z, n)...)), n-1, ...). That general expression means that if a zero is ever present, take the previous entries and recurse them by the answer they would give if they were their own array, with n as the final entry of the final array. Replace the zero with this new set of arrays and reduce n by 1 in the original array. (a, b, 0, ...0, n, ...) is excluded from the reduction rules for any zero, but the recursions still apply.

(3, 3, 0, 3)=(3, 3, (3, 3, ...(3, 3, 3)...), 2)

(3, 3, 0, 0, 3)=(3, 3, (3, 3, ...(3, 3, 0)...), 0, 3)

I’m not good at growth rates as far as FGH is concerned, so I’d like anyone who is good at the FGH to help me out with this function’s strength. F(n) for this notation would be an n-length array of n’s.