User blog:TheMostAwesomer/Compressed BEAF: My own direction

I've been taking BEAF in an entirely different direction than multidimensional arrays. Much like those, these are technically just linear arrays of monstrous length (the number of entries itself being highly googological). Only these stay more or less linear themselves.

The Basics
At the heart of CBEAF is a simple array, ⟨a,b⟩. It has a simple definition; ⟨a,b⟩ = {a,a,a...a,a,a} with b a's. When it has more than 2 arguments, it follows BEAF's rules. Functions of this kind are called "Class A" for brevity's sake; when they have two arguments, a and b, they define the function a level below it with b a's. A level above ⟨a,b⟩ is ｢a,b｣, and above that is ⸤a,b⸥. ⸤a,b⸥ can be generalized as :a,b:, and anything above it in level with multiples of colons (::a,b:: is ⸤a,a,a...a,a,a⸥ w/b a's, for example). These can all be generalized into integers, with BEAF at zero.

This infinite set, (0,1,2,3,4...) can be given a function {[a,b,c]}; a and b are the a and b of the defined function, and c defines the level. It covers levels 0 through ω and is itself superlevel 1; superlevel 0 is BEAF. {[a,b,c]} follows the "Class B" rules. When there are more than three arguments, it follows its own rule set. When its third argument is 1, no matter what is in the later arguments, it's equal to ⟨a,b⟩. When the fourth is 1, the function equals {[a,b,c,e^(e-1),f^(f-1)...]}. Everything is moved 1 closer to the front and taken to the power of itself, minus one. When any argument further than 4 equals one, the expression equals {[a,b,c,d]}. When none of this is true, it's broken down to {[a,b,{[a,b,c-1,d,e,f...]},d-1,e,f...]}. Got it? This is where I stopped at first, and it can give some wild numbers.

Minor Interlude
First, mostly for my sake, I'll explain a second way of defining the levels (aka what I did in the Google document). In the doc, I set {[a,b,c,d...]} as the first level of super-compression, the soon-to-be explained /a,b/ as the second, //a,b// as the third, and so on. Levels of super-compression correspond to superlevels. Above these are levels of duper-compression, then levels of truper-compression, and so on, then levels of hyper-compression. These are really just naming differences.

Going Deeper
We can continue further into the superlevels. Superlevel 2 is /a,b/, which is Class A, and is {[a,a,a...a,a,a]}, with b a's. Superlevel 3 is //a,b//. This continues on until superlevel ω, or duperlevel 1. Once again, duperlevel 0 is BEAF. Duperlevel 1 is <[a,b,c]>, a Class B expression that can call from every superlevel. Once again, an infinite set could be used to simplify this concept. Duperlevel 2 is \a,b\, or <[a,a,a...a,a,a]>, duperlevel 3 is \\a,b\\, or \a,a,a...a,a,a\ w/b a's, and so on, all the way up to duperlevel ω, or truperlevel 1. Truperlevel 1 is )[a,b,c](, and can call from every duperlevel. All of these superlevel variants can themselves be reduced into an even more compressed function, at hyperlevel 1. Hyperlevel 1 can call from every -perlevel below it; superlevel, duperlevel, truperlevel, etc. It's the first to have four arguments naturally.

Hyperlevel 1
Hyperlevel 1 is ᄂa,b,c,dᄀ. A, b, and c are the a, b, and c of the called -perlevel, and d is the -perlevel being called. When d = 1, it's {[a,b,c]}, when d = 2, it's <[a,b,c]>, when d = 3, it's )[a,b,c](. When d = 4, however, it's instead noted as super4(a,b,c) (in the doc the 4 was subscript, I'll work on adding it here, too, perhaps). ᄂa,b,c,dᄀ can be generalized as superd(a,b,c). It's not defined for more than 4 arguments yet, but it will be a Class B-prime (or Class B').