SuperAcid Array Notation

SuperAcid Array Notation

This is my own version of Wythagoras's "Dollar Function"

I define a "block" X as a string of ['s, ]'s, and natural numbers.

I give the following rules, where X is any block:

a::0X = a::X
a::bX = a::(a::(a:...(a::X)...)::X)::X)::X)::X with b nestings
a::X = a::Y where Y is X's expanded form

Block expansion work as following:

Every single time you call one of the following rules, "a" increase by 1

Let X_k be a group delimited by []'s
Let copy(S,1)=S and copy(S,n+1) = S+copy(S,n) where + is string concatenation

Each bracket ] can have a level k (we note ]_k)
The level of the last bracket of any group is the level of the group (noted X^level)

a) [0]⁰ = a+1

If the first "..." does not contain any natural number:
b) [...X₁X₂X₃0...] = copy("[...X₁X₂X₃...]",a)
c) [...X₁X₂X₃(k+1)...] = copy("[kX₁kX₂kX₃k...]",a)

If there isn't any natural in the block and X₁ does not has the form [0]^k:
d) [X₁X₂X₃...] = copy("[Y₁X₂X₃...]",a) where Y₁ is X₁'s expanded form (we keep using/increasing the same "a" while expanding X₁)

If X₁ has the form [0]^k+1:
Let S be the level-k block of all blocks right after the "a::"
Let S' be that array with [0]^k+1 remplaced by [0]^k
e) [[0]^k+1X₂X₃...] = copy("[[S']^kX₂X₃...]",a)

If X₁ has form [0]^Y where Y is a block:
Let S be the level-Z block of all blocks right after the "a::" where Z is Y's expanded form
Let S' be that array with [0]^Y remplaced by [0]^Z
f) [[0]^YX₂X₃...] = copy("[[S']^ZX₂X₃...]",a)

Now I define the following sequence:
S₀ = [[0]⁰]
S_k+1 = [[0]^S_k]

And [0 ; 1] = S_a

Now you can go for [[1 ; 1]] = [[0 ; 1][0 ; 1] ... [0 ; 1] [0 ; 1]] and so on (old rules still apply)

Beyond?

Let D₀ = [[0 ; 1]] and D_n+1 = [[0 ; 1]^D_n]
Then [[0 ; 2]] = D_a

From there we can go to [[0 ; 3]], [[0 ; 4]], etc, and [[0 ; X]] => [[0 ; Y]] where Y is X's expanded form

I'll stop here... for the moment

Now, a function:

T₀ = [0 ; 1]
T_n+1 = [0 ; T_n]^T_n
SuperAcid¹(k) = k::[T_k]
SuperAcidⁿ⁺¹(k) = SuperAcidⁿ(k::[T_{SuperAcidⁿ(k)}])

Also the SuperAcid Number:
SAN = SuperAcid^{SuperAcid100(1010100)}(10¹⁰¹⁰⁰)

which is, huh, very big :V

Bonus: Modified Ackermann function

This is my stronger version of the Ackermann function.
I define it as following:

- MAF(0,0,0) = 3
- MAF(x,0,0) = 3x+3
- MAF(0,0,z) = MAF(z,MAF(0,0,z-1),z-1)
- MAF(x,0,z) = MAF(x,MAF(x-1,0,z),z-1)
- MAF(0,y,z) = MAF(MAF(0,y-1,z),y-1,z)
- MAF(x,y,z) = MAF(MAF(x-1,y,z),y-1,z)

Rules can be simplified without too much problem, but I don't really care

I believe it to reach f_ω²(n) in the fast-growing hiearchy. It's pretty much nothing compared to SAN though :V