User:Ynought/: notation

\(:\) notation
This will (hopefully) be my most refined (recursive) notation

\(\#_k\) is the labeled rest of the notation

\(\{ \text{and} \}\) is any amount of brackets

\(a\) will always denote the number before the \(:\)

\(\#_i\rightarrow\#_j\) means \(\#_i\) turns into \(\#_j\)

\(f_0(c)\rightarrow c^2\) and \(f_k(c)\rightarrow f_{k-1}^c(c)\) where \(f^0_k(c)=c\) and \(f^{d}_k(c)=f_k(f_k^{d-1}(c))\)

Simple notation
This is the 1st part of ? parts and it is the most simple

\(a:=a\)

\(a:\{\#_1b\}\rightarrow f_a^a(b):\{\#_1\}\) here \(\#_1\) isnt empty if \(\{ \text{ and } \}\) have \(1+\) brackets and b isnt inbetween

\(a:\#_1b\rightarrow f_a^a(b):\#_1\)

Start looking from right to left until you find a number inside a pair of bracket.Call that number \(b\).If.you have a power tower then start looking from the top-most element.Then: "1.If \(b=0\) then \((0)\rightarrow a\)""2.If \(b>0\) then \((b)\rightarrow(b-1)(b-1)...(b-1)\) with (in total) \(a\) \((b-1)\)'s"

Expanding brackets
"1. \((b)^0\rightarrow (b)...(b)\) with a nests""2. \((b)^c\rightarrow (...(b)^{c-1}...)^{c-1}(...(b)^{c-1}...)^{c-1}...(...(b)^{c-1}...)^{c-1}\) with a nests"