User blog:Antares.I.G.Harrison/Graham Array Notation V1.1

This will be a cite source for the page update. Please do not post errors here, rather post in my last post about 7th place.

Fundamental laws

All reworks/old versions of GAN follow the fundamental laws here
 * An # means the rest of an array or just an array.
 * An & means any kind of symbol, let it be, ^ / \ A and so on.

1. Regarding 1's. [#,1,#] = [#]

2. Regarding 0's. [#,0,#] = [#,#]

3. Symbols. [a&1] = [a]

4. If the symbol is decomposable. [a&1] = [a&[a&0]]. The way the symbol decomposites can change regarding which symbol is used. It can be [a&1] = [a&[a&[.....[a&0]...]]] with a &'s or another definition.

5. If the symbol is decomposable. [a&0] = [a,a]

6. Any single argument. [a] = a

7. Any emtpy slot. [] = the nearest number, righ to left ([5.[].6] = [5,6,6]). Just an empty slot ([]) is 10.

Basic Rules

Rules can be changed/followed (by) the fundamental laws.

[a,b] = ab

[a,b,c] = a↑↑...↑↑b with c up-arrows

[a,b,c,d] = a{a{a...a{c}b...b}b}b with d {}'s using BEAF. ab = [a,a,a,b-1] (May contain errors)

Arrays

with 5 or more entries, GAN, like CCAN or BEAF, we define degenerate arrays

The last entry in the array is called the "Helper" The second-to-last entry in the array is called the "Leader"

This only applies to 5+ entrie arrays. 4 or lower entry arrays should follow the "Basic Rules" section.

e.g. [a,b,c,d,e] > d is the "leader" | e is the "helper"

[a,b,c,H,L] = [a,b,c,[a,b,c,H,L-1]L-1] (H means the "Helper" and L means the "Leader")

The process stops when L reaches 1 e.g. [3,3,3,3,3] = [3,3,3,[3,3,3,3,2],2] = [3,3,3,[3,3,3,[3,3,3,3,1],1],2] = [3,3,3,[3,3,3,[3,3,3,3]],2] = [3,3,3,HUGE NUMBER,2] -> [3,3,3,[3,3,3,HUGE NUMBER]]

Continues :

[3,3,3,3,3,3] = [3,3,3,3,[3,3,3,3,2],2] [5,6,4,2,8,9] = [5,6,4,2,[5,6,4,2,8,8],8]

So we can move on to the rules

Rules

[a^b] = [a,a,a,...,a] with b a's

[a^^b] = [a^a^...^a] with b a's

[a^^^b] = [a^^a^^...^^a] with b a's

[a(^)b] = [a^^...^^a] with b ^'s

[a(^^)b] = [a(^)a(^)a(^)...(^)a] with b a's

[a(^^^)b] = [a(^^)a(^^)a(^^)...(^^)a]

[a((^))b] = [a(^^...^^)a] with b ^'s

[a((^^))b] = [a((^))a((^))a((^))...((^))a]

[a((^^^))b] = [a((^^))a((^^))a((^^))...((^^))a]

[a/b] = [a((...((^^...^^))...))a] with b 's and b ^'s [a//b] = [a/a/a/.../a] with b a's

[a///b] = [a//a//a//...//a] with b a's

Dimensional arrays

Dimensional arrays are arrays that need 2 or more spaces to represent.

they are done by [ac], think of it as ab&c in BEAF

[a<2>b] is a 2-row 2-dimensional array of a,b =

a,b a,b

Which is a,b]^[a,b. All spaces are connected by the ^

n-dimensinal arrays are n-dimensional, n-row arrays with n-1 dimensional arrays connected by the ^

Rules (Continued)

We define a powerfull function, [a\b]

[a\b] = [aaaa...a][aaaa...a][aaaa...a]...<a>[a<b>a<b>a<b>a...<b>a\b-1]

with b a's in [a<b>a<b>a<b>a...<b>a] and a [a<b>a<b>a<b>a...<b>a]'s overall.

[a\\b] = [a\a\a...a] with b a's

And so on (You already know what [a\\\b] means, do you :3)

[A,a] = [a\\\...\\\a] with a \'s

[A,A,a] = [A,[A,a]]

[A,a,a] = [A,[a,a]]

[A2,a] = [A,A,...,A,a] with a A's

[An,a] = [An-1,An-1,...,An-1,a] with a An-1's