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This is a notation that is being made on the Discord Chat.

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Linear Chainer Notation

This is the first part of the notation.

Symbol = is called a chainer.

Arrays are of the form a=b=c=...=z for non-negative integers.

To decompose an array, use the following algorithm:

  • If the array is just 0=n, then 0=n= 10^n.
  • If the array begins with a 0, it can be removed.
  • If the array ends in a string of 0's, replace the "x,0" where x is the last nonzero entry of the array with "x-1,x".
  • Otherwise, find the last nonzero entry before the last entry. Decrease it by 1, and replace the next entry with a copy of the whole expression with the last entry decreased by 1. This works even if the last entry before the final one is nonzero.

Analysis

  • 0=n = 10^n
  • 1=n = 10^^(n+1)
  • 2=n ~ 10^^^(n+2)
  • 3=n ~ 10^^^^(n+2)
  • n=n ~ f_w(n)
  • 1=0=n ~ f_w+1(n)
  • 1=1=n ~ f_w+2(n)
  • 1=2=n ~ f_w+3(n)
  • 1=n=n ~ f_w*2(n)
  • 2=0=n ~ f_w*2+1(n)
  • 2=1=n ~ f_w*2+2(n)
  • 2=n=n ~ f_w*3(n)
  • 3=0=n ~ f_w*3+1(n)
  • 3=n=n ~ f_2*4(n)
  • n=n=n ~ f_w^2(n)
  • 1=0=0=n ~ f_w^2+1(n)
  • 1=0=1=n ~ f_w^2+2(n)
  • 1=0=n=n ~ f_w^2+w(n)
  • 1=1=0=n ~ f_w^2+w+1(n)
  • 1=1=n=n ~ f_w^2+w*2(n)
  • 1=2=n=n ~ f_w^2+w3(n)
  • 1=n=n=n ~ f_w^2*2(n)
  • 2=0=0=n ~ f_w^2*2+1(n)
  • 2=0=n=n ~ f_w^2*2+w(n)
  • 2=n=n=n ~ f_w^2*3(n)
  • n=n=n=n ~ f_w^3(n)
  • 1=0=0=0=n ~ f_w^3+1(n)
  • 1=0=0=n=n ~ f_w^3+1(n)
  • 1=0=n=n=n ~ f_w^3+w^2(n)
  • 1=n=n=n=n ~ f_w^3*2(n)
  • n=n=n=n=n ~ f_w^4(n)
  • n=n=n=n=n=n ~ f_w^5(n)
  • Limit is w^w

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