User blog:Nayuta Ito/"I think this is the fastest" function (Part 1+1/2)

Basic M function: M(linear) is same as Taro's multivariable Ackermann function. Anything can be the linear function, but Ackermann is nice because it has the growth of w^w.

Notations:
 * 1) You must start with M.
 * 2) A number must come after M.
 * 3) Last thing must be a number.
 * 4) ^'s and brackets, which means {}, comes between.

Rules:
 * 1) If you see ^'s before the last number
 * 2) If what is before the ^'s is a number:
 * 3) ;It is just an arrow notation.
 * 4) Else (,which means what is before is brackets):
 * 5) ;It must be "M sth {x}^^..^^y" (a ^'s).
 * 6) ;That is equal to M sth {x}^^..^{x}^^..^{x}...{x}^^..^{x}^^..^{x} (a-1 ^'s each and y {x}'s).
 * 7) If you see bracket(s) before the last number
 * 8) If the last bracket is {0}
 * 9) If the before {0} is a number
 * 10) ;You can use basic function rules, ignoring what is before x{0}y.
 * 11) If the before is bracket (,which means M sth{x}{0}y)
 * 12) ;Let (the last number in sth) (sb),
 * 13) ;It is "M sth{x}sth{x}...sth{x}sb" (y {x}'s).
 * 14) If the before is ^ (,which can be written as sth^^..^^{0}y (x ^'s)),
 * 15) ;It is "sth^^..^{0}sth^^..^{0}...sth^^..^{0}sth^^..^{0}sb". (y {0}'s)

I don't know if this is well-defined.

(being written)