User blog:MilkyWay90/Diagonalized Notation

Take any F(a, b, ..., c).

Define █(function) = number of parameters the function has (from the most recent mention of the function)

A(a, b, ..., c) = F(a, b, ..., c)

A(a, b, ..., c, ..., d) = A(A(...A(A(a, b, ..., c, ..., d - 1), b, ..., c, ..., d - 1)..., b, ..., c, ..., d - 1), A(a, A(a, ...A(a, A(a, b, ..., c, ..., d - 1), ..., c, ..., d - 1)..., ..., c, ..., d - 1))), ..., A(a, b, ..., A(a, b, ... A(a, b, ..., A(a, b, ..., c, ..., d - 1), ..., d - 1), ..., d - 1), ..., d - 1), ..., d - 1) with there being d * █(A) As

A(a, b, ..., c, ..., 1) = A(a, b, ..., c, ...)

B(a, b) = A(a, a, ..., a) with there being b as

B(a, b, ..., c) = B(B(...B(B(a, b, ..., c - 1), b, ..., c - 1)..., b, ..., c - 1), B(a, B(...B(B(a, b, ..., c - 1), b, ..., c - 1), ..., c - 1 ), ..., c - 1) with there being c * █(B) Bs

B(a, b, ..., 1) = B(a, b, ...)

C(a, b) = B(a, a, ..., a) with a bs

C(a, b, ..., c) = C(C(...C(C(a, b, ..., c - 1), b, ..., c - 1)..., b, ..., c - 1), C(a, C(...C(C(a, b, ..., c - 1), b, ..., c - 1), ..., c - 1 ), ..., c - 1) with there being c * █(C) Cs

C(a, b, ..., 1) = C(a, b, ...)

and so on...

a = {A(...), B(...), C(...), ..., Z(...), AA(...), AB(...), ..., AZ(...), BA(...), BB(...), ..., ZZ(...), AAA(...), ...} where A, B, C, ... means the functions we mentioned above.

☺(b, c, d) = a_b(c, d) where a is the set mentioned above and x_y is the yth element of x

☻(function, b, c, d) = ☺(b, c, d) where F(a, b, ..., c) = function.

My notation will be defined as ☻(function, b, c, d)

What would be my notation's growth rate if I put Ackermann function, BEAF, BB(n), etc. in the function parameter?

I'm sure I have a typo in here, so let me know if I get anything wrong!