Chained arrow notation

Chained Arrow Notation is a generalization of Arrow Notation devised by J. H. Conway and R. K. Guy. By Bird's Proof, Jonathan Bowers' Array Notation will generally produce larger values than Chained Arrow Notation.

Formula
\(a \rightarrow b \rightarrow c = a\underbrace{\uparrow\ldots\uparrow}_cb\) (in which Arrow Notation is used)

\(a \rightarrow\ldots\rightarrow b \rightarrow 1 = a \rightarrow\ldots\rightarrow b\)

\(a \rightarrow\ldots\rightarrow b \rightarrow 1 \rightarrow c = a \rightarrow\ldots\rightarrow b\)

\(a \rightarrow\ldots\rightarrow b \rightarrow (c + 1) \rightarrow (d + 1) = a \rightarrow\ldots\rightarrow b \rightarrow (a \rightarrow\ldots\rightarrow b \rightarrow c \rightarrow (d + 1) ) \rightarrow d\)

CG function
Using this notation, Conway and Guy created a new function similar to the Ackermann numbers that defines \(cg(n) = \underbrace{n \rightarrow n \rightarrow \ldots \rightarrow n \rightarrow n}_n\). Although it is equal to the Ackermann numbers for n from 1 to 3, \(cg(4) = 4\rightarrow 4\rightarrow 4\rightarrow 4 > \lbrace 4,4,3,2 \rbrace\), using Bird's Proof.

Growth rate of this function is about \(f_{\omega^2}(n)\) in the fast-growing hierarchy. The example of function with comparable growth rate is E(n), the Exploding Tree Function.