Chained arrow notation

{{function 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.
 * type=Multidimensional
 * base=Hyper-operators
 * fgh=\omega^2}

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.

Peter Hurford's extensions
Peter Hurford extends chained arrow notation by adding the following rule:

\(a \rightarrow_c b = \underbrace{a \rightarrow_{c-1} a \rightarrow_{c-1}\ldots\rightarrow_{c-1} a \rightarrow_{c-1}}_{b \rightarrow_{c-1}'s}\)

More formally:

\(a \rightarrow_c b = a \rightarrow_{c-1} (a \rightarrow_c (b-1))\)

\(a \rightarrow_c 1 = a \rightarrow_{c-1} a\)

All normal rules remains unchanged. Thus, they apply ignoring subscripts. Notice that expressions like \(3 \rightarrow_{2} 3 \rightarrow 3\) are illegal. Entire expressions must have single type of right-arrows. Also, Hurford shows that \(f(n) = n \rightarrow_n n\) is about \(f_{\omega^3}(n)\) in the fast-growing hierarchy.

Furthermore, he defines C(n) function as follows:

\(C(a) = a \rightarrow_a a\)

\(C(a,1) = a \rightarrow_{C(a)} a\)

\(C(a,b) = a \rightarrow_{C(a,b-1)} a\)

\(C(a,1,1) = C(a,C(a,a))\)

\(C(a,b,1) = C(a,C(a,b-1,1))\)

\(C(a,1,c) = C(a,C(a,a,c-1),c-1)\)

\(C(a,b,c) = C(a,C(a,b-1,c),c-1)\)

The function \(f(n) = C(n,n,n)\) is about \(f_{\omega^5}(n)\) in the fast-growing hierarchy.