Chained arrow notation

Chained Arrow Notation is a generalization of up-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
\(a \rightarrow b = a^{b}\)

\(a \rightarrow b \rightarrow c = a\underbrace{\uparrow\ldots\uparrow}_cb\) (in which up-arrow notation is used)

When last entry is 1, it will be ignored. \(a \rightarrow\ldots\rightarrow b \rightarrow 1 = a \rightarrow\ldots\rightarrow b\)

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

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

Examples
Here are some small examples of chained arrow notation in action?

\(3 \rightarrow 3 \rightarrow 2 = 3\uparrow\uparrow3 = 7,625,597,484,987\)

\(3 \rightarrow 3 \rightarrow 2 \rightarrow 2 = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 2 \rightarrow 1 \rightarrow 2) \rightarrow 1 = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 2 \rightarrow 1 \rightarrow 2) = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 2 \rightarrow 1 \rightarrow 2) = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow) = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow) = 3 \rightarrow 3 \rightarrow (3^{3}) = 3 \rightarrow 3 \rightarrow 27 = 3 underbrace{\uparrow\ldots\uparrow}_27 3 \)

CG function
Using chained arrow 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. Therefore cg(4) is much, much larger than the famous Graham's number.

Growth rate of this function is about \(f_{\omega^2}(n)\) in the fast-growing hierarchy. This is comparable to 4 entry linear array in BEAF or BAN.

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

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

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. If we allow to mix different types of arrows in the single expression, we get \(f(n) \approx f_{\omega^\omega}(n)\).

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 \(f(n) = C(n,n,n)\) grows about as fast as \(f_{\omega^3 + \omega}(n)\) in the fast-growing hierarchy.