Chained arrow notation

**Chained arrow notation** is a generalization of up-arrow notation devised by J. H. Conway and R. K. Guy.

Nathan Ho and Wojowu showed that chained arrow notation terminates — that is, its output exists for all input sequences.[dead link]

By Bird's Proof, chained arrow notation is roughly comparable to 4-entry arrays in Jonathan Bowers' linear array notation, but is far exceeded by arrays with 5 entries or more.

## Definition

1. \(a \rightarrow b = a^{b}\)
2. \(a \rightarrow b \rightarrow c = a\uparrow^cb = a\underbrace{\uparrow\ldots\uparrow}\_cb\) (in which up-arrow notation is used) = a[c+2]b (in which the bracket notation of hyper operator is used)
3. \(a \rightarrow\ldots\rightarrow b \rightarrow 1 = a \rightarrow\ldots\rightarrow b\) — When last entry is 1, it will be ignored.
4. \(a \rightarrow\ldots\rightarrow b \rightarrow 1 \rightarrow c = a \rightarrow\ldots\rightarrow b\)
5. \(a \rightarrow\ldots\rightarrow b \rightarrow (c + 1) \rightarrow (d + 1) = a \rightarrow\ldots\rightarrow b \rightarrow (a \rightarrow\ldots\rightarrow b \rightarrow c \rightarrow (d + 1) ) \rightarrow d\)

Note that the arrow \(\rightarrow\) is not an operator in the conventional sense; \(a\rightarrow b\rightarrow c\) is equal to neither of \(a\rightarrow(b\rightarrow c)\) or \((a\rightarrow b)\rightarrow c\). The entire chain (which could be thought of as an array) represents a single operation.

## Examples

Here are some small examples of chained arrow notation in action:

\(2 \rightarrow 2 \rightarrow 2 \rightarrow 2\) = \(2 \rightarrow 2 \rightarrow (2 \rightarrow 2 \rightarrow 1 \rightarrow 2) \rightarrow 1\) = \(2 \rightarrow 2 \rightarrow 4 \rightarrow 1\) = \(2\uparrow\uparrow\uparrow\uparrow2\) = 4

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

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

## Approximation

Conway wrote that Graham's number = G is between 3→3→64→2 and 3→3→65→2, and therefore less than 3→3→3→3. It can be confirmed as follows. Let g(x) = \(3\uparrow^{x}3 = 3\rightarrow3\rightarrow x\), and G = g64(4).

* 3→3→1→2 = 3→3→1 = g(1) = 27
* 3→3→2→2 = 3→3→(3→3→1→2)→1 = 3→3→(3→3→1) = g(g(1)) = g2(1) = 3→3→27 = g(27)
* 3→3→3→2 = 3→3→(3→3→2→2) = g3(1) = g2(27)
* 3→3→n→2 = gn(1) = gn-1(27)
* 3→3→64→2 = g64(1) < g64(4) = G
* 3→3→65→2 = g64(27) > g64(4) = G
* 3→3→3→3 = 3→3→(3→3→2→3)→2 > 3→3→65→2 > G

Using this approximation, a Moser is \(\approx g(10 \uparrow\uparrow 257) < g(g(3))\),

* 3→3→2→2 < Moser < 3→3→3→2

The chained arrow notation can be approximated with the Taro's multivariable Ackermann function as follows for x=1, y>1 ou x>1, y+z>0.

A
(
x
,
y
,
z
)
<

3
→
3
→
⋯
→
3

⏟

x
1

→
z
2
→
y
1
<
A
(
x
,
y
,
z
1
)

{\displaystyle A(x,y,z) < \underbrace{3 \rightarrow3 \rightarrow \cdots \rightarrow 3}\_{x+1} \rightarrow z+2 \rightarrow y+1 < A(x,y,z+1)}

It can be approximated with Fast-growing hierarchy with the fundamental sequence of Wainer hierarchy as follows.

* \(3 \rightarrow 3 \rightarrow n \sim f\_\omega (n) \)
* \(3 \rightarrow 3 \rightarrow 2 \rightarrow 2 = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 1) \sim f\_\omega (27) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 2 = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 2 \rightarrow 2) \sim f^2\_\omega (27) \)
* \(3 \rightarrow 3 \rightarrow 4 \rightarrow 2 = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 3 \rightarrow 2) \sim f^3\_\omega (27) \)
* \(3 \rightarrow 3 \rightarrow (n+1) \rightarrow 2 \sim f\_{\omega +1} (n) \)
* \(3 \rightarrow 3 \rightarrow 2 \rightarrow 3 = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 1) \rightarrow 2 \sim f\_{\omega +1} (26) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 3 = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 2 \rightarrow 3) \rightarrow 2 \sim f^2\_{\omega +1} (26) \)
* \(3 \rightarrow 3 \rightarrow 4 \rightarrow 3 = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 3 \rightarrow 3) \rightarrow 2 \sim f^3\_{\omega +1} (26) \)
* \(3 \rightarrow 3 \rightarrow (n+1) \rightarrow 3 \sim f\_{\omega +2} (n) \)
* \(3 \rightarrow 3 \rightarrow 2 \rightarrow 4 = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 1) \rightarrow 3 \sim f\_{\omega +2} (26) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 4 = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 2 \rightarrow 4) \rightarrow 3 \sim f^2\_{\omega +2} (26) \)
* \(3 \rightarrow 3 \rightarrow 4 \rightarrow 4 = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 3 \rightarrow 4) \rightarrow 3 \sim f^3\_{\omega +2} (26) \)
* \(3 \rightarrow 3 \rightarrow (n+1) \rightarrow 4 \sim f\_{\omega +3} (n) \)
* \(3 \rightarrow 3 \rightarrow (n+1) \rightarrow 5 \sim f\_{\omega +4} (n) \)
* \(3 \rightarrow 3 \rightarrow (n+1) \rightarrow 6 \sim f\_{\omega +5} (n) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow (n+1) \sim f\_{\omega \times 2} (n) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 2 \rightarrow 2 = 3 \rightarrow 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 3) \sim f\_{\omega \times 2} (f\_4 (3)) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow 2 = 3 \rightarrow 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 3 \rightarrow 2 \rightarrow 2) \sim f^2\_{\omega \times 2} (f\_4 (3)) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 4 \rightarrow 2 = 3 \rightarrow 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow 2) \sim f^3\_{\omega \times 2} (f\_4 (3)) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow (n+1) \rightarrow 2 \sim f\_{\omega \times 2 +1} (n) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow (n+1) \rightarrow 3 \sim f\_{\omega \times 2 +2} (n) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow (n+1) \rightarrow 4 \sim f\_{\omega \times 2 +3} (n) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow (n+1) \sim f\_{\omega \times 3} (n) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow (n+1) \rightarrow 2 \sim f\_{\omega \times 3 +1} (n) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow (n+1) \rightarrow 3 \sim f\_{\omega \times 3 +2} (n) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow (n+1) \rightarrow 4 \sim f\_{\omega \times 3 +3} (n) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow (n+1) \sim f\_{\omega \times 4} (n) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow (n+1) \sim f\_{\omega \times 5} (n) \)
* \(3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow 3 \rightarrow (n+1) \sim f\_{\omega \times 6} (n) \)
* \(\underbrace{3 \rightarrow 3 \rightarrow \ldots \rightarrow 3 \rightarrow 3}\_{n+2} \sim f\_{\omega^2}(n)\)

## References

1. ↑ 1.0 1.1 *The Book of Numbers*, by J. H. Conway and R. K. Guy
2. ↑ Chained Arrow Notation
3. ↑ termination
4. ↑ Fish (2017) 巨大数論 (googology) (in Japanese) p.79
5. ↑ Fish, Ackermann-chain theorem (in Japanese)