Arrow notation

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For other arrow notations, see down-arrow notation and chained arrow notation.

Arrow notation or up-arrow notation is a widely used notation for the hyper operators, devised by Donald Knuth in 1976 to represent large numbers.[1] It is defined by the following rules:

\begin{eqnarray} a \uparrow^1 b &=& a^b \\ a \uparrow^n 1 &=& a \\ a \uparrow^{n + 1} (b + 1) &=& a \uparrow^n (a \uparrow^{n + 1} b) \\ \end{eqnarray}

$$a \uparrow^{n} b$$ is a shorthand for $$a \uparrow\uparrow\cdots\uparrow\uparrow b$$ with n arrows (where n is a positive integer). So, for example, $$a \uparrow^2 b = a \uparrow\uparrow b$$. Arrow notation operators are right-associative; $$a \uparrow b \uparrow c$$ always means $$a \uparrow (b \uparrow c)$$.

Specifically, $$a \uparrow b$$ is exponentiation, $$a \uparrow\uparrow b$$ is tetration, $$a \uparrow\uparrow\uparrow b$$ is pentation, and so forth. In ASCII, these are written a^b, a^^b, a^^^b, ...

The function $$f(n) = n \uparrow^n n$$ is a fast-growing function that eventually dominates all primitive recursive functions, and can be approximated using the fast-growing hierarchy as $$f_\omega(n)$$.

Arrow notation has been generalized to other notations. A few notable ones are chained arrow notation, BEAF, and BAN. It has also been compared exactly with Notation Array Notation using the function (a{2, number of arrows}b).

Nathan Ho and Wojowu showed that arrow notation terminates — that is, $$a \uparrow^n b$$ exists for all $$a,b,n$$.[2]

Examples Edit

• $$2 \uparrow 3 = 2^3 = 8$$
• $$5 \uparrow 6 = 5^6 = 15625$$
• $$10 \uparrow 100 = 10^{100} =$$ googol
• $$3 \uparrow\uparrow 4 = 3 \uparrow 3 \uparrow 3 \uparrow 3 = 3 \uparrow 3 \uparrow 27 = 3^{7625597484987}$$
• $$5 \uparrow\uparrow 3 = 5 \uparrow 5 \uparrow 5 = 5^{5^5}$$
• $$2 \uparrow\uparrow\uparrow 2 = 2 \uparrow\uparrow 2 = 2 \uparrow 2 = 2^2 = 4$$
• $$2 \uparrow\uparrow\uparrow\uparrow 2 = 2 \uparrow\uparrow\uparrow 2 = 2 \uparrow\uparrow 2 = 2 \uparrow 2 = 2^2 = 4$$
• $$3 \uparrow\uparrow\uparrow 2 = 3 \uparrow\uparrow 3 = 3 \uparrow 3 \uparrow 3 = 3^{3^3} = 3^{27} = 7625597484987$$
• $$2 \uparrow\uparrow\uparrow 3 = 2 \uparrow\uparrow 2 \uparrow\uparrow 2 = 2 \uparrow\uparrow 4 = 2 \uparrow 2 \uparrow 2 \uparrow 2 = 2 \uparrow 2 \uparrow 4 = 2 \uparrow 16 = 65536$$
• $$3 \uparrow\uparrow\uparrow 3 = 3 \uparrow\uparrow 3 \uparrow\uparrow 3 = 3 \uparrow\uparrow 7625597484987 =$$ Tritri

See below nice examples of $$2 \uparrow^{n} 3$$ series with colors for $$n = 1,2,3,4,5$$:

Turing machine codeEdit

Input form: to represent $$a \uparrow^{c} b$$, place a 1's, then c+2 ^'s, then b 1's. For example, 111^^^^^111 computes tritri.

Turing machine code

0 * * r 0
0 _ _ l 1
1 1 _ l 2
2 ^ _ l 3
2 1 1 l 4
3 ^ _ l 3
3 1 _ l 2
4 1 1 l 4
4 ^ 1 l 4'
4 _ 1 l halt
4' ^ ^ l 5
4' 1 1 r 0
5 ^ ^ l 5
5 1 1 r 6
6 ^ x r 7
6 1 y r 9
6 | ^ l 12
7 * * r 7
7 _ | r 8
7 | | r 8
8 * * r 8
8 _ ^ l 11
9 * * r 9
9 | | r 10
10 * * r 10
10 _ 1 l 11
11 * * l 11
11 x ^ r 6
11 y 1 r 6
12 * * l 12
12 ^ ^ l 12'
12' ^ ^ l 12'
12' * * l 13
12' 1 x r 14
13 * * l 13
13 ^ ^ r 20
13 _ _ r 20
13 1 x r 14
14 * * r 14
14 ^ ^ r 15
15 ^ ^ r 15
15 x x r 16
15 1 x l 12
16 x x r 16
16 1 x l 12
16 ^ x r 17
17 ^ ^ r 17
17 1 ^ r 18
18 ^ 1 r 17
18 1 1 r 18
18 _ 1 l 19
19 * * l 19
19 x x l 12
20 x 1 r 20
20 ^ ^ r 21
21 ^ ^ r 21
21 x x r 22
22 x x r 22
22 1 _ r 23
22 ^ ^ l 30
23 1 _ r 23
23 ^ ^ l 24
24 _ ^ r 25
25 ^ ^ r 25
25 1 1 l 26
26 ^ 1 r 27
27 1 1 r 27
27 _ _ l 28
28 1 _ l 29
29 * * l 29
29 _ ^ r 25
29 x 1 l 30
30 x 1 l 30
30 ^ ^ r 31
31 * * r 31
31 _ _ l 32
32 1 _ l 1