Array notation

Array notation is googological function created by Jonathan Bowers. It was generalized to Extended Array Notation and eventually BEAF and BAN.

History
Bowers say he was inspired to explore googology by reading a book that discussed hyper operators. From them he developed the four-argument extended operators, which are as strong as chained arrow notation. Later, Bowers decided to extend his operators to far more arguments than four, leading to the invention of array notation.

Initially Rule 1 had \(\{a, b\} = a + b\). At Chris Bird's suggestion, he changed this to \(a^b\) so that Rules 1 and 2 would be compatible.

Bowers usually used angle brackets  for delimiters of arrays, but the notation originally appeared on the site used curly braces   instead, due to angle brackets causing problems with. Aware of this fact, Sbiis Saibian uses curly braces for the original \(a + b\) array notation, and angle brackets for the revised \(a^b\) array notation.

As a convention, the rest of this article will use curly braces for the new array notation.

Nathan Ho and Wojowu showed that array notation terminates — that is, its output exists for all input sequences.

Rules
An array is defined as a finite sequence of positive integers. Array notation is a function \(v(A)\) mapping these sequences to large positive integers. If \(A = (a_1, a_2, \ldots, a_n)\), we write \(\{a_1, a_2, \ldots, a_n\}\) as a shorthand for \(v(A)\).


 * 1) \(\{a\} = a\) and \(\{a,b\} = a^b\)
 * 2) \(\{a,b,c,\ldots,n,1\} = \{a,b,c,\ldots,n\}\)
 * 3) \(\{a,1,b,c,\ldots,n\} = a\)
 * 4) \(\{a,b,1,\ldots,1,c,d,\ldots,n\} = \{a,a,a,\ldots,\{a,b-1,1,\ldots,1,c,d,\ldots,n\},c-1,d,\ldots,n\}\). In other words, if the third element is 1, then:
 * 5) *all elements before the next non-1 element become the first element,
 * 6) *the last of the above becomes the original array with the second element decreased by 1, and
 * 7) *the said non-1 element is decreased by one.
 * 8) If rules 1 to 4 do not apply, \(\{a,b,c,d,\ldots,n\} = \{a,\{a,b-1,c,d,\ldots,n\},c-1,d,\ldots,n\}\)

Extended Operators
Bowers also invented a set of "extended" operators to complement Array Notation.

\[a \{c\} b = \{a,b,c\}\]

For example, \(3\{3\}3 = \{3,3,3\}\) = tritri.

In the array  { a,b,c,d,e,f,g,h}, a, b, and c are all shown in numeric form, with d sets of curly braces { }. To represent e, Bowers uses e - 1 sets of square brackets [ ] above and below the sentence, f is represented with f - 1 Saturn-like rings around the sentence, and g is represented by g - 1 sets of X-wing brackets around it. h is represented with square plates with opposite edges bent 90 degrees inwards (i.e. 3-dimensional versions of square brackets).

Below 3 entries
When there are less then 3 entries, the arrays are fairly trivial to solve.


 * The simplest array of all is the empty one: \(\{\} = 1\). This is technically not a valid operation (it is in BEAF), but it is consistent with the rest of the rules.
 * Arrays of size 1: \(\{a\} = a\) (not to be confused with the bracket operator).
 * Arrays of size 2: \(\{a,b\} = a^b\), which is a to the power of b (addition on the old version).

3 to 4 entries
Arrays of size 3 gives the same value with chained arrow notation: \(\{a,b,c\} = a \rightarrow b \rightarrow c\), just like arrays of size 1 and 2.

The 4 entry arrays, \(\{a,b,c,d\}\), does not continue with this pattern; in fact, according to Bird's Proof, \(\{a,b,c,d\}\) reaches a length d+2 chain.

Approximations in other notations
\(\lbrace n,a+1,b+1,c+1,d+1,... \rbrace\) is approximated as follows.

Turing machine code
Input format:  where A is array in form of strings of 1's divided by blanks. For example,  calculates tritri.

Turing machine code

0 * * r 0 0 _ _ r 1 0 | | r 0' 0' * * r 0' 0' _ _ r 1 0' | _ l h0 1 _ _ r 1 1 1 1 r 2 2 _ _ r 3 2 | _ l 4 2 1 1 r 5 3 * _ r 3 3 | _ l 4 4 * * l 4 4 1 _ l r0 5 1 1 r 5 5 | _ l 6 5 _ _ r 9 6 1 x l 7 6 _ 1 l s1 7 * * l 7 7 | | r 8 7 x x r 8 8 1 x r s0 8 _ 1 r s3 9 1 1 r 10 9 _ _ r 9 9 y y l b0 10 1 1 r 11 10 _ _ r 9 10 | _ l 12 11 1 1 r 11 11 _ _ r 9 11 | | l 15 12 1 _ l 13 13 _ | l 14 14 * * l 14 14 | | r 0 15 * * l 15 15 | | r c0 15 x x r c0 15 y y r c0

c0 1 x r c1 c0 _ y r c3 c0 | | r c6 c1 * * r c1 c1 | | r c2 c2 * * r c2 c2 _ x l c5 c3 * * r c3 c3 | | r c4 c4 * * r c4 c4 _ y l c5 c5 * * l c5 c5 | | l 15 c6 x 1 r c6 c6 y _ r c6 c6 _ | l 16

16 * * l 16 16 | | r 17 17 1 1 r 17 17 _ _ r 18 18 1 1 r 18 18 _ 1 l 19 18 | _ l 20 19 1 _ r 18 20 1 | l 21 21 * * l 21 21 | | l 22 22 x 1 l 22 22 y _ l 22 22 | | r 23

23 1 1 r 23 23 _ _ r 24 24 1 1 r 24 24 _ _ r 25 25 1 y r 26 26 1 1 l 27 26 _ _ r 25 27 y _ l 27 27 _ _ l 28 28 1 1 l 28 28 y y l 27 28 _ _ r 29 29 1 x r 30 30 * * r 30 30 | | r 0

r0 _ _ l r0 r0 1 _ l r1 r0 | _ l r7 r1 * * l r1 r1 x 1 r r2 r2 1 x r r3 r2 _ x r r5 r2 | | r h0 r3 * * r r3 r3 | | r r4 r4 1 1 r r4 r4 _ _ l r0 r5 _ _ r r6 r5 1 _ r r10 r5 y _ r r13 r6 * * r r6 r6 | | r r4 r7 1 | l r8 r8 1 1 l r8 r8 _ 1 l r9 r8 x x r 32 r9 1 _ l r8 r9 _ _ l r9 r9 y _ l r14 r9 x _ r r9' r9' _ _ l 31 r9' * * l 14 r10 _ 1 r r11 r10 1 1 r r10 r10 | 1 r r12 r11 _ _ r r11 r11 1 _ r r10 r12 _ _ l r7 r12 1 | l r1 r13 _ y r r5 r14 _ y l r9

b0 * * l b0 b0 | | r b0' b0' 1 x r b0 b0 x 1 r b1 b1 _ _ r b7 b1 1 x r b2 b2 * * r b2 b2 y 1 r b3 b3 _ y r b4 b4 y _ r b3 b4 1 _ r b5 b5 1 1 r b5 b5 _ 1 r b4 b5 | 1 r b6 b6 _ | l b0 b7 * * r b7 b7 y 1 r 9

h0 * * l h0 h0 | 1 l h1 h1 * _ r halt

s0 * * r s0 s0 x x l 6 s1 * * l s1 s1 x x r s2 s2 1 _ r s4 s3 1 1 r s3 s3 x _ r s4 s4 * 1 r s4 s4 _ _ l s5 s5 * * l s5 s5 x 1 l s5 s5 | | r p0

p0 1 1 l p0a p0 _ _ r p19 p0a | | r p0b p0b 1 _ r p1 p1 1 _ r p21 p1 _ _ r p10 p1 x _ r p11 p2 * * r p2 p2 _ _ r p3 p3 1 _ r p4 p3 x _ l p8 p4 1 1 r p4 p4 _ x r p5 p4 x x r p5 p5 1 1 r p5 p5 _ 1 l p6 p6 1 1 l p6 p6 * * l p7 p7 1 1 l p7 p7 _ 1 r p3 p8 1 1 l p8 p8 x x l p8 p8 _ x l p9 p9 1 1 l p8 p9 _ _ r p10 p10 x _ r p1 p10 1 1 l p20 p11 1 1 r p11 p11 x _ r p11 p11 _ _ l p12 p12 1 1 l p12 p12 _ _ r m0 m0 1 _ r m1 m0 _ _ r m9 m1 1 1 r m1 m1 _ _ r m2 m2 1 _ r m3 m2 _ _ l m7 m3 1 1 r m3 m3 _ _ r m4 m4 1 1 r m4 m4 _ 1 l m5 m5 1 1 l m5 m5 _ _ l m6 m6 1 1 l m6 m6 _ 1 r m2 m7 1 1 l m7 m7 _ _ l m8 m8 1 1 l m8 m8 _ _ r m0 m9 1 _ r m9 m9 _ 1 r p13 p13 1 1 r p13 p13 _ _ l p14 p14 1 _ l p15 p15 1 1 l p15 p15 _ 1 l p16 p16 1 1 r p17 p16 _ _ r p13 p16 x _ r p18 p16 | | r p22 p17 1 _ l p12 p18 1 _ r halt p19 1 _ r p19 p19 _ 1 r halt p20 * * l p20 p20 x _ r halt p21 * * r p2 p21 _ x r p3 p22 1 1 r p22 p22 _ _ l p23 p23 1 _ l r0

31 _ x r 32 32 * * r 32 32 | _ l r7