Array notation

Array notation is a notation for evaluating 1-dimensional arrays created by Jonathan Bowers. Later, he decided to create a notation that evaluated multidimensional arrays (Extended Array Notation) and finally rolled both into one, creating BEAF.

Rules

 * 1) \(\{a\} = a\) and \(\{a,b\} = a^b\). The latter was originally \(a+b\), but this was changed to avoid confusion with the second rule; i.e. \(\{a,1\} = a+1 \not= a = \{a,1\}\).
 * 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, \(3 \lbrace3 \rbrace3 = \lbrace 3,3,3 \rbrace\) = 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.)

Turing machine code
0 * * r 0 0 _ _ r 1 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 | 1 l 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 r9 1 _ l r8 r9 _ _ l r9 r9 y _ l r14 r9 x _ 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 1 _ 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 r0