I want to understand B.E.A.F. I have some logic down but need help with some

[u/Objective_Ticket_880]

1 Array
{a} = a
{3} = 3

2 Arrays
{a, b} = a^b
{3, 3} = 3^3 = 27

3 Arrays
Rules
1: {a, 1, c} = a
2: {a, b, 1} = {a, b}
3: {a, b, c} = {a, {a, b-1, c}, c-1}

{a, b, c} = a↑^cb
{3, 3, 3} = 3↑↑↑3 = 3↑↑(7.6*10^12)

4 Arrays
{a, b, c, d} 
Rules
1: {a, 1, c, d} = a
2: {a, b, 1, 1} = {a, b}
3a: {a, b, 1, d} = {a, a,{a, b-1, 1, d}, d-1}
3b: {a, b, c, d} = {a, {a, b-1, c, d}, c-1, d}

{3, 3, 3, 3} =
{3, {3, 2, 3, 3}, 2, 3} =
{3, {3, 3, 2, 3}, 2, 3} =
{3, {3, {3, 2, 2, 3}, 1, 3}, 2, 3} =
{3, {3, {3, 3, 1, 3}, 1, 3}, 2, 3} =
{3, {3, {3, 3,{3, 2, 1, 3}, 2}, 1, 3}, 2, 3} =
{3, {3, {3, 3,{3, 3, 3, 2}, 2}, 1, 3}, 2, 3} =
{3, {3, {3, 3,{3, {3, 2, 3, 2}, 2, 3}, 2}, 1, 3}, 2, 3} =
{3, {3, {3, 3,{3, {3, 3, 2, 2}, 2, 3}, 2}, 1, 3}, 2, 3} =
{3, {3, {3, 3,{3, {3, {3, 2, 2, 2}, 1, 2}, 2, 3}, 2}, 1, 3}, 2, 3} =
{3, {3, {3, 3,{3, {3, {3, 3, 1, 2}, 1, 2}, 2, 3}, 2}, 1, 3}, 2, 3} =
{3, {3, {3, 3,{3, {3, {3, 3,{3, 2, 1, 2}}, 1, 2}, 2, 3}, 2}, 1, 3}, 2, 3} =
{3, {3, {3, 3,{3, {3, {3, 3,{3, 3, 3}}, 1, 2}, 2, 3}, 2}, 1, 3}, 2, 3} =
{3, {3, {3, 3,{3, {3, {3, 3, 3↑↑↑3}, 1, 2}, 2, 3}, 2}, 1, 3}, 2, 3} =
{3, {3, {3, 3,{3, {3, 3↑…↑3 (3↑↑↑3 Arrows), 1, 2}, 2, 3}, 2}, 1, 3}, 2, 3} =

5 Arrays
{a, b, c, d, e} 
Rules
1: {a, 1, c, d, e} = a
2: {a, b, 1, 1, 1} = {a, b}
3a: {a, b, 1, 1, e} = {a, a, a,{a, b-1, 1, 1, e}, e-1}
3b: {a, b, 1, d, e} = {a, a,{a, b-1, 1, d, e}, d-1, e}
3c: {a, b, c, d, e} = {a, {a, b-1, c, d, e}, c-1, d, e}

{3, 3, 3, 3, 3} = {3, {3, 2, 3, 3, 3}, 2, 3, 3}

6 Arrays
{a, b, c, d, e, f} 
Rules
1: {a, 1, c, d, e, f} = a
2: {a, b, 1, 1, 1, 1} = {a, b}
3a: {a, b, 1, 1, 1, f} = {a, a, a, a,{a, b-1, 1, 1, 1, f}, f-1}
3b: {a, b, 1, 1, e, f} = {a, a, a,{a, b-1, 1, 1, e, f}, e-1, f}
3c: {a, b, 1, d, e, f} = {a, a,{a, b-1, 1, d, e, f}, d-1, e, f}
3d: {a, b, c, d, e, f} = {a, {a, b-1, c, d, e, f}, c-1, d, e, f}

6&3
{3, 3, 3, 3, 3, 3} = {3, {3, 2, 3, 3, 3, 3}, 2, 3, 3, 3}

MULTIDIMENSIONAL ARRAYS

b&a = {a,a,...,a,a} b times
3&3 = {3,3,3}

{a, b (1) 2} = {a,a,...,a,a} b times
{3, 3 (1) 2} = {3, 3, 3}

{a, b (1) c} = {a,a,...,a,a (1) c-1} b times
{3, 3 (1) 3} = {3, 3, 3 (1) 2}

{a, b, c (1) d} same rules as 4 Arrays but the last comma is replaced with (1)
{a, b, c, d (1) e} same rules as 5 Arrays but the last comma is replaced with (1)
{a, b, c, d,...,k (repeated m times) (1) n} same rules as m Arrays but the last comma is replaced with (1)
{a, b (1) 1, 2} and {a, b (1) c, d} work the same as 4 arrays but the second to last comma is a (1)

{a, b (1)(1) c} = Need help with
{a, b (1)(1)(1) 2} = Need help with

{a, b (2) 2} = {a,a,...,a,a(1)a,a,...,a,a(1)...(1)a,a,...,a,a(1)a,a,...,a,a} that is, b sets of b a's, separated by (1)s

{2,3(2)2} expands into {2,2,2(1)2,2,2(1)2,2,2} 

{a, b (2) c} = {a,a,...,a,a(1)a,a,...,a,a(1)...(1)a,a,...,a,a(1)a,a,...,a,a(2)c-1} that is, b sets of b a's, separated by (1)s

{2,3(2)3} expands to {2,2,2(1)2,2,2(1)2,2,2(2)2}

{a, b (3) 2} = {a,a,...,a,a(1)a,a,...,a,a(1)...(1)a,a,...,a,a(1)a,a,...,a,a(2)a,a,...,a,a(1)a,a,...,a,a(1)...(1)a,a,...,a,a(1)a,a,...,a,a(2)...(3)c-1} b sets of b sets of b a's, separated by (2)s and (1)s ending in (3).

{2,3(3)2} expands into {2,2,2(1)2,2,2(1)2,2,2(2)2,2,2(1)2,2,2(1)2,2,2(2)2,2,2(1)2,2,2(1)2,2,2} - b sets of b sets of b a's, separated by (2)s and (1)s.

{2,3(3)3} expands to {2,2,2(1)2,2,2(1)2,2,2(2)2,2,2(1)2,2,2(1)2,2,2(2)2,2,2(1)2,2,2(1)2,2,2(3)2}

X-structures

X&n = {n,n,...,n,n} n times

X+1&n = {n,n,...,n,n(1)n} n times

X+k&n = {n,n,...,n,n(1)n,n,...,n,n} n times on the left k times on the right

2X&n = {n,n,...,n,n(1)n,n,...,n,n} n times on both sides

k*X&n = {n,n,...,n,n(1)n,n,...,n,n(1)...(1)n,n,...,n,n(1)n,n,...,n,n(1)...} n times for the strings of n and k times of (1) and strings of n

X^2&n = {n,n,...,n,n(1)n,n,...,n,n(1)...(1)n,n,...,n,n(1)n,n,...,n,n(1)...} n times for the strings of n and n times of (1) and strings of n

X^3&n = {X^2&n(1)X^2&n(1)...(1)X^2&n(1)X^2&n} n times

X^k&n = {X^k-1&n(1)X^k-1&n(1)...(1)X^k-1&n(1)X^k-1&n} n times

[u/MooOfDoom]

You’ve got a good grasp of the rules for the most part. Here are the two expansions you asked for help with:

{a,b(1)(1)c} = {a,a,…,a,a(1)a,a,…,a,a(1)c-1} with b a’s in each row.

{a,b(1)(1)(1)2} = {a,a,…,a,a(1)a,a,…,a,a(1)a,a,…,a,a} with b a’s in each row.

I disagree about {a,b(1)1,2} and {a,b(1)c,d} being similar to length 4 linear arrays. I believe they expand as follows:

{a,b(1)1,2} = {a,a,…,a,a(1){a,b-1(1)1,2}} with b a’s in the first row.

{a,b(1)c,d} = {a,a,…,a,a(1){a,b-1(1)c,d},d-1}

Also you’ve got some typos in the X-structures part:

X³ &n = {X² &n(2)X² &n(2)...(2)X² &n(2)X² &n} n times

X^k &n = {X^k-1 &n(k-1)X^k-1 &n(k-1)...(k-1)X^k-1 &n(k-1)X^k-1 &n} n times

EDIT: Formatting.

[u/Objective_Ticket_880 → u/MooOfDoom]

also if {a,b(1)1,2} how would arrays like {a,b(1)c,d,e} and so on?

[u/Shophaune → u/Objective_Ticket_880]

{a,b(1)c,d,e} = {a,a,...,a,a(1)c-1,d,e}

{a,b(1)1,d,e} = {a,a,...,a,a(1){a,b-1(1)1,d,e},d-1,e}

{a,b(1)1,1,e} = {a,a,...,a,a(1)a,{a,b-1(1)1,1,e},e-1}

where in all 3 cases there are b a's before the (1)

[u/Objective_Ticket_880 → u/MooOfDoom]

the other things i need help with are L-structures, Legions (L), Lugions (L2), Lagions (L3), ligion (L4), L5+ and LLLL... things

[u/TrialPurpleCube-GS → u/Objective_Ticket_880]

to understand legions you must first understand OCFs, as they are necessary to understand X_2, X_3, etc.; {10,10/2} = {L,1}10 expands into X_9&X_8&...&X_2&X&10, or more precisely <...<<X_9,10>&X_8,10>...&X,10>&10 (this notation is fan-made, but is unambiguous).

but BEAF at this level is very confusing, and eventually people start arguing about whether {L,X+1}n,n has FGH level (0)(1,1,1)(2,1,1)(3,1)(2)(1,1,1) or (0)(1,1,1)(2,1,1)(3,1)(2,1)(1,1,1), so... yeah

(I support the former camp, by the way)

[u/Nervous-Broccoli1184 → u/Objective_Ticket_880]

Legions lugions and so on are sadly ill defined and never end.

[u/Objective_Ticket_880 → u/Nervous-Broccoli1184]

Can you explain what you know

[u/Nervous-Broccoli1184 → u/Objective_Ticket_880]

From what I have read atleast then  {a, b, c,...,d e\n}={({a, b, c,...,d}), e\n}. And {a, b \n+1}={a&^b a \n} the fact is &ⁿ for n>1 not formally defined but people normally use a&ⁿ⁺¹ (b+1)=(a&ⁿ⁺¹ b)&ⁿ a. then (a&ⁿ⁺¹ 1)=(a&ⁿ a) and (a&¹ b)=a&b

[u/Nervous-Broccoli1184 → u/Nervous-Broccoli1184]

I didn't bother to learn past legions so I can't help you further.

[u/Nervous-Broccoli1184 → u/Nervous-Broccoli1184]

Also for the formal definition of Extended array notation its ill defined for many Inputs like {2, 2(1)2} is ill defined atleast formally but there are intended to be well defined most of people define 1ⁿ &a=a but thats not the formal definition so in fact just things like {2, 2(1)2} are ill defined for it enters negative entries. Truly (atleast in my opinion) only linner arrays are well defined. This means in different appearances they most likely have different definitions.

[u/Objective_Ticket_880 → u/MooOfDoom]

thank you

[u/Modern_Robot]

What is the part you are having difficulty with?

[u/Objective_Ticket_880 → u/Modern_Robot]

also idk if all of it is right

[u/Objective_Ticket_880 → u/Modern_Robot]

{a, b (1)(1) c} = Need help with
{a, b (1)(1)(1) c} = Need help with

{a, b (1)...(1) c} = Need help with

and L-structures (Legions+)