Linear Array Notation

Linear Array Notation (LAN) is the first part of almighty array notation. A valid expression in LAN is of the form a*(#), where # is a non-empty sequence of positive integers separated by comma, and the linear array notation uses pretty much the same rules as BEAF.

Definition

The Linear Array Notation has the following form:

a*(a, b, c, d, e, f, ..., m, n) where a, b, c, d, e, f, ..., m, n ≥ 1. They can have any number of entries.

The first entry of the array is the base, the number after it is the iterator.

Rules

1. Empty array rule: a*() = 1, since a*(1) = a*() using the tailing rule.

2. Base rule: a*(a) = a and a*(a, b) = a^b

3. Tailing rule: a*(# 1) = a*(#) where # indicates the rest of array

4. Prime rule: a*(a, 1 #) = a

5. Recursion rule: a*(a, b, 1, 1, ..., 1, 1, c #) = a*(a, a, a, a, ..., a, a*(a, b - 1, 1, 1, ..., 1, 1, c #), c - 1 #) (w/ b number of a's) for a, b, c > 1 and the length of the array ≥ 4

6. If rules 1 - 5 do not apply: a*(a, b, c #) = a*(a, a*(a, b - 1, c #), c - 1 #)

7. If more than one rule(s) apply to an array, start from step 1, and find the nearest rule that applies to that array.
   So a*(2, 1, 1, 1, 1, 1, 2, 2) = a*(2) = 2 (by rule 4) instead of a*(2, 2, 2, 2, ..., 2, a*(2, 0, 1, 1, ..., 1, 1, 2, 2), 1, 2) (by rule 5), which is ill-formed due to the occurrence of 0.

Explaination

3-entry arrays

• a*(a, b, 2) = a^^b. FGH level 3.

• a*(a, b, 3) = a^^^b. FGH level 4.

• a*(a, b, 4) = a^^^^b. FGH level 5.

• a*(a, b, c)  = a {c} b. FGH level c - 1.

• a*(a, a, a)  = a {a} a. FGH level ω, which is the limit of the arrow notation.

4-entry arrays

• a*(a, b, 1, 2) = {a, b, 1, 2}. FGH level ω + 1.

• a*(a, b, 2, 2) = {a, b, 2, 2}. FGH level ω + 2.

• a*(a, b, 3, 2) = {a, b, 3, 2}. FGH level ω + 3.

• a*(a, b, 1, 3) = {a, b, 1, 3}. FGH level ω2.

• a*(a, b, 2, 3) = {a, b, 2, 3}. FGH level ω2 + 1.

• a*(a, b, 3, 3) = {a, b, 3, 3}. FGH level ω2 + 2.

• a*(a, b, 1, 4) = {a, b, 1, 4}. FGH level ω3.

• a*(a, b, 1, 5) = {a, b, 1, 5}. FGH level ω4.

• a*(a, b, 1, 6) = {a, b, 1, 6}. FGH level ω5.

• a*(a, b, c, d) = {a, b, c, d}. FGH level ω(d - 1) + (c - 1).

5-entry arrays

• a*(a, b, 1, 1, 2) = {a, b, 1, 1, 2}. FGH level ω^2.

• a*(a, b, 2, 1, 2) = {a, b, 2, 1, 2}. FGH level ω^2 + 1.

• a*(a, b, 3, 1, 2) = {a, b, 3, 1, 2}. FGH level ω^2 + 2.

• a*(a, b, 1, 2, 2) = {a, b, 2, 1, 2}. FGH level ω^2 + ω.

• a*(a, b, 2, 2, 2) = {a, b, 2, 2, 2}. FGH level ω^2 + ω + 1.

• a*(a, b, 1, 3, 2) = {a, b, 1, 3, 2}. FGH level ω^2 + ω2.

• a*(a, b, 1, 4, 2) = {a, b, 1, 4, 2}. FGH level ω^2 + ω3.

• a*(a, b, 1, 1, 3) = {a, b, 1, 1, 3}. FGH level (ω^2)2.

• a*(a, b, 2, 1, 3) = {a, b, 2, 1, 3}. FGH level (ω^2)2 + 1.

• a*(a, b, 1, 2, 3) = {a, b, 1, 2, 3}. FGH level (ω^2)2 + ω.

• a*(a, b, 1, 3, 3) = {a, b, 1, 3, 3}. FGH level (ω^2)2 + ω2.

• a*(a, b, 1, 1, 4) = {a, b, 1, 1, 4}. FGH level (ω^2)3.

• a*(a, b, 1, 1, 5) = {a, b, 1, 1, 5}. FGH level (ω^2)4.

6-entry arrays and beyond

• a*(a, b, 1, 1, 1, 2) = {a, b, 1, 1, 1, 2}. FGH level ω^3.

• a*(a, b, 2, 1, 1, 2) = {a, b, 2, 1, 1, 2}. FGH level ω^3 + 1.

• a*(a, b, 1, 2, 1, 2) = {a, b, 1, 2, 1, 2}. FGH level ω^3 + ω.

• a*(a, b, 1, 3, 1, 2) = {a, b, 1, 3, 1, 2}. FGH level ω^3 + ω2.

• a*(a, b, 1, 1, 2, 2) = {a, b, 1, 1, 2, 2}. FGH level ω^3 + ω^2.

• a*(a, b, 1, 2, 2, 2) = {a, b, 1, 2, 2, 2}. FGH level ω^3 + ω^2 + ω.

• a*(a, b, 1, 1, 3, 2) = {a, b, 1, 1, 3, 2}. FGH level ω^3 + (ω^2)2.

• a*(a, b, 1, 1, 4, 2) = {a, b, 1, 1, 4, 2}. FGH level ω^3 + (ω^2)3.

• a*(a, b, 1, 1, 1, 3) = {a, b, 1, 1, 1, 3}. FGH level (ω^3)2.

• a*(a, b, 1, 2, 1, 3) = {a, b, 1, 2, 1, 3}. FGH level (ω^3)2 + ω.

• a*(a, b, 1, 1, 2, 3) = {a, b, 1, 1, 2, 3}. FGH level (ω^3)2 + ω^2.

• a*(a, b, 1, 1, 3, 3) = {a, b, 1, 1, 3, 3}. FGH level (ω^3)2 + (ω^2)2.

• a*(a, b, 1, 1, 1, 4) = {a, b, 1, 1, 1, 4}. FGH level (ω^3)4.

• a*(a, b, 1, 1, 1, 5) = {a, b, 1, 1, 1, 5}. FGH level (ω^3)5.

• a*(a, b, 1, 1, 1, 1, 2) = {a, b, 1, 1, 1, 1, 2}. FGH level ω^4.

• a*(a, b, 1, 1, 1, 2, 2) = {a, b, 1, 1, 1, 2, 2}. FGH level ω^4 + ω^3.

• a*(a, b, 1, 1, 1, 1, 3) = {a, b, 1, 1, 1, 1, 3}. FGH level (ω^4)2.

• a*(a, b, 1, 1, 1, 1, 4) = {a, b, 1, 1, 1, 1, 4}. FGH level (ω^4)3.

• a*(a, b, 1, 1, 1, 1, 1, 2) = {a, b, 1, 1, 1, 1, 1, 2}. FGH level ω^5.

• a*(a, b, 1, 1, 1, 1, 1, 1, 2) = {a, b, 1, 1, 1, 1, 1, 1, 2}. FGH level ω^5.

In general, if the array has n 1's, the growth rate is ω^n in the FGH, and the limit of the notation level is ω^ω.