Nested Array Notation (NAN) is the forth part of almighty array notation. A valid expression in NAN is of the form a*(a, b X c X d X e X ... X n) where a, b, c, d, e, ..., n ≥ 1, and X are the separators.
The Nested Array Notation has the following form:
a*(a, b X c X d X e X ... X n) where a, b, c, d, e, ..., n ≥ 1, and X are the separators.
In NAN, The separator can come in many forms, such as {3}, {1, 4, 5}, {1, 1, 3 {2} 2}.
The first entry of the array is the base, the number after it is the iterator.
The {1} separator stands for comma.
Empty array rule: a*() = 1, since a*(1) = a*() using the tailing rule.
Base rule: a*(a) = a and a*(a, b) = a^b
Tailing rule 1: a*(# 1 X #) = a*(#) where # indicates the rest of array (X can be any separators)
Tailing rule 2: a*(# X 1) = a*(#)
Tailing rule 3: a*(# {% 1} #) = a*(# {%} #) where % indicates the rest of array in the separator
Prime rule: a*(a, 1 #) = a
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
If rules 1 - 7 do not apply: a*(a, b, c #) = a*(a, a*(a, b - 1, c #), c - 1 #)
Comma rule: a*(a, b {1} c #) = a*(a, b, c #)
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 6) instead of a*(2, 2, 2, 2, ..., 2, a*(2, 0, 1, 1, ..., 1, 1, 2, 2), 1, 2) (by rule 7), which is ill-formed due to the occurrence of 0.
Followed by some additional rules by following:
Reuse the rules for linear arrays to that row (after separators other than commas)
a*(a, b {2} c) a*(a, a, 1, 1, 1, ..., 1, 1, 2 {2} c - 1) with b string of 1's using comma as a separator.
a*(a, b {2} 1 {2} ... {2} 1 {2} c) = a*(a, a{2} 1 {2} ... {2} 1, 1, 1, ..., 1, 1, 2 {2} c - 1) with b string of 1's using comma as a separator.
a*(a, b {d} 1 {d} ... {d} 1 {d} c) = a*(a, a, {d - 1} 1 {d - 1} ... {d - 1} 1, 1, 1, ..., 1, 1, 2 {d - 1} c - 1) with b string of 1's using {d - 1} as a separator.
a*(a, b {1, 1, 1, ..., 1, 1, c, d%} 2) = a*(a, a {1, 1, 1, ..., 1, b, c - 1, d%} 2), which is different from 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 #).
a*(a, b {1 {2} c} 2) = a*(a, a {1, 1, 1, ..., 1, 1, 2 {2} c - 1} 2), and so on. Nested arrays reuse the rules 2 through 5.
a*(a, b {1 {2} 2} 2) = a*(a, a {1, 1, 1, ..., 1, 1, 2} 2). FGH level ω^ω^ω^ω.
a*(a, b {2 {2} 2} 2) = a*(a, a {1 {2} 2} 1 {1 {2} 2} ... {1 {2} 2} 1 {1 {2} 2} 2) with b string of 1's separated by {1 {2} 2}. FGH level ω^ω^(ω^ω + 1).
a*(a, b {1 {2} 3} 2) = a*(a, a {1, 1, 1, ..., 1, 1, 2 {2} 2} 2). FGH level ω^ω^((ω^ω)2).
a*(a, b {1 {2} 1, 2} 2) = a*(a, a {1 {2} b} 2). FGH ω^ω^ω^(ω + 2).
a*(a, b {1 {2} 1 {2} 2} 2) = a*(a, a {1 {2} 1, 1, 1, ..., 1, 1, 2} 2} 2). FGH level ω^ω^ω^ω2.
a*(a, b {1 {3} 2} = a*(a, a {1 {2} 1 {2} ... {2} 1 {2} 2} 2) with b string of 1's separated by {2}. FGH level ω^ω^ω^ω^2.
a*(a, b {1 {1, 2} 2} 2) = a*(a, a {1 {b} 2} 2). FGH level ω^ω^ω^ω^ω.
a*(a, b {1 {1 {2} 2} 2} 2) = a*(a, a {1 {1, 1, 1, ..., 1, 1, 2} 2} 2) with b string of 1's. FGH level ω^ω^ω^ω^ω^ω.
etc.
And finally, the limit of the nested arrays is a*(a, a {1 { ... 1 {1, 2} 2 ... } 2} 2), which is an ε0-level.