User blog:Hyp cos/Bird's array notation - Extension?

I'm afraid that Hierarchical nested-subscript array notation isn't as strong as Wythagoras thinks. If I use rules in Bird's Hierarchical hyper-nested array notation, and add a sub-rule, this notation will be not so strong.

This added rule is:

(Rules A5a-b do not apply, separator [Ai(pi)] = [d #1]M, where d ≥ 2 and #1 contains at least one H-hyperseparator in its base layer, where H begins with 1 and H ≠ ‘1’; in other words,

H = ‘1 [H1] 1 [H2] ... 1 [Hk] h #2’, where h ≥ 2, k ≥ 1 and each of [Hi] is a normal separator):

Si = ‘b ‹Ai(1)’› b [Ai(1)] b ‹Ai(2)’› b [Ai(2)] ... b ‹Ai(pi-1)’› b [Ai(pi-1)] Rb [d #1]M ci-1 #i’,

Rn = ‘b ‹Rn-1› b’,

R1 = ‘b [d-1 #1]1 [H 1] 1 [H2] ... 1 [Hk-1] 1 ‹Hk’› b (←m+b-1) b ’, where m (which may be 1) is the kth and final entry in the subscript array M when written as M = ‘m1 [H1] m2 [H2] ... mk-1 [Hk-1] m’.

Different H-hyperseparators can be placed at the same nested level in HNSAN, so when apply this rule, use the first H-hyperseparator.

In Bird's "separator level" system, a level-\(\alpha\) separator [A] can handle \(\omega^\alpha\) entries,and {n,n[A]2} has growth rate \(\omega^{\omega^\alpha}\) in FGH.

Now I reuse "●" for shorthand of [2/1,22]. and the [1●2] marks the limit of HHNAN. And this table compares their growth rates.