Multiple expanding array notation

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Multiple expanding array notation (mEAN) is the fourth part of my array notation. In this page, first I introduce higher expanding arrays to you, then I change the rules a little, to form the complete definition.

Higher expanding arrays

In step 3 of case B2, we find an innermost separator without grave accents for the grave accent. Now I introduce a new separator – the double grave accent (` `). We find an innermost separator with a grave accent for it, and it also reduces in the case B2 way. For example, s(a,b{1{1` `3^{`</sup>}3}3) = s(a,b{1{1{1…{1{1,2` `2<sup>`}}2` `2^{`</sup>}…2` `2<sup>`}}2` `2^{`</sup>}2{1` `3<sup>`}}2}2{1{1` `3^{`</sup>}3}2) with b-1 2` `2<sup>`}}′s.

The next step is separators with 2 grave accents at the left-superscript position of the rbrace. And the double grave accent is just a shorthand for {1^{` `}}. Nothing changes, but the only notable thing is that we find an innermost separator with exactly one grave accent for the double grave accent.

What’s the next thing is very clear now. Triple grave accent (` ` `), and separators with 3 grave accents at the left-superscript position of the rbrace. And so on, m-ple grave accent, and separators with m grave accents at the left-superscript position of the rbrace. Generally, an m-ple grave accent reduces in the case B2 way, but we find an innermost separator with m-1 grave accents for it.

So here comes the process.

Process

Case B1, B2 and B4 are terminal but case A and B3 are not. And note that red texts imply changes on the array. First start from the 3rd entry.

• Case A: If the entry is 1, then you jump to the next entry.
• Case B: If the entry n is not 1, look to your left:
  • Case B1: If the comma is immediately before you, then
    1. Change the “1,n” into “b,n-1” where n is this non-1 entry and the b is the iterator.
    2. Change all the entries at base layer before them into the base.
    3. The process ends.
  • Case B2: If the m-ple grave accent M (M = ` `…` or {1<sup>` `…`</sup>} with m grave accents) is immediately before you, then
    1. Let t be such that “M is at layer t”.
    2. Repeat this:
       1. Subtract t by 1.
       2. Let separator B_t be such that it’s at layer t, and M is inside it.
       3. If t = 1, then break the repeating, or else continue repeating.
    3. Find the maximal t such that A_t has exactly m-1 grave accents at the left-superscript position of the rbrace.
    4. Let string P and Q be such that B_t = “P M n Q”
    5. Change B_t into S_b, where b is the iterator, S₁ is comma, and S_i+1 = “P S_i 2 M n-1 Q”
    6. The process ends.
  • Case B3: If a separator K that is neither comma nor grave accent is immediately before you, then
    1. Change the “K n” into “K 2 K n-1”.
    2. Set separator A_t = K, here K is at layer t.
    3. Jump to the first entry of the former K.
  • Case B4: If an lbrace is immediately before you, then
    1. Change separator {n #} into string S_b, where b is the iterator, S₁ = “{n-1 #}” and S_i+1 = “S_i 1 {n-1 #}”.
    2. The process ends.

Explanation

The process has only a small change. Just note that if m = 1, the step 3 of case B2 also works correctly, because a separator with 0 grave accent is just a separator without any grave accents at the left-superscript position of the rbrace.

An m-ple grave accent searches for a separator with m-1 grave accents, and then it expands into b nests of it. That’s why this notation is named multiple expanding array notation.

Comparison

s(n,n{1{1` `2^`}2}2) has growth rate , so it eventually outgrows all functions provably recursive in Kripke-Platek set theory with the axiom of infinity (KP), Aczel’s constructive Zermelo-Fraenkel set theory (CZF), the theory of inductive definitions (ID₁), and Martin-Löf Type Theory with indexed W-Types (MLW). It’s also comparable to the H(n) in BAN. And then,

{1 {1` `2^{`</sup>} 3} has recursion level 
{1 {1` `2<sup>`}} 1`2} has recursion level 
{1 {1` `2<sup>`</sup>} 1 {1` `2^{`</sup>} 2} has recursion level 
{1 {2` `2<sup>`}} 2} has recursion level 
{1 {1`2` `2<sup>`</sup>} 2} has recursion level 
{1 {1`3` `2<sup>`</sup>} 2} has recursion level 
{1 {1`1`2` `2^{`</sup>} 2} has recursion level 
{1 {1 {2<sup>`}} 2` `2^{`</sup>} 2} has recursion level 
{1 {1 {1`2^{`</sup>} 2` `2<sup>`}} 2} has recursion level 
{1 {1 {1`1`2^{`</sup>} 2` `2<sup>`}} 2} has recursion level 
{1 {1 {1 {1`2<sup>`}} 2^{`</sup>} 2` `2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{`</sup>} 2` `2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{`</sup>} 3` `2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{`</sup>} 1`2` `2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>`}} 1 {1` `2^{`</sup>} 2` `2<sup>`}} 2} has recursion level 
{1 {1 {2` `2^{`</sup>} 2` `2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` `2^{`</sup>} 2` `2<sup>`}} 2` `2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1` `2<sup>`}} 2` `2^{`</sup>} 2` `2<sup>`}} 2` `2^{`</sup>} 2} has recursion level 
{1 {1` `3<sup>`}} 2} has recursion level 
{1 {1` `4^{`</sup>} 2} has recursion level 
{1 {1` `1,2<sup>`}} 2} has recursion level 
{1 {1` `1 {1 {1` `2^{`</sup>} 2} 2<sup>`}} 2} has recursion level 
{1 {1` `1`2<sup>`}} 2} has recursion level 
{1 {1` `2`2<sup>`</sup>} 2} has recursion level 
{1 {1` `1`3<sup>`</sup>} 2} has recursion level 
{1 {1` `1`1,2<sup>`</sup>} 2} has recursion level 
{1 {1` `1`1 {1 {1` `1`2^{`</sup>} 2} 2<sup>`}} 2} has recursion level 
{1 {1` `1`1`2^{`</sup>} 2} has recursion level 
{1 {1` `1 {2<sup>`}} 2^{`</sup>} 2} has recursion level 
{1 {1` `1 {1`2^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1` `1 {1`1`2^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1` `1 {1 {1`2<sup>`}} 2^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1` `1 {1` `2^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1` `2 {1` `2^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1` `1 {1` `2^{`</sup>} 3<sup>`}} 2} has recursion level 
{1 {1` `1 {2` `2^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1` `1 {1` `3^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1` `1 {1` `1`2<sup>`</sup>} 2^{`</sup>} 2} has recursion level 
{1 {1` `1 {1` `1 {1` `2<sup>`}} 2^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1` `1` `2^{`</sup>} 2} has recursion level 
{1 {1` `2` `2<sup>`}} 2} has recursion level 
{1 {1` `1`2` `2<sup>`</sup>} 2} has recursion level 
{1 {1` `1 {1` `2^{`</sup>} 2` `2<sup>`}} 2} has recursion level 
{1 {1` `1 {1` `1` `2^{`</sup>} 2` `2<sup>`}} 2} has recursion level 
{1 {1` `1 {1` `1` `2^{`</sup>} 3` `2<sup>`}} 2} has recursion level 
{1 {1` `1 {2` `1` `2^{`</sup>} 2` `2<sup>`}} 2} has recursion level 
{1 {1` `1 {1` `2` `2^{`</sup>} 2` `2<sup>`}} 2} has recursion level 
{1 {1` `1 {1` `1 {1` `1` `2^{`</sup>} 2` `2<sup>`}} 2` `2^{`</sup>} 2} has recursion level 
{1 {1` `1 {1` `1 {1` `2` `2<sup>`}} 2` `2^{`</sup>} 2` `2<sup>`}} 2} has recursion level 
{1 {1` `1` `3^{`</sup>} 2} has recursion level 
{1 {1` `1` `4<sup>`}} 2} has recursion level 
{1 {1` `1` `1`2<sup>`</sup>} 2} has recursion level 
{1 {1` `1` `1 {1` `1` `2^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1` `1` `1` `2^{`</sup>} 2} has recursion level 
{1 {1` `2` `1` `2<sup>`}} 2} has recursion level 
{1 {1` `1` `2` `2^{`</sup>} 2} has recursion level 
{1 {1` `1` `1` `3<sup>`}} 2} has recursion level 
{1 {1` `1` `1` `1` `2^{`</sup>} 2} has recursion level 
{1 {1` `1` `1` `1` `1` `2<sup>`}} 2} has recursion level 
{1 {1 {2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1` `2 {2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1` `1` `2 {2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {2<sup>` `</sup>} 3<sup>`}} 2} has recursion level 
{1 {1 {2^{` `}} 1`2<sup>`</sup>} 2} has recursion level 
{1 {1 {2^{` `}} 1 {1` `2^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {2^{` `}} 1 {1` `1` `2^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {2^{` `}} 1 {1 {2^{` `}} 2^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {2^{` `}} 1 {1 {2^{` `}} 1 {1 {2^{` `}} 2^{`</sup>} 2<sup>`}} 2^{`</sup>} 2} has recursion level 
{1 {1 {2<sup>` `</sup>} 1` `2<sup>`}} 2} has recursion level 
{1 {1 {2^{` `}} 1` `1` `2^{`</sup>} 2} has recursion level 
{1 {1 {2<sup>` `</sup>} 1 {2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {3^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1,2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1 {2^{` `}} 2^{`</sup>} 2} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1`2<sup>` `</sup>} 2<sup>`</sup>} 2} has recursion level 
{1 {1 {1`2<sup>` `</sup>} 3<sup>`</sup>} 2} has recursion level 
{1 {1 {1`2<sup>` `</sup>} 1`2^{`</sup>} 2} has recursion level 
{1 {1 {1`2^{` `}} 1 {1 {1`2<sup>` `</sup>} 2<sup>`}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1`2^{` `}} 1` `2^{`</sup>} 2} has recursion level 
{1 {1 {1`2^{` `}} 1` `1` `2^{`</sup>} 2} has recursion level 
{1 {1 {1`2^{` `}} 1 {2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1`2^{` `}} 1 {1`2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {2`2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1`3<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1`1`2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1`2^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` `2^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` `3^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` `1` `2^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {2^{` `}} 2^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1`2<sup>` `</sup>} 2<sup>`}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1 {1` `2<sup>`}} 2^{` `}} 2^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1 {1 {1 {1` `2^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2^{` `}} 2^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 2<sup>`}} 3} has recursion level 
{1 {1 {1` `2^{` `}} 2^{`</sup>} 1`2} has recursion level 
{1 {1 {1` `2^{` `}} 2^{`</sup>} 1 {1`2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 2<sup>`}} 1 {1 {2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 2<sup>`}} 1 {1 {1`2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{` `}} 2^{`</sup>} 1 {1 {1 {1 {2<sup>` `</sup>} 2<sup>`}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 2<sup>`}} 1 {1 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 2<sup>`}} 1 {1 {1` `2^{` `}} 2^{`</sup>} 1`2} has recursion level 
{1 {1 {1` `2^{` `}} 2^{`</sup>} 1 {1 {1` `2<sup>` `</sup>} 2<sup>`}} 1 {1 {2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 2<sup>`}} 1 {1 {1` `2^{` `}} 2^{`</sup>} 1 {1 {1` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {2 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1` `2 {1` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {2^{` `}} 2 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1`2^{` `}} 2 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` `2<sup>`}} 2^{` `}} 2 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {2<sup>` `</sup>} 2<sup>`}} 2^{` `}} 2 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1 {1 {2<sup>` `</sup>} 2<sup>`}} 2^{` `}} 2^{`</sup>} 2<sup>` `</sup>} 2 {1` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1` `2^{` `}} 2^{`</sup>} 2<sup>` `</sup>} 2 {1` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1 {1 {1` `2^{` `}} 2^{`</sup>} 2<sup>` `</sup>} 2 {1` `2<sup>` `</sup>} 2<sup>`}} 2^{` `}} 2 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 3<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{` `}} 1`2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{` `}} 1 {1 {1` `2^{` `}} 2^{`</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{` `}} 1` `2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 1 {2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{` `}} 1 {1`2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{` `}} 1 {1 {1 {2^{` `}} 2^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{` `}} 1 {1 {1 {1 {1 {2^{` `}} 2^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 1 {1 {1 {1` `2<sup>` `</sup>} 2<sup>`}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 1 {1 {1 {1` `2<sup>` `</sup>} 3<sup>`}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 1 {1 {1 {1` `2<sup>` `</sup>} 1` `2<sup>`}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 1 {1 {1 {1` `2<sup>` `</sup>} 1 {1 {1 {1` `2<sup>` `</sup>} 2<sup>`}} 2^{` `}} 2^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{` `}} 1 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 1 {1` `2<sup>` `</sup>} 2<sup>`}} 1`2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 1 {1` `2<sup>` `</sup>} 2<sup>`}} 1 {1 {1` `2^{` `}} 1 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {2 {1` `2<sup>` `</sup>} 1 {1` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{` `}} 2 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 1 {1 {1` `2<sup>` `</sup>} 1 {1` `2<sup>` `</sup>} 2<sup>`}} 2 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 1 {1` `2<sup>` `</sup>} 3<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{` `}} 1 {1` `2^{` `}} 1` `2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 1 {1` `2<sup>` `</sup>} 1 {1 {1` `2<sup>`}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 1 {1` `2<sup>` `</sup>} 1 {1 {1 {1` `2<sup>` `</sup>} 1 {1` `2<sup>` `</sup>} 2<sup>`}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2<sup>` `</sup>} 1 {1` `2<sup>` `</sup>} 1 {1` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `2^{` `}} 1 {1` `2^{` `}} 1 {1` `2^{` `}} 1 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {2` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1`2` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` `2^{`</sup>} 2` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {2` `2^{` `}} 2^{`</sup>} 2` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1 {1 {2` `2^{` `}} 2^{`</sup>} 2` `2<sup>` `</sup>} 2<sup>`}} 2` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `3<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `4^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `1,2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `1`2<sup>` `</sup>} 2<sup>`</sup>} 2} has recursion level 
{1 {1 {1` `1 {1 {1` `1`2<sup>` `</sup>} 2<sup>`</sup>} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `1 {1 {1` `1 {1 {1` `1`2^{` `}} 2^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `1` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `1` `2^{` `}} 1 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `1` `2<sup>` `</sup>} 1 {1` `1` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {2` `1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` `1` `3^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `1` `1` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {2^{` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1`2^{` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` `2<sup>` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` `2^{` `}} 2^{` `}} 1 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` `2<sup>` `</sup>} 2<sup>` `</sup>} 1 {1 {1` `2<sup>` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {2 {1` `2^{` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` `2 {1` `2<sup>` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` `2^{` `}} 3^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` `2<sup>` `</sup>} 1` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` `2^{` `}} 1 {1` `2^{` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {2` `2<sup>` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` `3^{` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` `1` `2<sup>` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1` `1` `2^{` `}} 2^{` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` ` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` ` `2^{` `}} 1 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` ` `2^{` `}} 1 {1` ` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {2` ` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `2<sup>` `</sup>} 2` ` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `2^{` `}} 1 {1` ` `2<sup>` `</sup>} 2` ` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {2` ` `2^{` `}} 2` ` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1` ` `2<sup>` `</sup>} 2` ` `2^{` `}} 2` ` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` ` `3<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` ` `1` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` ` `1` `3<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` ` `1` `1` `2<sup>` `</sup>} 2<sup>`</sup>} 2} has recursion level 
{1 {1 {1` ` `1 {1` `2<sup>` `</sup>} 2<sup>` `</sup>} 2<sup>`</sup>} 2} has recursion level 
{1 {1 {1` ` `1 {1 {1` `2<sup>` `</sup>} 2<sup>` `</sup>} 2<sup>` `</sup>} 2<sup>`</sup>} 2} has recursion level 
{1 {1 {1` ` `1 {1` ` `2^{` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` ` `1 {1` ` `1 {1` ` `2^{` `}} 2^{` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` ` `1` ` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1` ` `1` ` `1` ` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1`2<sup>` ` `</sup>} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` `2<sup>` ` `}} 3^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` `2<sup>` ` `}} 1` ` `2<sup>` `</sup>} 2<sup>`</sup>} 2} has recursion level 
{1 {1 {1 {1` `2^{` ` `</sup>} 1 {1` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {2` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` `3<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` `1` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1` `2<sup>` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1` ` `2^{` `}} 2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1 {2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1 {1 {1 {2<sup>` ` `}} 2^{` `}} 2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 3} has recursion level 
{1 {1 {1 {1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 1`2} has recursion level 
{1 {1 {1 {1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 1 {1`2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 1 {1 {1 {1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {2 {1 {1` ` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1` `2 {1 {1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `2<sup>` ` `}} 2^{` `}} 3^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 1` `2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `2<sup>` ` `}} 2^{` `}} 1 {1`2<sup>` `</sup>} 2<sup>`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `2<sup>` ` `</sup>} 2^{` `}} 1 {1 {1 {1 {1` ` `2<sup>` ` `</sup>} 2^{` `}} 2^{`</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `2<sup>` ` `</sup>} 2^{` `}} 1 {1` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 1 {1` ` `2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 1 {1 {1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {2 {1` ` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1` ` `2 {1` ` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `2^{` ` `</sup>} 3<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `2<sup>` ` `}} 1` ` `2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `2<sup>` ` `}} 1 {1` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `2<sup>` ` `}} 1 {1 {1 {1` ` `2<sup>` ` `</sup>} 2^{` `}} 2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `2<sup>` ` `</sup>} 1 {1` ` `2<sup>` ` `</sup>} 2^{` `}} 2<sup>`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `2<sup>` ` `</sup>} 1 {1` ` `2^{` ` `</sup>} 1 {1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {2` ` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1`2` ` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` `2` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `3<sup>` ` `}} 2^{` `}} 2<sup>`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `4^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `1` ` `2<sup>` ` `</sup>} 1 {1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` `1` ` `2^{` ` `</sup>} 1 {1` ` `1` ` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {2` ` `1` ` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `2` ` `2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `1` ` `3<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `1` ` `1` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {2^{` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1` ` `2^{` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {2 {1` ` `2^{` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1` ` `2 {1` ` `2<sup>` ` `}} 2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1` ` `2<sup>` ` `}} 3^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1` ` `2<sup>` ` `}} 1` ` `2<sup>` ` `}} 2^{` `}} 2<sup>`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1` ` `2<sup>` ` `</sup>} 1 {1` ` `2^{` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2<sup>`</sup>} 2} has recursion level 
{1 {1 {1 {1 {2` ` `2^{` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1` ` `3^{` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1` ` `1` ` `2<sup>` ` `}} 2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1 {1` ` `1` ` `2^{` ` `</sup>} 2<sup>` ` `}} 2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` ` `3^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1` ` ` `1` ` ` `2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {2^{` ` ` `}} 2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1` ` `2<sup>` ` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1` ` ` `2<sup>` ` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1` ` ` `2<sup>` ` ` `</sup>} 1 {1` ` ` `2<sup>` ` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {2` ` ` `2<sup>` ` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1` ` ` `3<sup>` ` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1` ` ` `1` ` ` `2<sup>` ` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1 {1` ` ` `2<sup>` ` ` `</sup>} 2<sup>` ` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^{`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1 {1` ` ` `1` ` ` `2<sup>` ` ` `</sup>} 2<sup>` ` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2<sup>`</sup>} 2} has recursion level 
{1 {1 {1 {1 {1` ` ` ` `2^{` ` ` `}} 2^{` ` `</sup>} 2<sup>` `</sup>} 2<sup>`}} 2} has recursion level 
{1 {1 {1 {1 {1 {1` ` ` ` ` `2^{` ` ` ` `</sup>} 2<sup>` ` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^`} 2} has recursion level 

Complete definition

Separators such as {1 {1 {1 {1 {1 {1` ` ` ` ` `2^{` ` ` ` `</sup>} 2<sup>` ` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^`} 2} look very complex, but we can simplify it by changing the process a little. Here’s the complete ruleset of mEAN.

First, let separators without any grave accents at the left-superscript position of the rbrace have GA-value 0, and separators with m grave accents at the left-superscript position of the rbrace have GA-value m (note that the m-ple grave accent also has GA-value m).

Rule and process

• Rule 1: (base rule – only 2 entries) s(a,b) = a^b
• Rule 2: (recursion rule – neither the 2nd nor 3rd entry is 1) s(a,b,c #) = s(a, s(a,b-1,c #) ,c-1 #)
• Rule 3: (tailing rule – the last entry is 1) s(# A 1) = s(#), {# A 1} = {#}, and {# A 1^{` `…`</sup>} = {#<sup>` `…`}}
• Rule 4: (if lv(A) < lv(B)) {# A 1 B #′} = {# B #′}

where # is a string of entries and separators, it can also be empty. A and B are separators.

If none of the 4 rules above applies, start the process shown below. Note that case B1, B2 and B4 are terminal but case A and B3 are not. And note that red texts imply changes on the array. First start from the 3rd entry.

• Case A: If the entry is 1, then you jump to the next entry.
• Case B: If the entry n is not 1, look to your left:
  • Case B1: If the comma is immediately before you, then
    1. Change the “1,n” into “b,n-1” where n is this non-1 entry and the b is the iterator.
    2. Change all the entries at base layer before them into the base.
    3. The process ends.
  • Case B2: If the m-ple grave accent M (M = ` `…` or {1<sup>` `…`</sup>} with m grave accents) is immediately before you, then
    1. Change the “M n” into “M 2 M n-1”. In following steps “M” refers to the former M.
    2. Let t be such that the M is at layer t. And Let separator A_t = B_t = M.
    3. Repeat this:
       1. Subtract t by 1.
       2. Let separator B_t be such that it’s at layer t, and M is inside it.
       3. If t = 1, then break the repeating, or else continue repeating.
    4. Find the maximal t such that A_t has GA-value lessequal to m-1.
    5. If A_t has GA-value m-1, then
       1. Let string P and Q be such that B_t = “P M Q”.
       2. Change B_t into S_b, where b is the iterator, S₁ is comma, and S_i+1 = “P S_i Q”.
       3. The process ends.
    6. If A_t has GA-value less than m-1, then
       1. Let string P and Q be such that B_t = “P B_t+1 Q”.
       2. Change B_t into “P {1 A_t+1 2^{` `…`}} Q” with m-1 grave accents at the left-superscript position of the rbrace.
       3. The process ends.
  • Case B3: If a separator K that is neither comma nor grave accent is immediately before you, then
    1. Change the “K n” into “K 2 K n-1”.
    2. Set separator A_t = K, here K is at layer t.
    3. Jump to the first entry of the former K.
  • Case B4: If an lbrace is immediately before you, then
    1. Change separator {n #} into string S_b, where b is the iterator, S₁ = “{n-1 #}” and S_i+1 = “S_i 1 {n-1 #}”.
    2. The process ends.

Level comparison

First, all arrays have the lowest and the same level. And note that the same separators have the same level. To compare levels of other separators A and B, we follow these steps.

1. Apply rule 3 to A and B until rule 3 cannot apply any more.
2. If GA-value of A < GA-value of B, then lv(A) < lv(B); if GA-value of A > GA-value of B, then lv(A) > lv(B); or else –
3. Let A = {a₁A₁a₂A₂…a_k-1A_k-1a_k^`…`} and B = {b₁B₁b₂B₂…b_l-1B_l-1b_l^`…`}
4. If k = 1 and l > 1, then lv(A) < lv(B); if k > 1 and l = 1, then lv(A) > lv(B); if k = l = 1, follow step 5; if k > 1 and l > 1, follow step 6 ~ 11
5. If a₁ < b₁, then lv(A) < lv(B); if a₁ > b₁, then lv(A) > lv(B); if a₁ = b₁, then lv(A) = lv(B)
6. Let , and .
7. If lv(A_maxM(A)) < lv(B_maxM(B)), then lv(A) < lv(B); if lv(A_maxM(A)) > lv(B_maxM(B)), then lv(A) > lv(B); or else –
8. If |M(A)| < |M(B)|, then lv(A) < lv(B); if |M(A)| > |M(B)|, then lv(A) > lv(B); or else –
9. Let A = {#₁ A_maxM(A) #₂} and B = {#₃ B_maxM(B) #₄}
10. If lv({#₂}) < lv({#₄}), then lv(A) < lv(B); if lv({#₂}) > lv({#₄}), then lv(A) > lv(B); or else –
11. If lv({#₁}) < lv({#₃}), then lv(A) < lv(B); if lv({#₁}) > lv({#₃}), then lv(A) > lv(B); if lv({#₁}) = lv({#₃}), then lv(A) = lv(B)

Explanation

The step 4, 5 and 6 of case B2 allow us to use separators like {1` ` ` ` ` `2} in EAN. For example, s(a,b {1 {1` ` `3<sup>` ` `</sup>} 3} 2) changes into s(a,b {1 {1` ` `3<sup>` ` `</sup>} 2 {1` ` `3<sup>` ` `</sup>} 2} 2) until we meet case B2, by step 1 it becomes s(a,b {1 {1` ` `2` ` `2<sup>` ` `</sup>} 2 {1` ` `3<sup>` ` `</sup>} 2} 2), and in step 4 A<sub>t</sub> = A<sub>1</sub> = {1 {1` ` `3<sup>` ` `</sup>} 3} has GA-value 0 (less than 2), so we jump into step 6, so s(a,b {1 {1` ` `3<sup>` ` `</sup>} 3} 2) = s(a,b {1 {1 {1` ` `3<sup>` ` `</sup>} 2<sup>` `</sup>} 2 {1` ` `3<sup>` ` `</sup>} 2} 2). The next time we start the process, in step 4 of case B2, A<sub>t</sub> will be A<sub>2</sub> = {1 {1` ` `3<sup>` ` `</sup>} 2<sup>` `</sup>} (has GA-value 2), so we jump to step 5, so s(a,b {1 {1 {1` ` `3<sup>` ` `</sup>} 2<sup>` `</sup>} 2 {1` ` `3<sup>` ` `</sup>} 2} 2) = s(a,b {1 {1 {1 …{1 {1 {1 {1,2` ` `2<sup>` ` `</sup>} 2<sup>` `</sup>} 2` ` `2<sup>` ` `</sup>} 2<sup>` `</sup>}… 2` ` `2<sup>` ` `</sup>} 2<sup>` `</sup>} 2 {1 {1` ` `3<sup>` ` `</sup>} 2<sup>` `</sup>} 1 {1` ` `3<sup>` ` `</sup>} 2} 2) with b-1 2` ` `2^{` ` `</sup>} 2<sup>` `}}′s.

Before this change, an m-ple grave accent needs to be inside a separator with GA-value = m-1, and this GA-value = m-1 separator needs to be inside a separator with GA-value = m-2, and so on. But now, a separator with GA-value = 0 is enough. And {1` ` ` ` ` `2} still have similar strength to {1 {1 {1 {1 {1 {1` ` ` ` ` `2^{` ` ` ` `</sup>} 2<sup>` ` ` `</sup>} 2<sup>` ` `}} 2^{` `}} 2^`} 2}.

Comparison

Let f(n) = s(n,n {1` `…`2} 2) with an n-ple grave accent, then f(n) eventually outgrows all expressions in mEAN, so it has growth rate . That’s the limit of mEAN, also eventually outgrows any function provable recursive in  comprehension () and the theory of finitely iterated inductive definitions (). It has similar growth rate to U(n) function in BAN, and it’s a lower bound for growth rate of subcubic graph number (SCG(n)) and simple subcubic graph number (SSCG(n)).