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Here is reboot version of my array notation, simpler than ever.

Simple

Growth rate: 0
The simplest and weakest part in AANR (Aarex's Array Notation Rebooted). There is only 1 rule in this extension:

  • r(a) = a+1

The limit of simple arrays corresponds to FGH level 0 because r(a) evaluates as the same of \(f_0(a)\).

2-entry

Growth rate: \(\omega\)
Let [A]^B = AAA... with Bs, but [A] evaluates, even inside of any [_]s.

  • r(a,0) = a+1
  • r(a,b) = [r(]aa[,[b-1])]a

If the syntax concentration is too hard for you, here is rules without syntax concentration:

  • r(a,0) = a+1
  • r(a,b)[0] = a
  • r(a,b)[n] = r(r(a,b)[n-1],b-1)[n]
  • r(a,b) = r(a,b)[a]

To solve r(2,2) (using without syntax concentration):

  • r(2,2) =
  • r(2,2)[2] =
  • r(r(2,2)[1],1) =
  • r(r(r(2,2)[0],1),1) =
  • r(r(2,1),1) =
  • r(r(2,1)[2],1) =
  • r(r(r(2,1)[1],0),1) =
  • r(r(r(r(2,1)[0],0),0),1) =
  • r(r(r(2,0),0),1) =
  • r(r(3,0),1) =
  • r(4,1) =
  • r(4,1)[4] =
  • r(r(4,1)[3],0) =
  • r(r(r(4,1)[2],0),0) =
  • r(r(r(r(4,1)[1],0),0),0) =
  • r(r(r(r(r(4,1)[0],0),0),0),0) =
  • r(r(r(r(4,0),0),0),0) =
  • r(r(r(5,0),0),0) =
  • r(r(6,0),0) =
  • r(7,0) =
  • 8

Therefore, r(2,2) = 8. Here is the analysis of 2-entry.

  • r(a,1) = r(a,1)[a] = r(r(...r(a,0)...,0),0) with a r's, corresponds to FGH level 1
  • r(a,2) = r(r(...r(a,1)...,1),1) with a r's, corresponds to FGH level 1
  • r(a,3) corresponds to FGH level 3
  • r(a,n) corresponds to FGH level n

The limit of 2-entry arrays corresponds corresponds to FGH level \(\omega\).

Linear

Growth rate: \(\omega^{\omega}\)
Let # be the rest of the array, and stay the same while outputting. For now on, let define rules without syntax concentration:

Here is the rules without syntax concentration:

  • r(a,0) = a+1
  • r(#,0) = r(#)
  • r(a,b#)[0] = a
  • r(a,b#)[n] = r(r(a,b#)[n-1],b-1#)[n]
  • r(a,b#) = r(a,b#)[a]
  • Find the second non-zero entry c, then decrease c by 1 and change its' previous entry by a.

To solve r(2,0,2):

  • r(2,0,2) =
  • r(2,2,1) =
  • r(r(2,1,1),1,1) =
  • r(r(r(2,0,1),0,1),1,1) =
  • r(r(r(2,2),0,1),1,1) =
  • r(r(8,0,1),1,1) =
  • r(r(8,8),1,1) =
  • r(f_8(8),1,1) =
  • Too large to show in next step!

Here is the analysis of linear.

  • r(a,0,1) corresponds to FGH level \(\omega\)
  • r(a,1,1) corresponds to FGH level \(\omega+1\)
  • r(a,2,1) corresponds to FGH level \(\omega+2\)
  • r(a,n,1) corresponds to FGH level \(\omega+n\)
  • r(a,0,2) corresponds to FGH level \(\omega 2\)
  • r(a,1,2) corresponds to FGH level \(\omega 2+1\)
  • r(a,0,3) corresponds to FGH level \(\omega 3\)
  • r(a,0,4) corresponds to FGH level \(\omega 4\)
  • r(a,0,n) corresponds to FGH level \(\omega n\)
  • r(a,0,0,1) corresponds to FGH level \(\omega^2\)
  • r(a,1,0,1) corresponds to FGH level \(\omega^2+1\)
  • r(a,0,1,1) corresponds to FGH level \(\omega^2+\omega\)
  • r(a,0,0,2) corresponds to FGH level \(\omega^22\)
  • r(a,0,0,3) corresponds to FGH level \(\omega^23\)
  • r(a,0,0,n) corresponds to FGH level \(\omega^2n\)
  • r(a,0,0,0,1) corresponds to FGH level \(\omega^3\)
  • r(a,0,0,0,2) corresponds to FGH level \(\omega^32\)
  • r(a,0,0,0,0,1) corresponds to FGH level \(\omega^4\)
  • r(a,0,0,0,0,0,1) corresponds to FGH level \(\omega^5\)
  • r(a,0,0...0,0,1) /w n 0s corresponds to FGH level \(\omega^n\)

The limit of linear arrays corresponds to FGH level \(\omega^{\omega}\).

Dimensional

Growth rate: \(\omega^{\omega^{\omega}}\)
Let {0} is first and weakest separator as comma is and {a+1} is next and stronger separator than {a}.

  • r(a,0) = a+1
  • r(#0) = r(#)
  • r(a,b#)[0] = a
  • r(a,b#)[n] = r(r(a,b#)[n-1],b-1#)[n]
  • r(a,b#) = r(a,b#)[a]
  • Find the second non-zero entry c and then define d such that there is a array '0{d}c':
    • Decrease c by 1.
    • If there is comma before you, then change its' previous entry by a.
    • Else replace it's previous entry by Aa, where A0 = '1' and An+1 = '0{d-1}An'.

Also, r(a{b}#) is same as r(a,0{b}#) only if b > 0.

Here how to expand to weaker separators: r(3{3}1) = r(3,0{3}1) = r(3,0{2}0{2}0{2}1) = r(3,0{2}0{2}0{1}0{1}0{1}1) = r(3,0{2}0{2}0{1}0{1}0,0,1) = r(3,0{2}0{2}0{1}0{1}0,3) = ...

Here is the analysis of dimensional.

  • r(a{1}1) corresponds to FGH level \(\omega^{\omega}\)
  • r(a,1{1}1) corresponds to FGH level \(\omega^{\omega}+1\)
  • r(a,0,1{1}1) corresponds to FGH level \(\omega^{\omega}+\omega\)
  • r(a,0,0,1{1}1) corresponds to FGH level \(\omega^{\omega}+\omega^2\)
  • r(a,0,0...0,0,1{1}1) /w n 0s corresponds to FGH level \(\omega^{\omega}+\omega^n\)
  • r(a{1}2) corresponds to FGH level \(\omega^{\omega}2\)
  • r(a{1}3) corresponds to FGH level \(\omega^{\omega}3\)
  • r(a{1}n) corresponds to FGH level \(\omega^{\omega}n\)
  • r(a{1}0,1) corresponds to FGH level \(\omega^{\omega+1}\)
  • r(a,0,1{1}0,1) corresponds to FGH level \(\omega^{\omega+1}+\omega\)
  • r(a{1}1,1) corresponds to FGH level \(\omega^{\omega+1}+\omega^{\omega}\)
  • r(a{1}0,2) corresponds to FGH level \(\omega^{\omega+1}2\)
  • r(a{1}0,0,1) corresponds to FGH level \(\omega^{\omega+2}\)
  • r(a{1}0{1}1) corresponds to FGH level \(\omega^{\omega 2}\)
  • r(a{1}0{1}0{1}1) corresponds to FGH level \(\omega^{\omega 3}\)
  • r(a{1}0{1}0...0{1}1) /w n 0s corresponds to FGH level \(\omega^{\omega n}\)
  • r(a{2}1) corresponds to FGH level \(\omega^{\omega^2}\)
  • r(a{1}1{2}1) corresponds to FGH level \(\omega^{\omega^2}+\omega^{\omega}\)
  • r(a{2}2) corresponds to FGH level \(\omega^{\omega^2}2\)
  • r(a{2}0{2}1) corresponds to FGH level \(\omega^{\omega^22}\)
  • r(a{3}1) corresponds to FGH level \(\omega^{\omega^3}\)
  • r(a{4}1) corresponds to FGH level \(\omega^{\omega^4}\)
  • r(a{n}1) corresponds to FGH level \(\omega^{\omega^n}\)

The limit of dimensional arrays corresponds to FGH level \(\omega^{\omega^{\omega}}\).

Nested

Growth rate: \(\varepsilon_0\)
Now we could have arrays instead of numbers in the separators.

  • r(a,0) = a+1
  • r(#0) = r(#)
  • r(a,b#)[0] = a
  • r(a,b#)[n] = r(r(a,b#)[n-1],b-1#)[n]
  • r(a,b#) = r(a,b#)[a]
  • Find the second non-zero entry c and then define d such that there is a array '0{d}c':
    • Decrease c by 1.
    • If there is comma before you, then change its' previous entry by a.
    • Else replace it's previous entry by Aa, where A0 = '1' and An+1 = '0{d-1}An' only if the first entry is not zero in the separator.
    • Else replace it's previous entry by '0{d}1', then jump to d in the previous entry. Then repeat this process.

If a(a,A) corresponds to FGH level \(B\) then a(a{A}1) corresponds to FGH level \(\omega^{\omega^{B}}\). Therefore, the limit of linear arrays corresponds to FGH level \(\varepsilon_0\).

Hyperseparators

Growth rate: \(\psi(\varepsilon_{\Omega+1})\)
Now we introduce separator layers: The entries and separators (except all that in the separator) in r() are in separator layer 0, and in any separator, that is separator layer a, are in separator layer a+1.

Let define : (shorthand of :{0}) as a new separator, and there is more separators by defining :{a} as like {1}, but with :{a-1}.

  • r(a,0) = a+1
  • r(#0) = r(#)
  • r(a,b#)[0] = a
  • r(a,b#)[n] = r(r(a,b#)[n-1],b-1#)[n]
  • r(a,b#) = r(a,b#)[a]
  • Find the second non-zero entry c and then define d such that there is a array '0{d}c':
    • Note: :{a} still works, so {A} and :{A} separators are vaild in this rule.
    • Decrease c by 1.
    • If there is comma before you, then change its' previous entry by a.
    • Else replace it's previous entry by Aa, where A0 = '1' and An+1 = '0{d-1}An' only if the first entry is not zero in the separator.
    • Else replace it's previous entry by '0{d}1', then jump to d in the previous entry, only if d is not zero. Then repeat this process.
    • Else:
      • Define Ac = : and Ad = separator which includes Ad-1, for all d > c and for c, it is such that : is in layer c.
      • Find the maximum of e that Ae < :.
      • Let B and C such that Be = 'B:C'.
      • Replace Ae with Da, where D0 = , and Dn+1 = 'B Dn C'.

Level-comparing

Find the first highest-level separator in the array, then:

  • If A > B, then C,A > D,B, else compare C and D.
  • If A > B, then C1{A}C2 > D1{B}D2, else compare C1,C2 and D1,D2.
  • : is greater level than all separators excluding all :{A} separators.

Example

The first and only example that is vaild in Hyperseparators AANR or beyond is r(3{0:{0:1}1}1).

  • r(3{0:{0:1}1}1) =
    • c = 3
    • A3 = :,A2 = :{0:1}, A1 = {0:{0:1}1}, A0 = 3{0:{0:1}1}1 (for all vaild d's)
    • e = 1 ({0:{0:1}1} < :)
    • A1 = {0:{0:1}1}, B = {0:{0, C = 1}1}
    • A1 replaces to D3, D0 = comma, D1 = {0:{0,1}1}, D2 = {0:{0{0:{0,1}1}1}1}, D3 = {0:{0{0:{0{0:{0,1}1}1}1}1}1}
  • r(3{0:{0{0:{0{0:{0,1}1}1}1}1}1}1) =
  • r(3{0:{0{0:{0{0:{3}1}1}1}1}1}1) =
  • r(3{0:{0{0:{0{0:{2}0:{2}0:{2}1}1}1}1}1}1) =
  • r(3{0:{0{0:{0{0:{2}0:{2}0:{1}0:{1}0:{1}1}1}1}1}1}1) =
  • r(3{0:{0{0:{0{0:{2}0:{2}0:{1}0:{1}0:0:0:1}1}1}1}1}1) =
  • r(3{0:{0{0:{0{0:{2}0:{2}0:{1}0:{1}0:0:0{0:{2}0:{2}0:{1}0:{1}0:0:0{0:{2}0:{2}0:{1}0:{1}0:0:0,1}1}1}1}1}1}1}1) =
    • c = 6
    • A6 = :, A5 = {0{0:{2}0:{2}0:{1}0:{1}0:0:0:1}1}, A4 = {0:{0{0:{2}0:{2}0:{1}0:{1}0:0:0:1}1}1}, A3 = {0:{0{0:{2}0:{2}0:{1}0:{1}0:0:0:1}1}1}1}, A2 = {0{0:{0{0:{2}0:{2}0:{1}0:{1}0:0:0:1}1}1}1}, A1 = {0:{0{0:{0{0:{2}0:{2}0:{1}0:{1}0:0:0:1}1}1}1}1}, A0 = 3{0:{0{0:{0{0:{2}0:{2}0:{1}0:{1}0:0:0:1}1}1}1}1}1
    • e = 5 ({0:{2}0:{2}0:{1}0:{1}0:0:0:1} < :)
    • A5 = {0:{2}0:{2}0:{1}0:{1}0:0:0:1}, B = {0:{2}0:{2}0:{1}0:{1}0:0:0, C = 1}
    • A5 replaces to D3, D0 = comma, D1 = {0:{2}0:{2}0:{1}0:{1}0:0:0,1}, D2 = {0:{2}0:{2}0:{1}0:{1}0:0:0{0:{2}0:{2}0:{1}0:{1}0:0:0,1}1}, D3 = {0:{0:{2}0:{2}0:{1}0:{1}0:0:0{0:{2}0:{2}0:{1}0:{1}0:0:0{0:{2}0:{2}0:{1}0:{1}0:0:0,1},1}1}
  • ...

Analysis

Here is the analysis, and : is sightly faster than ` in strong array notation. Therefore, HSAANR > EAN.

Define 'A has recursion level \(B\)' such that r(a{A}1) corresponds to FGH level \(\omega^{\omega^B}\)

  • {0:1} has recursion level \(\varepsilon_0\)
  • {1:1} has recursion level \(\varepsilon_0+1\)
  • {0,1:1} has recursion level \(\varepsilon_0+\omega\)
  • {0{0,1}1:1} has recursion level \(\varepsilon_0+\omega^{\omega^{\omega}}\)
  • {0{0:1}1:1} has recursion level \(\varepsilon_02\)
  • {0{0:1}0{0:1}1:1} has recursion level \(\varepsilon_0^2\)
  • {0{0{0:1}1:1}1:1} has recursion level \(\varepsilon_0^{\varepsilon_0}\)
  • {0{0{0:1}1:1}0{0:1}1:1} has recursion level \(\varepsilon_0^{\varepsilon_0+1}\)
  • {0{0{0:1}1:1}0{0{0:1}1:1}1:1} has recursion level \(\varepsilon_0^{\varepsilon_02}\)
  • {0{0{0:1}2:1}1:1} has recursion level \(\varepsilon_0^{\varepsilon_0^2}\)
  • {0{0{0:1}0{0:1}1:1}1:1} has recursion level \(\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}\)
  • {0{0{0:1}1{0:1}1:1}1:1} has recursion level \(\varepsilon_0^{\varepsilon_0^{\varepsilon_0+1}}\)
  • {0{0{0:1}0{0:1}2:1}1:1} has recursion level \(\varepsilon_0^{\varepsilon_0^{\varepsilon_02}}\)
  • {0{0{0:1}0{0:1}0{0:1}1:1}1:1} has recursion level \(\varepsilon_0^{\varepsilon_0^{\varepsilon_0^2}}\)
  • {0:2} has recursion level \(\varepsilon_1\)
  • {0{0:2}1:2} has recursion level \(\varepsilon_12\)
  • {0{0{0:2}1:2}1:2} has recursion level \(\varepsilon_1^{\varepsilon_1}\)
  • {0:1} has recursion level \(\varepsilon_2\)
  • {0:0,1} has recursion level \(\varepsilon_{\omega}\)
  • {0:0{0:1}1} has recursion level \(\varepsilon_{\varepsilon_0}\)
  • {0:0:1} has recursion level \(\zeta_0\)
  • {0:1:1} has recursion level \(\varepsilon_{\zeta_0+1}\)
  • {0:0{0:0:1}1:1} has recursion level \(\varepsilon_{\zeta_02}\)
  • {0:0:2} has recursion level \(\zeta_1\)
  • {0:0:0{0:0:1}1} has recursion level \(\zeta_{\zeta_0}\)
  • {0:0:0:1} has recursion level \(\eta_0\)
  • {0:0:0:1} has recursion level \(\varphi(4,0)\)
  • {0:{1}1} has recursion level \(\varphi(\omega,0)\)
  • {0:{0{0:1}1}1} has recursion level \(\varphi(\varepsilon_0,0)\)
  • {0:{0:1}1} has recursion level \(\Gamma_0\)
  • {0:{0{0:{0:1}1}1}1:{0:1}1} has recursion level \(\varphi(\Gamma_0,1)\)
  • {0:{0:1}2} has recursion level \(\Gamma_1\)
  • {0:{0:1}0:1} has recursion level \(\varphi(1,1,0)\)
  • {0:{0:1}0:{0:1}1} has recursion level \(\varphi(2,0,0)\)
  • {0:{0{0:{0:1}1}1:1}1} has recursion level \(\varphi(\Gamma_0,0,0)\)
  • {0:{0:2}1} has recursion level \(\varphi(1,0,0,0)\)
  • {0:{0:2}0:{0:2}1} has recursion level \(\varphi(2,0,0,0)\)
  • {0:{0:3}1} has recursion level \(\varphi(1,0,0,0,0)\)
  • {0:{0:1:1}1} has recursion level \(\psi(\Omega^{\Omega^{\Omega +1}})\)
  • {0:{0:0:2}1} has recursion level \(\psi(\Omega^{\Omega^{\Omega 2}})\)
  • {0:{0:{0:1}1}1} has recursion level \(\psi(\Omega^{\Omega^{\Omega^{\Omega}}})\)

The limit of hyperseparators arrays corresponds to FGH level \(\psi(\varepsilon_{\Omega+1})\).

Multiple hyperseparators

Growth rate: \(\psi(\Omega_{\omega})\)
We can extend the concept of hyperseparators with expanding to lower hyperseparators.

  • r(a,0) = a+1
  • r(#0) = r(#)
  • r(a,b#)[0] = a
  • r(a,b#)[n] = r(r(a,b#)[n-1],b-1#)[n]
  • r(a,b#) = r(a,b#)[a]
  • Find the second non-zero entry c and then define d such that there is a array '0{d}c':
    • Decrease c by 1.
    • If there is comma before you, then change its' previous entry by a.
    • Else replace it's previous entry by Aa, where A0 = '1' and An+1 = '0{d-1}An' only if the first entry is not zero in the separator.
    • Else replace it's previous entry by '0{d}1', then jump to d in the previous entry, only if d is not zero. Then repeat this process.
    • Else: (Let (n) be n colons)
      • Define Ac = (f) and Ad = separator which includes Ad-1, for all d > c and for c, it is such that (f) is in layer c.
      • Find the maximum of e that Ae < (f).
      • Let B and C such that Be = 'B(f)C'.
      • Replace Ae with Da, where D0 = , and Dn+1 = 'B Dn C'.

Reduced expressions

  • r(a,0) = a+1
  • r(#0) = r(#)
  • r(a,b#)[0] = a
  • r(a,b#)[n] = r(r(a,b#)[n-1],b-1#)[n]
  • r(a,b#) = r(a,b#)[a]
  • Find the second non-zero entry c and then define d such that there is a array '0{d}c':
    • Decrease c by 1.
    • If there is comma before you, then change its' previous entry by a.
    • Else replace it's previous entry by Aa, where A0 = '1' and An+1 = '0{d-1}An' only if the first entry is not zero in the separator.
    • Else replace it's previous entry by '0{d}1', then jump to d in the previous entry, only if d is not zero. Then repeat this process.
    • Else: (Let (n) be n colons)
      • Define Ac = (f) and Ad = separator which includes Ad-1, for all d > c and for c, it is such that (f) is in layer c.
      • Find the maximum of e that Ae < (f).
      • ?

Planned parts

  • Poly-hyperseparators
  • 2nd-order hyperseparators
  • Higher-order hyperseparators
  • Poly-order hyperseparators
  • Dropping

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