Notation des tableaux

La notation des tableaux est une fonction googologique créée par Jonathan Bowers inventée en 2002. Jonathan Bowers a étendu la fonction en BEAF (Bowers Exploding Array Function) et Chris Bird a étendu la fonction en BAN (Bird's array notation).

Définition

An array is defined as a finite sequence of positive integers. Array notation is a function v(A) mapping these sequences to large positive integers. If A = (a₁, a₂, ..., a_n), we write {a₁, a₂, ..., a_n} as a shorthand for v(A).

1. {a} = a and {a,b} = a^b
2. {a,b,c,...,n,1 = a,b,c,...,n}
3. {a,1,b,c,...,n} = a
4. If the third element is 1, {a,b,1,...,1,c,d,...,n} = {a,a,a,...,{a,b-1,1,...,1,c,d,...,n,c-1,d,...,n}.
   • all elements before the next non-1 element become the first element,
   • the last of the above becomes the original array with the second element decreased by 1
   • the said non-1 element is decreased by one.
5. If rules 1 to 4 do not apply, {a,b,c,d,...,n} = {a,{a,b-1,c,d,...,n,c-1,d,...,n}

Calcul

Arrays of length 3 gives the same value with the notation des flèches chaînées and therefore related to the notation des flèches chaînées as

{a,b,c} = a→b→c = a↑^cb

Arrays of length 1 and 2 are also consistent with the chained arrow notation.

Arrays of length 4 {a,b,c,d} with a > 2, b > 1, c > 0, d > 1, Chris Bird proved the following relationship with the chained arrow notation.

{a, b, c, d} > $\underbrace{a \rightarrow a \rightarrow \ldots \rightarrow a}_d$ → (b - 1) → (c + 1)

Fish compared the recursion level of the array notation with the fonction d'Ackermann multivariable as

{n+1,2,a₀+1,a₁+1,...,a_k+1} ≈ A(a_k,...,a₁,a₀,n)

and by defining f(n) = A(a_k, ..., a₂, a₁, a₀, n),

{n+1,m+1,a₀+1,a₁+1,...,a_k+1} ≈ f^m(n)

Therefore, by using the relationship of the multivaribale Ackermann function with the fast-growing hierarchy, the growth rate of the array notation can be approximated with fast-growing hierarchy with Wainer hierarchy as

{n+1,m+1,a₀+1,a₁+1,...,a_k+1} ≈ $f_{\omega^k \times a_k + ... + \omega^2 \times a_2 + \omega \times a_1 +a_0}^m(n)$

Googolismes

In the googolisms defined by Bowers, numbers in the googol group, giggol group, gaggol group, tridecal, boogol group, corporal group, biggol group, tetratri, baggol group, general group, pentadecalgroup, hexadecal group and iteral group are expressed with linear array notation.

Numbers defined with {n,n,n} are the nombres d'Ackermann, and as written in the article of the Ackermann numbers, {3,3,3} = 3→3→3 = 3↑↑↑3 is tritri, {4,4,4} = 4→4→4 = 4↑↑↑↑4 is tritet, and so on.

Numbers defined with multiple 3 are as follows.

• {3,3,3} = tritri
• {3,3,3,3} = tetratri
• {3,3,3,3,3} = pentatri
• {3,3,3,3,3,3} = hexatri
• {3,3,3,3,3,3,3} = heptatri
• {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} (with 27 3's) = ultatri

From the comparison in the last section, pentatri = {3,3,3,3,3} is approximated as f²(n) for A(2,2,2,n), i.e., A(2,2,2(A(2,2,2,2)). It is therefore approximated with the recursion level of A(1,0,0,0,3) = A(2,2,2,3).