@ notation

View full site to see MathJax equation

For the operator in BEAF, see Bowers Exploding Array Function#Multiple legions, legiattic arrays.

@ notation is a notation to generate a large number. The term was coined by Googology Wiki user mumuji.

Definition

(note that all terms and things used in this are in the set of Positive Whole Numbers)

First, define the @: a@b = \(a\uparrow^{a\uparrow^{a^b}b}b\) Then,\(a@^{c}b = a@^{c-1}(a@^{c-1}b)\) until c is equal to one, in that case, \(a@^{c}b = a@b\)

Then, introduce a shorthand for the @ notation: \(a[b]c = a@^{b}c\)

I use the modified FIH, hereby I will declare it's base rule as "\(f_0(n) =\) "insert function name"

Suppose a large countable ordinal such to that every finite number a satisfies a < u. A fast iteration hierarchy of functions, \(f_a:\mathbb N \rightarrow \mathbb N\)

\(f_0(n)=\)"insert function" \(f_{n+1}(n) = f_{n}\underbrace{\rightarrow n \cdots \rightarrow n}_{\text{n copies of 'n', where array only has n.}}(n)\) until n = 0, in which case, refer to the top rule. Where the \(\uparrow\) is defined as follows:

\begin{eqnarray*} f_x \uparrow^n b(c) = \left\{ \begin{array}{ll} {f_x}^b(c) & (n = 1) \\ f_x(c) & (n > 1, b = 1) \\ f_x \uparrow^{n-1} (f_x \uparrow^{n-1} (f_x \uparrow^{n-1} (f_x \uparrow^n (b-1)(c))(c) & (n > 1, b > 1) \end{array} \right. \end{eqnarray*}

\(a \rightarrow b = a^{b}\) Where the chained arrows is as follows:

\(f_x \rightarrow b \rightarrow c = a\uparrow^cb = f_x\underbrace{\uparrow\ldots\uparrow}_cb\) (in which up-arrow notation is used) = a[c+2]b (in which the bracket notation of hyper operator is used)

\(f_x \rightarrow\ldots\rightarrow b \rightarrow 1 = f_x \rightarrow\ldots\rightarrow b\) — When last entry is 1, it will be ignored.

\(f_x \rightarrow\ldots\rightarrow b \rightarrow 1 \rightarrow c = f_x \rightarrow\ldots\rightarrow b\)

\(f_x \rightarrow\ldots\rightarrow b \rightarrow (c + 1) \rightarrow (d + 1) = f_x \rightarrow\ldots\rightarrow b \rightarrow (f_x \rightarrow\ldots\rightarrow b \rightarrow c \rightarrow (d + 1) ) \rightarrow d\)

\(a[1]^{c}b = f_{c-1}(a@b),f_0(n) = a[b]c\)
\(a[b]^{d}c = f_{d-1}(a@^{b}c)), f_0(n) = a[b]^{d-1}c\), until d = 1, in which case refer below \(a[b]^{1}c = a[b]c\)

 Then, define two rules: \(a[b]c = [a,b,c]\) and \(a[b]^{d}c = [a,b,c,d]\)  After that, continue to extend for arrays of length n.

\([\underbrace{x_1,x_2,\cdots,x_{n-1},x_n}_{n},1] = [\underbrace{x_1,x_2,\cdots,x_{n-1},x_n}_{n}]\)

\([1,\underbrace{x_1,x_2,\cdots,x_{n-1},x_n}_{n}] = [\underbrace{x_1,x_2,\cdots,x_{n-1},x_n}_{n}]\)

\([\underbrace{x_1,x_2,\cdots,x_{n-1},x_n}_{n}] = [\underbrace{x_1-1,x_2*(n-1),\cdots,x_{n-1}*(n-1),x_n*(n-1)}_{n}]\)(apply only if \(x_1\) is larger than 1.

\([x_1,x_2,\underbrace{1,\cdots, 1}_{p\,1s},x_{n-1},x_n] = [x_1*(n-1),x_2*(n-1),\underbrace{1,\cdots,1}_{(p-1)1s},x_{n-1}*(n-1),x_n*(n-1)]\)

The * operator is defined below:

\(a*b = [\underbrace{a,a,\cdots,a,a}_{\text {b copies of a, and no other term other than a.}}]\)

Approximation of speed

For \(a@b\):

Notation	Approximation
Fast-growing hierarchy	\(f^2_\omega(a^b)\)

For \(a@^c b\)

Notation	Approximation