Chained array notation

Chained array notation is a notation defined by Rune-tu Plycien (Japanese: 閏土=プリシエン) and kaosu on July 21, 2017. It is an extension of Conway's chained arrow notation.

Definition
It uses array of \([n_1,n_2,...,n_k]\) instead of arrow \(\rightarrow\) in the chained arrow notation. The chained array

\(a_1[n_1,n_2,...,n_k]a_2[n_1,n_2,...,n_k]...[n_1,n_2,...,n_k]a_m\)

has length of \(m\). Calculation rule is defined as

\begin{eqnarray*} a[1,Y]b &=& a^b \\ a[n+1,X_2](b+1) &=& a[n,X_2]a[n+1,X_2]b \\ a[Y,1,m+1,X_2](b+1) &=& a[Y,a[Y,1,m+1,X_2]b,m,X_2]a \\ X_1[n,X_2](a+1)[n,X_2](b+1) &=& X_1[n,X_2](X_1[n,X_2]a[n,X_2](b+1))[n,X_2]b \\ X_1[n,X_2]1 &=& X_1 \\ X_1[n,X_2]1[n,X_2]X_1' &=& X_1 \end{eqnarray*}

where
 * \(X_1, X_1'\): Chained array with length larger than 0
 * \(X_2\): vector of integers larger than 0, with the length larger than or equal to 0 (ex., [3, 1, 1, 1])
 * \(Y\): vector of 1 with the length larger than or equal to 0

Approximation
[1] exactly matches the arrow \(\rightarrow\) of Conway's chained arrow. Therefore

\[3[1]3[1]3[1]3 = 3\rightarrow 3\rightarrow 3\rightarrow 3\]

[n] matches \(\rightarrow_n\) of Hurford's extension, except that in the Hurford's extension there is always one kind of arrows, but in this system there can be different kinds of arrows. In that case, calculation begins from the rightmost block of "same kind of arrows", for example:

3[2]4 = 3[1]3[2]3 = 3[1]3[1]3[2]2 = 3[1]3[1]3[1]3[2]1 = 3[1]3[1]3[1]3

Approximation with FGH follows.

\begin{eqnarray*} 3[n]n &\approx& f_{\omega^3}(n) \\ 3[n,1]n &\approx& f_{\omega^3}(n) \\ n[1,2]n &=& n[n[1,2](n-1),1]n \approx f_{\omega^3+1}(n) \\ n[2,2]3 &=& n[1,2]n[1,2]n \approx f_{\omega^3+\omega}(n) \\ n[2,2]4 &=& n[1,2]n[1,2]n[1,2]n \approx f_{\omega^3+\omega 2}(n) \\ n[2,2]n &\approx& f_{\omega^3+\omega^2}(n) \\ n[3,2]n &\approx& f_{\omega^3+\omega^2 2}(n) \\ n[n,2]n &\approx& f_{\omega^3 2}(n) \\ n[1,3]n &=& n[n[1,3](n-1),2]n \approx f_{\omega^3 2+1}(n) \\ n[2,3]n &\approx& f_{\omega^3 2 +\omega^2}(n) \\ n[n,3]n &\approx& f_{\omega^3 3}(n) \\ n[n,n]n &\approx& f_{\omega^4}(n) \\ n[n,n,1]n &\approx& f_{\omega^4}(n) \\ n[n,1,2]n &=& n[n,n[n,1,2](n-1),1]n \approx f_{\omega^4 +1}(n) \\ n[n,1,3]n &=& n[n,n[n,1,3](n-1),2]n \approx f_{\omega^4 + \omega^3}(n) \\ n[n,1,n]n &\approx& f_{\omega^4 2}(n) \\ n[1,2,n]n &=& n[n[1,2,n](n-1),1,n]n \approx f_{\omega^4 2 + 1}(n) \\ n[1,3,n]n &=& n[n[1,3,n](n-1),2,n]n \approx f_{\omega^4 2 + \omega^3}(n) \\ n[n,n,n]n &\approx& f_{\omega^4 3}(n) \\ n[n,n,n,1]n &\approx& f_{\omega^4 3}(n) \\ n[n,n,1,n]n &\approx& f_{\omega^4 4}(n) \\ n[n,1,n,n]n &\approx& f_{\omega^4 5}(n) \\ n[n,n,n,n]n &\approx& f_{\omega^4 6}(n) \\ n[\underbrace{n,...,n}_{n}]n &\approx& f_{\omega^5}(n) \\ \end{eqnarray*}