Taro's multivariable Ackermann function

Taro's multivariable Ackermann function
Based on	successor function
Growth rate	\(f_{\omega^\omega}(n)\)

Taro's multivariable Ackermann function \(A(x_1,x_2, …, x_n)\) is a recursive function invented by a Japanese googologist Taro (2007) and described by Fish (2013).

It is similar to Ackermann function with 2 variables, and with more than 3 variables grows at similar rate with Array notation, despite its simple definition.

Definition

• X : vector of integers larger than or equal to 0, with the length larger than or equal to 0 (ex., [3, 1, 0, 0])
• Y : vector of 0 with the length larger than or equal to 0 (ex., [0, 0, 0, 0, 0])
• a, b : integer larger than or equal to 0

\begin{eqnarray*} A(Y, a) & = & a+1 \\ A(X, b+1, 0) & = & A(X, b, 1) \\ A(X, b+1, a+1) & = & A( X, b, A(X, b+1, a) ) \\ A(X, b+1, 0, Y, a ) & = & A(X, b, a, Y, a) \end{eqnarray*}

In other words:

• If all but/and the last variable are zeros,the function value is the successor of the last variable. (1st case)
• If the second to last variable is a non-zero: (2nd or 3rd case)
  • If the last variable is zero: (2nd case)
    • Replace the last variable to 1. 
    • Reduce the second to last variable in 1.
  • Else: (3rd case)
    • Replace the last variable with value of the function, but with its last variable reduced by 1.
    • Reduce the second to last variable in 1.
• Else (the second to last entry is a end of a Y vector (which will be named Z)): (4th case)
  • Replace the first variable of Z with the last variable.
  • Reduce the last variable before Z by 1.

Calculation

The secret of the fast growth rate of this function is in the 4th case of the definition. In order to reduce the "b" variable by 1, it diagonizes the right variable from 0 to a. It is enough to make strong recursion. If we count the variables from right to left, 2nd variable recurses the 1st variable (primitive recursion in Ackermann function), 3rd variable recurses the second variable, where recursion of primitive recursion makes double recursion, 4th variable recurses the 3rd variable. As the nth variable recurses the (n-1)th variable, the "level" of recursion steadily goes up.

Example calculation is shown to compare this with Graham's number with help of Conway's chained arrow notation. With 2 variable, it is similar to standard Ackermann function and can be compared as

\begin{eqnarray*} A(x,y) & \approx & 3 \rightarrow y \rightarrow x-2 \\ \end{eqnarray*}

With 3 variables,

\begin{eqnarray*} A(1,1,0) & = & A(1,0,1) = A(0,1,1) = A(1,1) = 3 \\ A(1,1,1) & = & A(1,0,A(1,1,0)) = A(1,0,3) = A(3,3) = 61 \\ A(1,1,2) & = & A(1,0,61) = A(61,61) > 3 \rightarrow 3 \rightarrow 2 \rightarrow 2 \\ A(1,1,3) & \approx & A(1,0,3 \rightarrow 3 \rightarrow 2 \rightarrow 2) \approx 3 \rightarrow 3 \rightarrow 3 \rightarrow 2 \\ A(1,1,4) & \approx & A(1,0,3 \rightarrow 3 \rightarrow 2 \rightarrow 3) \approx 3 \rightarrow 3 \rightarrow 4 \rightarrow 2 \\ A(1,1,x) & \approx & 3 \rightarrow 3 \rightarrow x \rightarrow 2 \\ A(1,1,65) & \approx & 3 \rightarrow 3 \rightarrow 65 \rightarrow 2 > G \\ A(1,2,0) & = & A(1,1,1) = 61 \\ A(1,2,1) & = & A(1,1,61) > 3 \rightarrow 3 \rightarrow 61 \rightarrow 2 \\ A(1,2,2) & \approx & A(1,1,3 \rightarrow 3 \rightarrow 61 \rightarrow 2) > 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 61 \rightarrow 2) \rightarrow 2 > A(1,1,65) \end{eqnarray*}

And therefore A(1,2,2) > A(1,1,65) > Graham's number.

The following relationship was calculated:

For \(x=1, y>1\) or \(x>1, y+z>0\)

\[A(x,y,z) < \underbrace{3 \rightarrow3 \rightarrow \cdots \rightarrow 3}_{x+1 \text{reps of} 3s} \rightarrow z+2 \rightarrow y+1 < A(x,y,z+1)\]

Therefore function A(x,y,z) has recursive level corresponding to x+3 variables of Conway's chained arrow notation, and A(1,0,1,2) exceeds Conway's chained arrow notation with Graham's numbers length.

\begin{eqnarray*} A(1,0,1,0) & = & A(1,0,0,1) = A(1,0,1) = A(1,1) = 3 \\ A(1,0,1,1) & = & A(1,0,0,A(1,0,1,0)) \\ & = & A(1,0,0,3) = A(3,0,3) = A(2,3,3) \\ A(1,0,1,2) & = & A(1,0,0,A(1,0,1,1)) \\ & = & A(1,0,0,A(2,3,3)) \\ & = & A(A(2,3,3),0,A(2,3,3)) \approx \underbrace{3 \rightarrow 3 \rightarrow 3 ... 3 \rightarrow 3 \rightarrow 3}_{A(2,3,3)} \end{eqnarray*}

The growth rate of this function was compared with fast-growing hierarchy as follows.

\begin{eqnarray*} A(n, n) & \approx & f_{\omega}(n) \\ A(1, 0, n) & \approx & f_{\omega}(n) \\ A(a, 0, n) & \approx & f_{\omega･a}(n) \\ A(n, 0, n) & \approx & f_{\omega^2}(n) \\ A(1, 0, 0, n) & \approx & f_{\omega^2}(n) \\ A(a, 0, 0, n) & \approx & f_{\omega^2･a}(n) \\ A(n, 0, 0, n) & \approx & f_{\omega^3}(n) \\ A(1, 0, 0, 0, n) & \approx & f_{\omega^3}(n) \\ A(a, 0, 0, 0, n) & \approx & f_{\omega^3･a}(n) \\ A(n, 0, 0, 0, n) & \approx & f_{\omega^4}(n) \\ A(..., a3, a2, a1, a0, n) & \approx & f_{... + \omega^3･a3 + \omega^2･a2 + \omega･a1 + a0}(n) \\ A(\underbrace{1,1,...,1}_n) & \approx & f_{\omega^\omega}(n) \\ \end{eqnarray*}

Approximations in other notations

\(A(..., d, c, b, a, n)\) is approximated as follows.

Notation	Approximation
Bowers' Exploding Array Function	\(\lbrace n,2,a+1,b+1,c+1,d+1,... \rbrace\)
Fast-growing hierarchy	\(f_{... \omega^3 d + \omega^2 c + \omega b + a}(n)\)
Hardy hierarchy	\(H_{\omega^{... \omega^3 d + \omega^2 c + \omega b + a} }(n)\)

See also

• Joyce's More Generalized Exponential Notation

Googology in Asia

Indian counting system: Lakh · Crore · Tallakshana · Uppala · Dvajagravati · Paduma · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra · Nirabhilapya nirabhilapya parivarta · Jaghanya Parīta Asaṃkhyāta
Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
See also: Template:Googology in Japan