User:Kyodaisuu/Sandbox

This is a personal sandbox to prepare for publishing at this wiki. ja:利用者:Kyodaisuu/砂場

Fish number 4 (F4), is an uncomputational number defined by Japanese googologist Fish in 2002. It is one of Fish numbers.

s'(1) map is a functional mapping from function \(f\) to function \(g\), as follows.


 * Function \(g\)is a busy beaver function with an having an oracle which calculates function \(f\). That is, with Turing machine + function \(f\), the maximum possible numbers of 1 that can be set with n-state machine is \(g(n)\).

By comparing to order-x busy beaver function \(\Sigma_x(n)\), which has growth rate of \(f_{\omega^\text{CK}_x}(n)\) in FGH, let \(f(x) = x+1\) and

\begin{eqnarray*} s'(1)f & = & \Sigma_x(1) \approx f_{\omega^\text{CK}_1}(n) \\ s'(1)^2f & = & \Sigma_x(2) \approx f_{\omega^\text{CK}_2}(n) \\ s'(1)^3f & = & \Sigma_x(3) \approx f_{\omega^\text{CK}_3}(n) \\ s'(1)^xf & = & \Sigma_x(n) \approx f_{\omega^\text{CK}_n}(n) \\ s'(1)^nf & = & \Sigma_n(n) \approx f_{\omega^\text{CK}_\omega}(n) \end{eqnarray*}

After this, definition is similar to Fish number 3;

\begin{eqnarray*} s'(n)f & = & s'(n-1)^{x}f(x) (\text{if } n>1) \\ ssʹ(1)f & = & sʹ(x)f(x) \\ ssʹ(n)f & = & [ssʹ(n − 1)^{x}]f(x) (\text{if } n>1) \\ F_4(x) & = & ssʹ(2)^{63}f; f(x) = x + 1 \\ F_4 & = & F_4^{63}(3) \end{eqnarray*}

As \(F_3(n) \approx f_{(\omega^{\omega+1}) 63}(n)\), \(F_4(n) \approx f_{\omega^\text{CK}_{(\omega^{\omega+1}) 63}}(n)\).

Therefore, \(F_4 \approx f_{[\omega^\text{CK}_{(\omega^{\omega+1}) 63}] + 1}(63)\).