Fish number 4

Fish number 4 (F4) is a number defined by Japanese googologist Fish in 2002.Fish, Googology in Japan - exploring large numbers (2013) It is the smallest of the Fish numbers that is defined using an uncomputable function.
s'(1) map is a function which maps functions to functions, as follows.
:Function \(s'(1)f\) is a busy beaver function for an having an oracle which calculates function \(f\). That is, the maximum possible numbers of ones that can be written with an n-state, two-color oracle Turing machine is \(s'(1)f(n)\).
By comparing with the order-n busy beaver function \(\Sigma_n(x)\), let \(f\) be a computable function. Then it's easy to see that (exponents mean iteration of the map here):
\begin{eqnarray*}
s'(1)f & = & \Sigma_1(x)\\
s'(1)^2f & = & \Sigma_2(x)\\
s'(1)^3f & = & \Sigma_3(x)\\
s'(1)^nf & = & \Sigma_n(x)\\
s'(1)^xf & = & \Sigma_x(x) & = & \Sigma_{\omega}(x)\end{eqnarray*}
For \(n>1\), \(s'(n)\) map is defined similar to the s(n) map,
\begin{eqnarray*}
s'(n)f & = & s'(n-1)^{x}f(x) (\text{for } n>1) \\
\end{eqnarray*}
Its analysis is exactly the same except for ordinals being indices of oracle and not of FGH:
\begin{eqnarray*} s'(2)f & = & s'(1)^xf(x) & = & \Sigma_{\omega}(x) \\ s'(1)s'(2)f & = & \Sigma_{\omega+1}(x) \\ s'(2)^2f & = & \Sigma_{\omega \times 2}(x) \\ s'(3)f & = & \Sigma_{\omega^2}(x) \\ s'(4)f & = & \Sigma_{\omega^3}(x) \\ s'(n)f & = & \Sigma_{\omega^{n-1}}(x) \\ s'(x)f & = & \Sigma'_{\omega^\omega}(x) \end{eqnarray*}
where \(\Sigma'_\alpha(x)\) is oracle busy beaver function, where Wainer hierarchy is used with a single modification: \(\omega^\omega [n] = \omega^{n-1}\).
After this, the definition is similar to Fish number 3;
\begin{eqnarray*}
ssʹ(1)f & = & sʹ(x)f(x) \\
ssʹ(n)f & = & [ssʹ(n − 1)^{x}]f(x) (\text{for } n>1) \\
F_4(x) & = & ssʹ(2)^{63}f; f(x) = x + 1 \\
F_4 & = & F_4^{63}(3) & = & \Sigma'^{63}_{\omega^{\omega+1}63}(3)
\end{eqnarray*}
== Sources ==
== See also ==
ja:ふぃっしゅ数バージョン4
Category:Numbers
Category:Googology in Asia
Category:Uncomputable
Category:Eponymous numbers
Category:Non-powers
Category:Fish number
Category:Numbers by Fish
Category:Numbers based off of the busy beaver