SS map

SS map is a function which maps "a set of a natural number and a function and S map" to "a set of a natural number and a function and S map". It was defined by Japanese googologist Fish in 2002Fish, Googology in Japan - exploring large numbers (2013) and used in the definition of Fish number 1 and Fish number 2.
== SS map in \(F_1\) ==
\begin{eqnarray*}
SS:[m,f(x),S]→[n,g(x),S2] 
\end{eqnarray*}
Here, S2, n, and g(x) are defined as follows.
\begin{eqnarray*}
S2 & = & S^{f(m)} \\
S2 & : & [m,f(x)]→[n,g(x)]
\end{eqnarray*}
== SS map in \(F_2\) ==
\begin{eqnarray*}
SS:[m,f(x),S]→[n,g(x),S2]
\end{eqnarray*}
Here, S2, n, and g(x) are defined as follows.
\begin{eqnarray*}
S2 & = & S^{f(m)} \\
S2:[m,f(x)] & → & [n,p(x)] \\
S2^x:[m,f(x)] & → & [q,g(x)] \\
\end{eqnarray*}
== Difference of SS map in \(F_1\) and \(F_2\) ==
In \(F_1\), SS map repeats S map \(f(m)\) times, while in \(F_2\), SS map diagonizes the repetition time of S map. It looks similar but actually this is a big difference, because the SS map in \(F_1\) has the growing level of \(A(0,1) → A(1,0,1,1) → A(1,0,1,2) → A(1,0,1,3)\), while the SS map in \(F_2\) has the growing level of \(A(0,1)→A(1,0,0,n)→A(2,0,0,n)→A(3,0,0,n)\).
As the essential growing mechanism in \(F_2\) is to "diagonizing the functional", in Fish number 3, the definition was simplified. The idea of SS map, i.e., making higher order functional to produce fast-growing function, was not effective in \(F_1\) and \(F_2\) and therefore discarded in \(F_3\); and it was finally realized effectively in Fish number 5.
== Sources ==
== See also ==
ja:SS変換
Category:Functions
Category:Googology in Asia
Category:Notations
Category:Numbers by Fish