S map

S map is a function which maps "a pair of a natural number and a function" to "a pair of a natural number and a function". 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. It is defined as
\begin{eqnarray*}
S:[m,f(x)]→[g(m),g(x)]
\end{eqnarray*}
which means that when a pair of \(m \in \mathbb{N}\) and a function \(f(x)\) is given as input variables of S map, a pair of \(g(m) \in \mathbb{N}\) and a function \(g(x)\) is obtained as return values, where \(g(x)\) is defined as
\begin{eqnarray*}
B(0,n) & = & f(n) \\ 
B(m+1,0) & = & B(m, 1) \\ 
B(m+1,n+1) & = & B(m, B(m+1, n)) \\
g(x) & = & B(x,x) 
\end{eqnarray*}
and \(g(m)\) is calculated by substituding \(x=m\) to \(g(m)\).
\(B(m,n)\) is similar to Ackermann function except \(B(0,n) = f(n)\).
== Approximation in other notation ==
S map is similar to Taro's multivariable Ackermann function with 3 variables. By applying S map n times to [3,x+1], we get a number \(A(n,1,1)\) and a function \(A(n-1,x,x)\). Therefore, S map corresponds to adding \(\omega\) to the ordinal of FGH. At the time when \(F_1\) was developed, people at Japanese BBS didn't know FGH or multivariable Ackermann function (which was developed in 2007), but it was soon calculated that applying S map is similar to adding one to the length of the arrow of Chained arrow notation.Log of Japanese BBS, October 5, 2002
S map is used in \(F_1\) and \(F_2\), but not in Fish number 3, where s(n) map is used instead. \(F_1\) and \(F_2\) is based on S map, but later Fish found that s(2) map, which is similar to S map, is obtained with the definition of s(n) map, and the Ackermann function is not necessary in the definition.
== Sources ==
== See also ==
ja:S変換
Category:Functions
Category:Googology in Asia
Category:Notations
Category:Numbers by Fish