## FANDOM

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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 2002[1] and used in the definition of Fish number 1 and Fish number 2. It is defined as

\begin{eqnarray*} S:[m,f(x)]â[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 Edit

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.[2]

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.