【ふぃっしゅ数】巨大数の探索スレ【ばーど数】

ロバートさんからの返答を書く前に、ロバートさんに送ったふぃっしゅ数
バージョン３をそのままのせておきます。

********** Definition of Fish number and function **********

(1) First step: definition of s(1) mapping

S(1) mapping is a mapping from a pair of a natural number
and a function to a pair of a natural number and a function.
One of S(1) mapping, s(1), is defined by the following rule:

s(1):[m,f(x)] --> [g(m),g(x)]

where

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)

When I denote mapping in general, I use large letter S as
in S(1) mapping, and when I denote a special mapping, I use
small letter s as in s(1). This distinction will follow.

s(1):[m,x+1] --> [g(m),g(x)] gives the Ackermann function
g(x)=ac(x,x). Applying s(1) mapping to the Ackermann function
itself will yield even larger function.

(2) Second step: definition of s(2) mapping

S(2) mapping is a mapping from a set of a natural number and
a function and an S(1) mapping to a set of a natural number and
a function and an S(1) mapping. The mapping rule of s(2), one
of the S(2) mapping, is defined by:

s(2):[m,f(x),s(1)] --> [n,g(x),s'(1)]

where

s'(1)=s(1)^f(m)
s'(1):[m,f(x)] --> [n,p(x)]
s'(1)^y:[m,f(x)] --> [q(y),r(x,y)]
g(x)=r(x,x)

(3)が２重かきこで書き込めない。ちょっと一休み。

(3) Third step: definition of s(n) mapping (n>1)

In fact (2) is included here, when n=2.

S(n) mapping (n>1) is a mapping from a set of a natural
number and a function and S(1),S(2),...,S(n-1) mappings to a
set of a number and a function and S(1),...,S(n-1) mappings.
The s(n) mapping, one of the S(n) mapping, is defined by:

s(n):[m,f(x),s(1),s(2),...,s(n-1) -->
[n,g(x),s'(1),s'(2),...,s'(n-1)]

where

s'(n-1)=s(n-1)^f(m)
s'(n-1):[m,f(x),s(1),s(2),...,s(n-2) -->
[n,p(x),s'(1),s'(2),...,s'(n-2)]
s'(n-1)^y:[m,f(x),s(1),s(2),...,s(n-2) -->
[q(y),r(x,y),s''(1)(y),s''(2)(y),...,s''(n-2)(y)]
g(x)=r(x,x)

(4) Forth step: definition of ss(1) mapping

ss(1) mapping is a kind of S(1) mapping and defined by

ss(1):[m,f(x)] --> [n,g(x)]

where

s(m+1)^f(m):[m,f(x),s(1),s(2),...,s(m)] -->
[n,g(x),s'(1),s'(2),...,s'(m)]

g(x) is a function obtained by applying s(m+1) mapping to
[m,f(x),s(1),s(2),...,s(m)] f(m) times.

(5) Last step: definition of Fish number and Fish function:

Apply s(2) mapping 63 times to [3,x+1,ss(1)]:

S(2)^63:[3,x+1,ss(1)] --> [F,F(x),ss'(1)]

We get the Fish number F and the Fish function F(x).