User blog:B1mb0w/Generalised S Function

The Generalised S Function (substitution function) Version 4
The Generalised S function generates very large string sequences (representing large numbers). It is the fourth version of my substitution functions and it has a growth rate \(\approx f_{LVO}(n)\).

This version is based on my earlier three versions substitution functions and it has the fastest growth rate of them all. Refer to my other blogs for more information on my work.

What are Generalised S Function strings
Strings can be recursively defined with two functions \(S\) and \(g\) which use string substitution procedures only. They do not explicitly use any mathematical or transfinite ordinal theory. The \(S\) function behaves the same as it does in in each of the other versions, but the \(g\) function has been defined to create a faster growth rate.

The growth rate of the Generalised S Function is beyond \(f_{LVO}(n)\).

The S function is defined recursively as follows:

\(S(a,b,c)\) where \(a,b,c\) can be finite integer \(n\), an \(S\) function or a \(g\) function

\(S(a,b,0) = a\) for any \(a\) or \(b\)

The \(g\) function is defined recursively as follows:

\(g(p_1;p_2;p_3; ... p_i)\) where \(p\) is a partition of finitely many parameters \((d_1,d_2, ... d_j)\).

Each parameter \(d\) can be a finite integer \(n\), an \(S\) function or a \(g\) function

Definition of valid strings
\(S\) function strings can be equivalent (=) or in an ascending order (<). This is evaluated between arbitrary S function strings using two string substitution procedures:

\(sub\)

\(dec\)

Refer to blogs for the other versions for more detail.

The \(g\) Function
The \(g\) Function supports multiple parameters and multiple partitions (of multiple parameters). The ruleset for the \(g\) Function is:

\(g(1) = S(g(0),g(0),1)\)

\(g(2) = S(g(1),S(g(1),g(1),1),1) = S^2(g(1),g(1)_*,1)\)

or

\(T(d) = S(T(dec(d)),S(dec(d)),1)\)

The ruleset for the extended T Function with multiple parameters is:

\(T(1,0) = T^{T(0)}(0)\)

\(T(1,m + 1) = T^{T(1,m)}(T(1,m))\)

\(T(2,0) = T^{T(1,0)}(1,0_*)\)

\(T(2,m + 1) = T^{T(2,m)}(1,0_*)\)

\(T(k + 1,0) = T^{T(k,0)}(k,0_*)\)

\(T(k + 1,m + 1) = T^{T(k + 1,m)}(k,0_*)\)

\(T(1,0,0) = T^{T(1,0)}(T(0)_*,T(0))\)

\(T(1,0,1) = T^{T(1,0,0)}(T(0)_*,T(0))\)

\(T(1,1,0) = T^{T(1,0,0)}(1,0,0_*)\)

\(T(2,0,0) = T^{T(1,0,0)}(1,0_*,0)\)

\(T(2,0,1) = T^{T(2,0,0)}(1,0_*,0)\)

\(T(2,1,0) = T^{T(2,0,0)}(2,0,0_*)\)

The ruleset for the extended T Function with multiple partitions of parameters is:

\(T(0;0) = T(0)\)

\(T(1;0) = T(0;1,0_{[T(1,0)]}) = T(1,0_{[T(1,0)]})\)

\(T(1;m + 1) = T(1,0_{[T(1,m)]})\)

\(T(1;1,0) = T^{T(1;0)}(1;0_*)\)

\(T(1;1,1) = T^{T(1;1,0)}(1;0_*)\)

\(T(1;2,0) = T^{T(1;1,0)}(1;1,0_*)\)

\(T(2;0) = T(1;1,0_{[T(1,0)]})\)

\(T(j + 1;0) = T(j;1,0_{[T(j,0)]})\)

\(T(1;0;0) = T(1,0_{[T(1,0)]};0)\)

Some Example SeT Function strings
Here are some example SeT Function strings in ascending order:

\(0\)

\(1\)

\(2\)

\(3\)

\(S(3,0,1)\)

\(S(3,0,2)\)

\(S(3,1,1)\)

\(S(S(3,1,1),0,1)\)

\(S(S(3,1,1),0,2)\)

\(S(S(3,1,1),0,S(3,0,1))\)

\(S(3,1,2)\)

\(S(3,2,1)\)

\(S(3,T(0),1)\)

\(S(3,T(0),2)\)

\(S(3,S(T(0),0,1),1)\)

\(S(3,S(T(0),0,1),2)\)

\(S(3,S(T(0),0,2),1)\)

\(S(3,S(T(0),1,1),1)\)

\(S(3,S(T(0),2,1),1)\)

\(S(3,T(1),1)\)

\(S(3,S(T(1),0,T(0)),1)\)

\(S(3,S(T(1),T(0),0),1)\)

\(S(3,T(2),1)\)

\(S(3,T^2(0),1)\)

\(S(3,T(1,0),1)\)

\(S(3,T(1,0,0),1)\)

\(S(3,T(1,0_{[T(0)]}),1)\)

\(S(3,T(1,0_{[T(1)]}),1)\)

\(S(3,T(1,0_{[T(2)]}),1)\)

\(S(3,T(1,0_{[T^2(0)]}),1)\)

\(S(3,T(1;0),1)\)

\(S(3,T(1;1),1)\)

\(S(3,T(1;T(0)),1)\)

\(S(3,T(1;T^2(0)),1)\)

\(S(3,T(1;1,0),1)\)

\(S(3,T(1;1,0,0),1)\)

\(S(3,T(1;1,0_{[T(0)]}),1)\)

\(S(3,T(1;1,0_{[T(1)]}),1)\)

\(S(3,T(1;1,0_{[T(2)]}),1)\)

\(S(3,T(1;1,0_{[T^2(0)]}),1)\)

\(S(3,T(2;0),1)\)

\(S(3,T(T(0);0),1)\)

\(S(3,T(T^2(0);0),1)\)

\(S(3,T(1,0;0),1)\)

\(S(3,T(1,0,0;0),1)\)

\(S(3,T(1;0;0),1) = S(3,T(1,0_{[T(1,0)]};0),1) = S(3,T(1,0_{[T^3(0)]};0),1)\)

Growth Rate of the SeT Function
The number of SeT Function strings that has a growth rate beyond \(f_{LVO}(n)\). Here are some example calculations:

\(S(n,T(0),1) = f_{\omega}(n)\)

\(S(n,S(T(0),3,1),1) > f_{\varphi(1,0)}(n) = f_{\epsilon_0}(n)\)

\(S(n,T(1),1) \approx f_{\varphi(\omega,0)}(n)\)

\(S(n,T(2),1) \approx f_{\varphi^2(1_*,0)}(n) = f_{\varphi(\varphi(1,0),0)}(n)\)

\(S(n,T(T(0)),1) \approx f_{\varphi(1,0,0)}(n)\)

\(S(n,T(1,0),1) \approx f_{\varphi(1,1,0)}(n)\)

\(S(n,T^{T(0)}(T(1,0)),1) \approx f_{\varphi(1,2,0)}(n)\)

\(S(n,T(1,1),1) \approx f_{\varphi^2(1,1_*,0)}(n)\)

\(S(n,T(1,m + 1),1) \approx f_{\varphi(1,T(1,m),0)}(n)\)

\(S(n,T(1,T(0)),1) \approx f_{\varphi(2,0,0)}(n)\)

\(S(n,T(1,T(m)),1) \approx f_{\varphi(2,0,T(m))}(n)\)

\(S(n,T(2,T(0)),1) \approx f_{\varphi(3,0,0)}(n)\)

\(S(n,T(T(0),T(0)),1) > f_{\varphi(\omega,0,0)}(n)\)

\(S(n,T^{T(0)}(T(0)_*,T(0)),1) > f_{\varphi(1,0,0,0)}(n)\)

\(S(n,T^{S(T(0),1,1)}(T(0)_*,T(0)),1) > f_{\varphi(1,0,0,1)}(n)\)

\(S(n,T(1,0,0),1) > f_{\varphi(1,0,0,\varphi(1,1,0))}(n)\)

\(S(n,T(1,0_{[m]}),1) > f_{\varphi(1,0_{[m + 1]})}(n)\)

The SeT Function is one of the Fastest Computable functions:

tree(n) function \(\approx f_{\vartheta(\Omega^\omega)}(n)\)

\(S(n,T(1,0_{[T(0)]}),1) > f_{\varphi(1,0_{[\omega]})}(n) = f_{svo}(n) = f_{\vartheta(\Omega^\omega)}(n)\)

\(S(n,T(1,0_{[S(T(0),1,1)]}),1) > f_{\varphi(1,0_{[\omega.2]})}(n)\)

\(S(n,T(1,0_{[S(T(0),1,2)]}),1) > f_{\varphi(1,0_{[\omega.4]})}(n)\)

\(S(n,T(1,0_{[S(T(0),2,1)]}),1) > f_{\varphi(1,0_{[\omega^2]})}(n)\)

\(S(n,T(1,0_{[T(1)]}),1) > f_{\varphi(1,0_{[\varphi(\omega,0)]})}(n)\)

\(S(n,T(1,0_{[T^2(0)]}),1) > f_{\varphi(1,0_{[\varphi(1,0,0)]})}(n)\)

\(S(n,T(1;0),1) = S(n,T(1,0_{[T(1,0)]}),1) > f_{\varphi(1,0_{[\varphi(1,1,0)]})}(n)\)

TREE(n) function \(≥ f_{\vartheta(\Omega^\omega\omega)}(n)\)

\(S(n,T(1;1),1) = S(n,T(1,0_{[T(1;0)]}),1) > f_{\varphi(1,0_{svo})}(n)\)

Large Veblen Ordinal (LVO) \(≥ f_{\vartheta(\Omega^\Omega)}(n)\)

Bird's H(n) function \(\approx f_{\vartheta(\varepsilon_{\Omega+1})}(n)\)

Further References
Further references to relevant blogs can be found here: User:B1mb0w