The SeT Function (substitution function) Version 3[]
The SeT function offers a way to generate very large string sequences (representing large numbers). It is the third version of my substitution functions and it has a growth rate \(\approx f_{LVO}(n)\).
The SeT function is based on my other substitution function called The S Function (Version 2). Refer to my other blogs for more information on my work.
What are SeT Function strings[]
The SeT function is recursively defined set of two functions \(S()\) and \(T()\) which use string substitution procedures only. They do not explicitly use any mathematical or transfinite ordinal theory. In the SeT function, the \(S()\) function behaves the same as it does in The S Function (Version 2), but the \(T()\) function has been extended. The new substitution function is called S and extended T function or SeT for short.
The growth rate of the S Function is \(f_{svo}(n)\) and the SeT Function grows at a rate 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 \(T()\) function
\(S(a,b,0) = a\) for any \(a\) or \(b\)
The extended T function is defined recursively as follows:
\(T(p_1;p_2;p_3; ... p_i)\) 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 \(T()\) function
Definition of SeT Function strings[]
In the SeT function, the \(S()\) function behaves the same as it does in The S Function (Version 2). Refer to that blog for more detail.
\(S()\) 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 my other blog The S Function (Version 2) for more detail.
The extended T Function[]
The T Function is extended to support multiple parameters and multiple partitions (of multiple parameters). The ruleset for the T Function is:
\(T(1) = S(T(0),T(0),1)\)
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