The Generalised S Function (substitution function) Version 4[]
The Generalised S function generates very large numbers. It has a growth rate \(\approx f_{LVO}(n)\).
It is the fourth version of my substitution functions. It is based on the earlier three versions but it has the fastest growth rate of them all. Refer to my other blogs for more information on my work.
What is the Generalised S Function[]
The Generalised S Function is actually two functions \(S()\) and \(g()\) where the \(S()\) function behaves the same as it does 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
The \(g()\) function uses indexes as well in the form:
\(g_m(p_1;p_2;p_3; ... p_i)\) where \(p\) is a partition of finitely many parameters \((d_1,d_2, ... d_j)\).
and \(m\) can be a finite integer \(n\), an \(S()\) function or a \(g()\) function
Ruleset for the S Function[]
The ruleset for the S Function can be defined easily:
\(S(n,0,0) = n\)
\(S(n,0,1) = n + 1\)
\(S(n,a,b) = S^b(n_*,a,1)\)
\(S(n,a,n) = S(n,S(a,0,1),1)\)
Refer to blogs for the other versions for more detail.
Ruleset for the g Function[]
The ruleset for the g Function involves more complex rules:
\(S(n,n,1) = S(n,g(0),1)\) where \(g(0)\) represents a 'wildcard' substitution for the value \(n\).
\(g(m + 1) = S^{g(m)}(g(m),g(m)_*,g(m))\)
\(g(1,0) = g^{g(0)}(g(0))\)
\(g(1,m + 1) = g^{g(1,m)}(g(1,m))\)
\(g(m + 1,0) = g^{g(m,0)}(m,g(m,0)_*)\)
\(g(m + 1,p + 1) = g^{g(m + 1,p)}(m,g(m + 1,p)_*)\)
\(g(1,0,0) = g^{g(1,0)}(g(1,0)_*,g(1,0))\)
\(g(1,0,q + 1) = g^{g(1,0,q)}(g(1,0,q)_*,g(1,0,q))\)
\(g(1,p + 1,0) = g^{g(1,p,0)}(1,p,g(1,p,0)_*)\)
\(g(1,p + 1,q + 1) = g^{g(1,p + 1,q)}(g(1,p + 1,q)_*,g(1,p + 1,q))\)
\(g(m + 1,0,0) = g^{g(m,0,0)}(m,g(m,0,0)_*,g(m,0,0))\)
\(g(m + 1,0,q + 1) = g^{g(m + 1,0,q)}(m,g(m + 1,0,q)_*,g(m + 1,0,q))\)
\(g(m + 1,p + 1,q + 1) = g^{g(m + 1,p + 1,q)}(m + 1,p,g(m + 1,p + 1,q)_*)\)
The ruleset for the \(g()\) Function with multiple partitions of parameters is:
\(g(0;0) = g(0)\)
\(g(1;0) = g(g(1,0),0_{[g(1,0)]})\)
\(g(1;m + 1) = g(g(1;m),0_{[g(1;m)]})\)
\(g(1;1,0) = g^{g(1;0)}(1;g(1;0)_*)\)
\(g(m + 1;0) = g(m;g(m;0),0_{[g(m;0)]})\)
\(g(1,0;0) = g^{g(1;0)}(g(1;0)_*;0)\)
The ruleset for the \(g()\) Function with indices is:
\(g_0(0) = g(0)\)
\(g_{m + 1}(0) = g_m(g_m(1;0);0_{[g_m(1;0)]})\)
Growth Rate of the Generalised S Function ... to \(\Gamma_0\)[]
The Generalised S Function behaves like the FGH function up to a point:
\(S(n,g,h) = f_g^h(n)\)
\(S(n,g(0),1) = f_{\omega}(n)\)
\(S(n,S(g(0),g(0),1),1) \approx f_{\varphi(\omega,0)}(n)\)
\(S(n,S^2(g(0),g(0)_*,1),1) > f_{\varphi^2(1_*,0)}(n)\)
\(S(n,g(1),1) > S(n,S^{g(0)}(g(0),g(0)_*,1),1) > f_{\varphi^{\omega}(1_*,0)}(n) = f_{\varphi(1,0,0)}(n) = f_{\Gamma_0}(n)\)
Growth Rate ... to svo[]
The Generalised S Function will eventually reach and surpass the small Veblen ordinal (svo):
\(S(n,S(g(1),g(0),1),1) \approx f_{\varphi(\omega,\varphi(1,0,0)+1)}(n)\)
\(S(n,S(g(1),S(g(0),g(0),1),1),1) \approx f_{\varphi(\varphi(\omega,0),\varphi(1,0,0)+1)}(n)\)
\(S(n,S(g(1),g(1),1),1) \approx f_{\varphi(1,0,1)}(n)\)
\(S(n,S(g(1),S(g(1),1,1),1),1) \approx f_{\varphi(1,0,2)}(n)\)
\(S(n,S(g(1),S(g(1),2,1),1),1) \approx f_{\varphi(1,0,\omega^2)}(n)\)
\(S(n,S(g(1),S(g(1),3,1),1),1) \approx f_{\varphi(1,0,\varphi(1,0))}(n)\)
\(S(n,S^2(g(1),g(1)_*,1),1) = S(n,S(g(1),S(g(1),g(1),1),1),1) \approx f_{\varphi(1,0,\varphi(1,0,0))}(n) = f_{\varphi^2(1,0,0_*)}(n)\)
\(S(n,S^{g(0)}(g(1),g(1)_*,1),1) \approx f_{\varphi^{\omega}(1,0,0_*)}(n) = f_{\varphi(1,1,0)}(n)\)
\(S(n,S^{S(g(0),1,1)}(g(1),g(1)_*,1),1) \approx f_{\varphi(1,2,0)}(n)\)
\(S(n,g(2),1) > S(n,S^{g(1)}(g(1),g(1)_*,1),1) \approx f_{\varphi^2(1,0_*,0)}(n)\)
\(S(n,S(g(2),g(2),1),1) \approx f_{\varphi(1,\varphi(1,0,0),\varphi^2(1,0_*,0))}(n) = f_{\varphi^2(1,\varphi(1,0,0),0_*)}(n)\)
\(S(n,S^{g(0)}(g(2),g(2)_*,1),1) \approx f_{\varphi(1,\varphi(1,0,0)+1,0)}(n)\)
\(S(n,S^{g(1)}(g(2),g(2)_*,1),1) \approx f_{\varphi(1,\varphi(1,0,0).2,0)}(n)\)
\(S(n,S^{g(1)}(g(2),g(2)_*,1),1) \approx f_{\varphi(1,\varphi(1,0,0).2,0)}(n)\)
\(S(n,g(3),1) > S(n,S^{g(2)}(g(2),g(2)_*,1),1) \approx f_{\varphi^3(1,0_*,0)}(n)\)
\(S(n,g^2(0),1) = S(n,g(g(0)),1) \approx f_{\varphi(2,0,0)}(n)\)
\(S(n,g^3(0),1) \approx f_{\varphi^2(2,0,0_*)}(n)\)
\(S(n,g(1,0),1) = S(n,g^{(g(0))}(g(0)),1) \approx f_{\varphi(2,1,0)}(n)\)
\(S(n,g^{g(0)}(g(1,0)),1) \approx f_{\varphi(2,1,1)}(n)\)
\(S(n,g^{S(g(0),1,1)}(g(1,0)),1) \approx f_{\varphi(2,1,2)}(n)\)
\(S(n,g^{S(g(0),2,1)}(g(1,0)),1) \approx f_{\varphi(2,1,\omega^2)}(n)\)
\(S(n,g^{S(g(0),g(0),1)}(g(1,0)),1) \approx f_{\varphi(2,1,\varphi(\omega,0))}(n)\)
\(S(n,g^{g(1)}(g(1,0)),1) \approx f_{\varphi(2,1,\varphi(1,0,0))}(n)\)
\(S(n,g(1,1),1) = S(n,g^{g(1,0)}(g(1,0)),1) \approx f_{\varphi(2,1,\varphi(2,1,0))}(n) = f_{\varphi^2(2,1,0_*)}(n)\)
\(S(n,g(1,g(0)),1) \approx f_{\varphi(2,2,0)}(n)\)
\(S(n,g(1,g(1,0)),1) \approx f_{\varphi^2(2,0_*,0)}(n)\)
\(S(n,g^{g(0)}(1,g(1,0)_*),1) \approx f_{\varphi(3,0,0)}(n)\)
\(S(n,g^{S(g(0),0,1)}(1,g(1,0)_*),1) \approx f_{\varphi(2,\varphi(3,0,0)+1,0)}(n)\)
\(S(n,g^{S(g(0),1,1)}(1,g(1,0)_*),1) \approx f_{\varphi(3,0,1)}(n)\)
\(S(n,g^{S(g(0),2,1)}(1,g(1,0)_*),1) \approx f_{\varphi(3,0,\omega^2)}(n)\)
\(S(n,g^{g(1)}(1,g(1,0)_*),1) \approx f_{\varphi(3,0,\varphi(1,0,0))}(n)\)
\(S(n,g^{g(2)}(1,g(1,0)_*),1) \approx f_{\varphi(3,0,\varphi^2(1,0_*,0))}(n)\)
\(S(n,g(2,0),1) > S(n,g^{g(1,0)}(1,g(1,0)_*),1) \approx f_{\varphi(3,0,\varphi(2,1,0))}(n)\)
\(S(n,g(2,1),1) \approx f_{\varphi^2(3,0,0_*)}(n)\)
\(S(n,g(2,g(0)),1) \approx f_{\varphi(3,1,0)}(n)\)
\(S(n,g(2,S(g(0),1,1)),1) \approx f_{\varphi(3,1,1)}(n)\)
\(S(n,g(2,g(1)),1) \approx f_{\varphi(3,1,\varphi(1,0,0))}(n)\)
\(S(n,g^2(2,0_*),1) \approx f_{\varphi^2(3,1,0_*)}(n)\)
\(S(n,g^{g(0)}(2,0_*),1) \approx f_{\varphi(3,2,0)}(n)\)
\(S(n,g^{g(1)}(2,0_*),1) \approx f_{\varphi(3,\varphi(1,0,0),0)}(n)\)
\(S(n,g(3,0),1) > S(n,g^{g(2,0)}(2,0_*),1) \approx f_{\varphi^2(3,0_*,0)}(n)\)
\(S(n,g(3,1),1) \approx f_{\varphi^3(3,0_*,0)}(n)\)
\(S(n,g(3,g(0)),1) \approx f_{\varphi^{\omega}(3,0_*,0)}(n) = f_{\varphi(4,0,0)}(n)\)
\(S(n,g(4,1),1) \approx f_{\varphi^2(4,0_*,0)}(n)\)
\(S(n,g(4,g(0)),1) \approx f_{\varphi(5,0,0)}(n)\)
\(S(n,g(g(0),g(0)),1) \approx f_{\varphi(\omega,0,0)}(n)\)
\(S(n,g^2(g(0),g(0)),1) \approx f_{\varphi^2(1_*,0,0)}(n)\)
\(S(n,g(1,0,0),1) > S(n,g^{g(0)}(g(0),g(0)),1) \approx f_{\varphi(1,0,0,0)}(n)\)
\(S(n,g(1,0_{[g(0)]}),1) > S(n,g(1,0_{[n-1]}),1) \approx f_{\varphi(1,0_{[n]})}(n) = f_{svo}(n)\)
Growth Rate ... to LVO[]
The Generalised S Function is one of the Fastest Computable functions:
\(g(0) \approx \omega = \vartheta(0)\)
\(S(g(0),3,1) \approx \epsilon_0 = \varphi(1,0) = \vartheta(1)\)
\(g(1) \approx \Gamma_0 = \varphi(1,0,0) = \vartheta(\Omega^2)\)
\(g(1;0) > g(1,0_{[g(0)]}) \approx svo = \vartheta(\Omega^\omega)\)
\(g(1;0_{[g(0)]}) \approx \vartheta(\Omega^\omega\omega)\)
TREE(n) function \(≥ f_{\vartheta(\Omega^\omega\omega)}(n)\)
\(g_1(0) > g(1;0_{[g(1;0)]}) \approx \vartheta(\Omega^{\omega+1})\)
\(g_1(1) \approx \vartheta(\Omega^{\omega+2})\)
\(g_1^2(0) > g_1(g_1(0)) \approx \vartheta(\Omega^{\omega.2})\)
\(g_1(1,0) \approx \vartheta(\Omega^{\omega.3})\)
\(g_1(1;0) \approx \vartheta(\Omega^{\omega^2})\)
\(g_1(1;0_{[2]}) = g_1(1;0;0) \approx \vartheta(\Omega^{\omega^3})\)
\(g_1(1;0_{[g(0)]}) \approx \vartheta(\Omega^{\omega^{\omega}})\)
\(g_2(0) \approx \vartheta(\Omega^{\omega^{\omega^{\omega}}}) = \vartheta(\Omega^{\omega\uparrow\uparrow 3})\)
\(g_3(0) \approx \vartheta(\Omega^{\omega\uparrow\uparrow 4})\)
\(g_{g(0)}(0) \approx \vartheta(\Omega^{\omega\uparrow\uparrow\omega}) = \vartheta(\Omega^{\varphi(1,0)})\)
\(g_{S(g(0),1,1)}(0) \approx \vartheta(\Omega^{\varphi(1,1)})\)
\(g_{S(g(0),2,1)}(0) \approx \vartheta(\Omega^{\varphi(1,\omega^2)})\)
\(g_{S(g(0),3,1)}(0) \approx \vartheta(\Omega^{\varphi(1,\varphi(1,0))}) = \vartheta(\Omega^{\varphi^2(1,0_*)})\)
\(g_{S(g(0),g(0),1)}(0) \approx \vartheta(\Omega^{\varphi(2,0)})\)
\(g_{g(1)}(0) \approx \vartheta(\Omega^{\varphi(1,0,0)})\)
\(g_{g(1;0)}(0) \approx \vartheta(\Omega^{\Omega})\)
Large Veblen ordinal \(LVO ≥ f_{\vartheta(\Omega^\Omega)}(n)\)
\(g_{g(2;0)}(0) \approx \vartheta(\Omega^{\Omega^2})\)
Bird's H(n) function \(\approx f_{\vartheta(\varepsilon_{\Omega+1})}(n) = f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\)
Further References[]
Further references to relevant blogs can be found here: User:B1mb0w