User blog:B1mb0w/Growth Rate of the S Function

Growth Rate of the S Function
This blog will provide a detailed calculation and references for the growth rate of The S Function that I have developed. The growth rate is comparable to:

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

Introduction
The S function is recursively defined set of two functions \(S\) and \(T\) which use string substitution procedures only. S Functions can be either restricted or generalised. Refer to my main blog on The S Function for a full definition of how the function is constructed.

As a simple introduction it will be useful to compare a typical S function with more familiar functions:

\(S(3,2,1) = f_2(3)\) ordinal value

This equivalence is intentional. In fact:

\(S(n,g,h) = f_g^h(n)\) ordinal value

This equivalence will become more obvious if you refer to the definitions and rules for the function. I have intentionally defined The S Function to be a string substitution function with no explicit mathematical (e.g. algebraic) connotations. However the above equivalences will re-occur throughout the calculations. It is based on the definition that all restricted \(S\) functions belong to a well-ordered sequence and every individual restricted \(S\) function will have an assigned ordinal value equivalent to the result of the equivalent FGH function. In the case above; \(S(3,2,1)\) has the ordinal value 24.

The \(T\) function can be compared to various ordinals.

\(T(0) == \omega\) The similarity is intentional.

The generalised S Functions, the functions of the form \(S(T(0),g,h)\), are similar to the ordinals that arise from the various attempts to define an FGH of Omega function, therefore:

\(S(T(0),0,1) == \omega + 1\)

\(S(T(0),1,1) == \omega.2\)

The similarities are intentional. However, the similarity fades as we examine larger and larger inputs to the \(T\) function.

We can now start calculating the Growth Rate of the restricted S Function.

Summary of Growth Rate Calculations
Here is a summary of the Growth Rate Calculations that are explained in more detail in the following sections:

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

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

\(S(n,S(T(0),1,1),1) = f_{\omega.2}(n)\)

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

\(S(n,S(T(0),2,2),1) > f_{\omega\uparrow\uparrow 2}(n)\)

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

\(S(n,S(T(0),4,1),1) > f_{\varphi(2,0)}(n) = f_{\zeta_0}(n) = f_{\psi(\Omega)}(n)\)

\(S(n,S(T(0),5,1),1) > f_{\varphi(3,0)}(n) = f_{\psi(\Omega^2)}(n)\)

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

\(S(n,T(1),1)\) is comparable to \(f_{\varphi(\omega,0)}(n)\)

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

\(S(n,T(3),1) > f_{\varphi^3(1_*,0)}(n)\)

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

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

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

The S Function is one of the Fastest Computable functions:

S Function \(\approx f_{\vartheta(\Omega^\omega)}(n)\)

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

Growth Rate up to \(S(n,T(0),1)\)
All restricted S Functions belong to a well-ordered sequence and every individual restricted \(S\) function will have an assigned ordinal value. By definition:

\(S(0,0,0) = 0 = 0\) ordinal value

\(S(1,0,0) = 1 = 1\) ordinal value

\(S(n,0,0) = n = n\) ordinal value

\(S(n,g,h) = f_g^h(n)\) ordinal value

Therefore the Growth Rate of the S Function up to \(S(n,g,h)\) where \(g\) is a finite integer, grows at a rate comparable to \(f_{\omega}^h(n)\), and no additional proof should be required. In fact:

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

Growth Rate up to \(S(n,S(T(0),2,2),1)\)
From here we can refer to the various blogs for an FGH of Omega function to calculate the growth rate up to \(S(n,T(1),1)\).

We start by reviewing some general behaviour of The S Function:

\(S(n,S(T(0),p,T(0)),1) = S(n,S(T(0),p + 1,1),n)\)

\(S(n,S(S(T(0),p + 1,1),0,1),1) = S(n,S(T(0),p + 1,1),n)\)

\(S(n,S(S(T(0),p + 1,1),0,T(0)),1) = S(n,S(S(T(0),p + 1,1),0,n),1)\)

\(S(n,S(S(T(0),p + 1,1),0,S(T(0),p + 1,1)),1) = S(n,S(S(T(0),p + 1,1),1,1),1)\)

And some comparative behaviour:

\(S(n,S(T(0),1,1),1) = S(n,S(T(0),0,n-1),n)\)

\(S(n,S(T(0),1,1),1) = f_{\omega.2}(n)\) ordinal value

\(S(n,S(T(0),0,n-1),n) = f_{\omega + n-1}^n(n)\) ordinal value

Therefore the Growth Rate up to \(S(n,T(1),1)\) can be calculated as follows:

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

\(S(n,S(T(0),1,1),1) = f_{\omega.2}(n)\) ordinal value

\(S(n,S(S(T(0),1,1),0,1),1) = f_{\omega.2 + 1}(n)\) ordinal value

\(S(n,S(S(T(0),1,1),0,T(0)),1) = f_{\omega.2 + n}(n) = f_{\omega.3}(n)\) ordinal value

\(S(n,S(S(T(0),1,1),0,S(T(0),1,1)),1) = S(n,S(T(0),1,2),1) = f_{\omega.2 + \omega.2}(n) = f_{\omega.4}(n)\) ordinal value

\(S(n,S(T(0),2,1),1) >> f_{\omega.2^n}(n) >> f_{\omega.\omega}(n) = f_{\omega^2}(n)\) ordinal value

And

\(S(n,S(T(0),2,2),1) = S(n,S(S(T(0),2,1),1,S(T(0),2,1)),1) >> f_{\omega^{\omega}}(n)\) ordinal value

Because

\(S(n,S(T(0),2,2),1) >> f_{\omega^2.2^{\omega^2}}(n) >> f_{\omega^{\omega}}(n)\) ordinal value

where \(n^2.2^{n^2}(n) = n^2.2^{n.n} = n^2.(2^n)^n >> n^2.n^n = n^{n + 2} >> n^n\)

Growth Rate up to \(S(n,S(T(0),3,1),1)\)
From here we can show:

\(S(n,S(T(0),2,3),1) = S(n,S(S(T(0),2,2),1,S(T(0),2,2)),1) >> f_{\omega\uparrow\uparrow 3}(n)\) ordinal value

Because

\(S(n,S(T(0),2,3),1) >> f_{n^n.2^{n^n}}(n) >> f_{n\uparrow\uparrow 3}(n) = f_{\omega\uparrow\uparrow 3}(n)\) ordinal value

where \(n^{n + 2}.2^{n^{n + 2}} = n^{n + 2}.2^{n^n.n^2} = n^{n + 2}.(2^{n^2})^{n^n} >> n^{n + 2}.n^{n^n} = n^{n^n + n + 2} >> n^{n^n} = n\uparrow\uparrow 3\)

The general proof is as follows:

When

\(S(n,S(T(0),2,m),1) = f_{\omega^{(\omega\uparrow\uparrow m-1) + 1}}(n) >> f_{\omega^{\omega\uparrow\uparrow m-1}}(n) = f_{\omega\uparrow\uparrow m}(n)\) ordinal value

Then

\(S(n,S(T(0),2,m + 1),1) >> f_{\omega\uparrow\uparrow (m + 1)}(n)\) ordinal value

where \(n^{(n\uparrow\uparrow m-1) + 1}.2^{n^{(n\uparrow\uparrow m-1) + 1}} >> n.2^{n^{(n\uparrow\uparrow m-1) + 1}}\)

\(= n.2^{n^{n\uparrow\uparrow m-1}.n} = n.(2^n)^{n^{n\uparrow\uparrow m-1}} = n.(2^n)^{n\uparrow\uparrow m}\)

\(>> n.n^{n\uparrow\uparrow m} = n^{(n\uparrow\uparrow m) + 1} >> n^{n\uparrow\uparrow m} = n\uparrow\uparrow (m+1)\)

Until

\(S(n,S(T(0),3,1),1) >> f_{\omega\uparrow\uparrow n}(n) = f_{\varphi(1,0)}(n) = f_{\epsilon_0}(n) = f_{\psi(0)}(n)\) ordinal value

Here are some examples of this equality:

\(S(2,S(T(0),3,1),1) >> S(2,S(T(0),1,1),1) = f_{\omega.2}(2) = f_{\omega^2}(2) = f_{\epsilon_0}(2)\) ordinal value

\(S(3,S(T(0),3,1),1) >> S(3,S(S(T(0),2,2),1,15),1) = f_{\varphi(1,0)}(3) = f_{\epsilon_0}(3) = f_{\psi(0)}(3)\) ordinal value

\(S(4,S(T(0),3,1),1) >> S(3,S(S(S(T(0),2,2),1,442),2,1),1) >> f_{\varphi(1,0)}(4) = f_{\epsilon_0}(4) = f_{\psi(0)}(4)\) ordinal value

because

\(S(4,S(S(T(0),2,2),1,442),1) = f_{\omega\uparrow\uparrow 3}(4)\) ordinal value

\(S(4,S(T(0),2,3),1) >> f_{\omega^{(\omega\uparrow\uparrow 2) + 1}}(4)\) ordinal value

\(S(4,S(T(0),3,1),1) = S(4,S(T(0),2,T(0)),1) = S(4,S(T(0),2,4),1) >> f_{\omega^{(\omega\uparrow\uparrow 3) + 1}}(4)\)

\(>> f_{\omega^{\omega\uparrow\uparrow 3}}(4) = f_{\varphi(1,0)}(4) = f_{\epsilon_0}(4)\)

Growth Rate up to \(S(n,T(1),1)\)
Based on the definition of:

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

We can calculate:

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

and

\(S(4,T(1),1) = S(4,S(T(0),T(0),1),1) = S(4,S(T(0),4,1),1) > f_{\varphi(2,0)}(4)\)

\(S(5,T(1),1) = S(5,S(T(0),T(0),1),1) = S(5,S(T(0),5,1),1) > f_{\varphi(3,0)}(5)\)

then

\(S(5,S(T(1),0,1),1) = S(5,T(1),5) > S(f_{\varphi(3,0)}(5),T(1),4) \approx f_{\varphi(\omega,0)+1}(5)\)

and

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

Growth Rate up to \(S(n,T(2),1)\)
Based on the definition of:

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

We can calculate:

\(S(n,T(2),1) = S(n,S(T(1),T(1),1),1) = S(n,S(T(1),S(T(0),T(0),1),1),1)\)

\(= S(n,S(T(1),S(T(0),n,1),1),1)\)

From previous calculations:

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

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

\(S(n,S(T(1),1,2),1) \approx f_{\varphi(\omega,0).4}(n)\)

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

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

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

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

\(S(n,S(T(1),2,S(T(0),1,2)),1) \approx f_{\varphi(1,\varphi(\omega,0)+4)}(n)\)

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

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

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

\(S(n,S(T(1),2,S(T(0),4,1)),1) \approx f_{\varphi(1,\varphi(\omega,0)+\varphi(1,0))}(n)\)

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

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

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

and

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

Growth Rate up to \(S(n,T(m),1)\)
Based on the definition of:

\(T(m + 1) = S(T(m),T(m),1)\)

We can calculate:

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

\(S(n,T(3),1) \approx f_{\varphi^3(1_*,0)}(n)\)

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

Growth Rate up to \(S(n,T^m(0),1)\)
The growth rate is comparable to \(f_{svo}(n)\) :

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

\(S(n,T^m(0),1) \approx f_{\varphi(1,0_{[m]})}(n) = f_{svo}(n)\)

Alternative Calculations of \(S\) Growth Rates
I tried some alternative calculations but have since given up on them. The following blogs are no longer relevant:
 * Comparing T Functions to Ordinals
 * Omega Count

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