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The S Function (substitution function)

I have written a Version 2 of this function that generalises the function and significantly increases its growth rate. Please refer to that version instead.

The S function offers a way to generate very large string sequences (representing large numbers) which grows at a faster rate than $$f_{SVO}(n)$$, and may be possible to extend its range to grow at a comparable rate to Ordinal Collapsing Functions.

Refer to my The Alpha Function blogs for more information on my work.

What is the S Function

The S function is a string substitution function, and does not explicitly use any mathematical or transfinite ordinal theory. It is effectively a computer algorithm that can be converted to a program. My next version of my Alpha Function will do this and present the results.

The S function is defined recursively as follows:

$$S_b^c(a)$$ where $$a,b,c$$ are S functions

Definition of the S Function

Starting with:

$$S_b^c(a)$$ where $$a,b,c$$ are S functions

We define a valid S function recursively as $$a_m$$ where:

$$a_0 = S_{b_0 = d_p}^{c_o < n}(n)$$

• and
• $$d_0 = S_{b_0 < h_q}^{c_o < k_r}(n$$ or $$s_n)$$
• where
• $$h_q$$ is a valid S function with this additional restriction that it uses either:
• $$n$$
• smaller subscript $$n$$ in $$s_n$$ than that used in $$d_0$$
• $$k_r$$ is a valid S function with this additional restriction $$c_0 = 0$$
• then $$a_n$$ and $$d_m$$ can be constructed using this substitution procedure on $$j_n$$:
• $$j_{n+1} = S_{b_{n+1} < b_n}^{c_{n+1} < j_n}(j_n)$$

Note that $$d_0$$ and $$d_p$$ are not valid S functions. Therefore a valid S function $$a_m$$ will never be equivalent to $$d_0$$ or $$d_p$$.

However $$d_0$$ and $$d_p$$ are valid components of a valid S function. Therefore they will appear inside a valid S function $$a_m$$.

Any S Function of the form ($$c = 0)$$ collapses to:

$$S_b^0(a) = a$$

The S Function is familiar with but not identical to FGH Functions.

Definition of the wildcard character $$s_n$$

A wildcard character $$s_n$$ is used to enable further string substitutions in the S Function. There is a hierarchy of $$s_n$$ substitution wildcards, starting with $$s_0$$. Typically $$s_0$$ is substituted for a finite number $$n$$.

The role of $$s_0$$ substitution wildcard, is similar to the role of $$\omega$$ in FGH functions.

An Example of an S Function

An example of a valid S Function $$a_m$$ is:

• Let $$m = 0, n = 3, b_0 = d_0, c_0 = 1$$

$$a_m = a_0 = S_{d_0}^{1}(3) = S_{d_0}(3)$$

• Where
• $$d_0 = S_{b_0 < h_q}^{c_o < k_r}(n$$ or $$s_n)$$
• Let
• $$n = 0, b_0 = h_0, c_0 = k_0$$

$$a_m = S_{S_{h_0}^{k_0}(s_0)}(3)$$

• Where
• $$h_0$$ is a valid S function with this additional restriction that it uses either:
• $$n$$
• smaller subscript $$n$$ in $$s_n$$ than that used in $$d_0$$
• Let
• $$n = 5, c_0 = 0$$

$$a_m = S_{S_{5}^{k_0}(s_0)}(3)$$

• Where
• $$k_0$$ is a valid S function with this additional restriction $$c_0 = 0$$
• Let
• $$n = 1$$

$$a_m = S_{S_{5}^{1}(s_0)}(3) = S_{S_{5}(s_0)}(3)$$

It will be shown later in this blog that the number of valid S Functions that can be constructed up to this example is greater than $$f_{\zeta_0}(3)$$.

String Substitution Rule-Set

The S function only uses string substitution. Substitutions can occur using one of two functions (or substitution algorithms):

Sub()

Dec()

Rule-set for the Sub() Algorithm: The Sub() algorithm performs string substitutions on one S Function, resulting in equivalent S Functions. For any arbitrary S Function $$S_b^c(a)$$ where $$c,b,a$$ are any S Functions:

$$c$$ $$b$$ $$Sub(S_b^c(a))$$
$$s_{n>0}$$ any $$= S_b^{Sub(c)}(a)$$
$$s_0$$ any $$= S_b^a(a)$$
$$1$$ $$s_{n>0}$$ $$= S_{Sub(b)}(a)$$
$$s_0$$ $$= S_a(a)$$
$$S()$$ $$= S_{Dec(b)}(a)$$
$$S()$$ any $$= S_{Sub(b)}(S_b^{Dec(c)}(a))$$

Rule-set for the Dec() Algorithm: For any arbitrary S Function $$S()$$, the Dec() algorithm involves two steps:

1. Recursively apply the Sub() function on $$S()$$ until:

$$S() = Sub^{z}(S()) = S_0^{m}(S_b^c(a))$$

2. Decrement $$m$$ by 1:

$$Dec(S()) = Dec(Sub^{z}(S())) = S_0^{m'}(S_b^c(a))$$ where

$$m' = Dec(m)$$ when $$m$$ is an S Function or $$m' = m - 1$$ when $$m$$ is a finite integer.

Rule-set for applying Sub() Algorithm to $$s_{n>0}$$ substitution wildcards: The Rule-set for Sub() explains how $$s_0$$ wildcards are substituted. The rule-set is extended for all other wildcards as follows:

$$Sub(s_{n>0}) = S_{S_{n}^2(s_{n-1}}(s_{n-1})$$

Growth Rate of the S Function

WORK IN PROGRESS

The number of valid S Function sequences that can be constructed has a growth rate faster than $$f_{SVO}(n)$$. Here are the growth rates for sub-sets of valid S Function sequences:

$$S_{s_1}(n) >> f_{\varphi(\omega,0)}(n)$$

$$S_{S_3^2(s_0)}(n) >> f_{\varphi(1,0)}(n) = f_{\epsilon_0}(n)$$

$$S_{S_5(s_0)}(n) >> f_{\varphi(2,0)}(n) = f_{\zeta_0}(n)$$

$$S_{s_1}(n) >> f_{\varphi(\omega,0)}(n)$$

$$S_{s_2}(n) >> f_{\varphi(1,0,0)}(n) = f_{\Gamma_0}(n)$$

$$S_{s_m}(n) >> f_{\varphi(1,0_{[m]})}(n)$$