User blog comment:B1mb0w/The S Function Version 2/@comment-27513631-20160617114926/@comment-10262436-20160619091320

S(3,0,1) ...

Hi. I have intentionally defined the S function as a 'string substitution' function with no mathematical (algebraic) connotations. Therefore S(3,0,1) does not reduce any further. It does hoever have a well defined position in an ascending sequence of all 'restricted' S Functions with an initial value of a_0 = 3. The position of this number (its ordinal value) is 2, after, S(3,0,0) which has an ordinal value of 1. BTW, S(3,0,0) does reduce to the string '3' according to the rules in the blog, but will still have the same ordinal value of 1.

The rest of my blog shows how the growth rates of the number of strings that the S function can construct grows very fast (beyond LVO etc.).

Definition of S ...

I have tidied up this section to simplify it as much as possible. The rules for defining a resticted or generalised S function are very succinct. There are only 3 induction rules, and 2 additional constraints that apply to the different S Function types.

S_i still appears in the definition of the dec procedure. This is simply a placeholder notation for the input parameter 'a' in the final S function string, after the S function has been reduced as many times as neccessary (call this 'z' times) so that the input parameter 'b' is equal to 0. I will tidy this up further when I think of better notation.

Hope this explanation helps.