User blog:Bubby3/BM2.3 vs R function

Here, I am going to show a version of my BM2.3 analysis, but instead of psi functions, which are ambiguous, I am going to compare it to R function (not version 2 or 2.1 of it), which is unambiguous. I am also going to start from the very beginning. I am going to define a FGH-like version of BMS (can apply to any version).

The definition of BMS FGH versions is this: R function is being modifier so that the base function is n+1 instead of 10^n. With comparabilithty to R function, I am using BMS-FGH, where the matrix x[n] has a similar growth rate to the string nRm, and that ordinal can be represented elsewhere by {m}. So,here is the analysis.
 * If  the rightmost columm is all zeroes (or has no bad part) cut it off, and then iterate the matrix n times. In symbol form, #Z[n], where Z is the columm of all zeroes, becomes #[#[...#[#[n]]...]], where # is iterated n times.
 * If the matrix is empty, [n] = n+1
 * Otherwise, foerform the step as usual, but do not change the value of n, and only copy the bad part n times.

The left side represents BMS, and the right side represents R function.

Primitive sequence system

 * (empty matrix) = 0
 * 0 = 1
 * 0,0 = 2
 * 0,0,0 = 3
 * 0,0,0,0 = 4
 * 0,0,0,0... = n
 * 0,1 = {0}
 * 0,1,0 = 1{0}
 * 0,1,0,0 = 2{0}
 * 0,1,0,0... = n{0}
 * 0,1,0,1 = {0}{0}
 * 0,1,0,1,0,1 = {0}{0}{0}
 * 0,1,0,1,0,1... = {0}{0}{0}...
 * 0,1,1 = {1}
 * 0,1,1,0,1 = {0}{1}
 * 0,1,1,0,1,1 = {1}{1}
 * 0,1,1,1 = {2}
 * 0,1,1,1,0,1,1,1 = {2}{2}
 * 0,1,1,1,1 = {3}
 * 0,1,1,1,1... = {n}
 * 0,1,2 =
 * 0,1,2,1 = {1{0}}
 * 0,1,2,1,1 = {2{0}}
 * 0,1,2,1,2 =
 * 0,1,2,1,2,1,2 =
 * 0,1,2,1,2,1,2,1,2 =
 * 0,1,2,2 =
 * 0,1,2,2,1,2 =
 * 0,1,2,2,1,2,2 =
 * 0,1,2,2,2 =
 * 0,1,2,2,2,2 =
 * 0,1,2,3 =
 * 0,1,2,3,2 =
 * 0,1,2,3,2,3 =
 * 0,1,2,3,3 =
 * 0,1,2,3,4 =
 * 0,1,2,3,4,4 =
 * 0,1,2,3,4,5 =
 * 0,1,2,3,4,5,6 =

Pair sequence system

 * (0,0)(1,1) = {0,1}
 * (0,0)(1,1)(0,0) = 1{0,1}
 * (0,0)(1,1)(0,0)(1,0) = {0}{0,1}
 * (0,0)(1,1)(0,0)(1,0)(1,0) = {1}{0,1}
 * (0,0)(1,1)(0,0)(1,0)(2,0) = {0,1}
 * (0,0)(1,1)(0,0)(1,1) = {0,1}
 * (0,0)(1,1)(0,0)(1,1)(0,0)(1,1) = {0,1}
 * (0,0)(1,1)(1,0) = {1{0,1}}{0,1}
 * (0,0)(1,1)(1,0)(1,0) = {2{0,1}}{0,1}
 * (0,0)(1,1)(1,0)(2,0) = {0,1}
 * (0,0)(1,1)(1,0)(2,0)(2,0) = {0,1}
 * (0,0)(1,1)(1,0)(2,0)(3,0) = {0,1}
 * (0,0)(1,1)(1,0)(2,1) = {0,1}
 * (0,0)(1,1)(1,0)(2,1)(2,0) = {0,1}
 * (0,0)(1,1)(1,0)(2,1)(2,0)(3,0) = {0,1}
 * (0,0)(1,1)(1,0)(2,1)(2,0)(3,1) = {0,1}
 * (0,0)(1,1)(1,1) = {0,1}{0,1}
 * (0,0)(1,1)(1,1)(1,0) = {1{0,1}{0,1}}{0,1}{0,1}
 * (0,0)(1,1)(1,1)(1,0)(2,1 )= {0,1}{0,1}
 * (0,0)(1,1)(1,1)(1,0)(2,1)(2,1) = {0,1}{0,1}
 * (0,0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)(3,1)(3,1) = {0,1}{0,1}
 * (0,0)(1,1)(1,1)(1,1) = {0,1}{0,1}{0,1}
 * (0,0)(1,1)(1,1)(1,1)(1,1) = {0,1}{0,1}{0,1}{0,1}
 * (0,0)(1,1)(1,1)(1,1)(1,1)(1,1) = {0,1}{0,1}{0,1}{0,1}{0,1}
 * (0,0)(1,1)(2,0) = {1,1}
 * (0,0)(1,1)(2,0)(0,0)(1,1) = {1,1}
 * (0,0)(1,1)(2,0)(0,0)(1,1)(1,1) = {1,1}
 * (0,0)(1,1)(2,0)(0,0)(1,1)(2,0) = {1,1}
 * (0,0)(1,1)(2,0)(1,0) = {1{1,1}}{1,1}
 * (0,0)(1,1)(2,0)(1,0)(2,1) = {1,1}
 * (0,0)(1,1)(2,0)(1,0)(2,1)(3,0) = {1,1}
 * (0,0)(1,1)(2,0)(1,0)(2,1)(3,0)(2,0)(3,1)(4,0) = {1,1}
 * (0,0)(1,1)(2,0)(1,1) = {0,1}{1,1}
 * (0,0)(1,1)(2,0)(1,1)(1,1) = {0,1}{0,1}{1,1}
 * (0,0)(1,1)(2,0)(1,1)(2,0) = {1,1}{1,1}
 * (0,0)(1,1)(2,0)(1,1)(2,0)(1,1)(2,0) = {1,1}{1,1}{1,1}
 * (0,0)(1,1)(2,0)(2,0) = {2,1}
 * (0,0)(1,1)(2,0)(2,0)(2,0) = {3,1}
 * (0,0)(1,1)(2,0)(3,0) = {{0},1}
 * (0,0)(1,1)(2,0)(3,0)(3,0) = {{1},1}
 * (0,0)(1,1)(2,0)(3,0)(4,0) = {,1}
 * (0,0)(1,1)(2,0)(3,1) = {,1}
 * (0,0)(1,1)(2,0)(3,1)(0,0)(1,1)(2,0)(3,1) = {,1}
 * (0,0)(1,1)(2,0)(3,1)(1,0) = {1{,1}}{,1}
 * (0,0)(1,1)(2,0)(3,1)(1,0)(2,1)(3,0)(4,1) = {,1}
 * (0,0)(1,1)(2,0)(3,1)(1,1) = {0,1}{,1}
 * (0,0)(1,1)(2,0)(3,1)(1,1)(2,0)(3,1) = {{{0,1}},1}{{{0,1}},1}
 * (0,0)(1,1)(2,0)(3,1)(2,0) = {1{{0,1}},1}
 * (0,0)(1,1)(2,0)(3,1)(2,0)(3,1) = {{{0,1}}{{0,1}},1}
 * (0,0)(1,1)(2,0)(3,1)(3,0) = {{1{0,1}},1}
 * (0,0)(1,1)(2,0)(3,1)(3,0)(3,0) = {{2{0,1}},1}
 * (0,0)(1,1)(2,0)(3,1)(3,0)(4,0) = {,1}
 * (0,0)(1,1)(2,0)(3,1)(3,0)(4,1) = {,1}
 * (0,0)(1,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1) = {{{0,1}},1}
 * (0,0)(1,1)(2,0)(3,1)(3,1) = {,1}