User blog:Rgetar/Taranovsky's C - BB correspondence

Today I read translation of AAAgoogology's user page "TaranovskyのC表記の解析" by Rpakr and koteitan about Taranovsky's C, and I thought that there may be some correspondence between Taranovsky's C and my booster-base (BB) expressions.

I noticed some similarities between Taranovsky's C and BB in the first half of 2019, but it was before I came up with "L" and "R" in July 2019, so I thought that there is not much similarity.

Converting Taranovsky's C into BB and backwards
Taranovsky's C expression ↔ BB expression:
 * C(a, b) ↔ [a]b
 * 0 ↔ empty string
 * Ωn ↔ [Cn], where n ≥ 1

That is
 * Ω1 ↔ Ω
 * Ω2 ↔ L
 * Ω3 ↔ R
 * Ω4 ↔ S

Because
 * Ωn = C(Ωn + 1, 0)

that is
 * Ωn = [Ωn + 1] (here Ωx are not converted into BB)
 * Ω1 = [Ω2], Ω2 = [Ω3], Ω3 = [Ω4], ...

but
 * Ω = [L], L = [R], R = [S], ...

Fundamental sequences
In the translation it is said that Ωn does not have a fundamental sequence (similar is also true for some C(a, b) expressions), but in BB all expressions have a fundamental sequence. I think that expressions without a fundamental sequence in Taranovsky's C correspond to BB expressions with uncountable cofinality.

Examples

 * 0 = 0 = empty string
 * 1 = C(0, 0) = []
 * 2 = C(0, C(0, 0)) = [][] = []1
 * ω = C(C(0, 0), 0) = = [1]
 * ω + 1 = C(0, C(C(0, 0), 0)) = [] = []ω
 * ω2 = C(C(0, 0), C(C(0, 0), 0)) = = [1]ω
 * ω2 = C(C(0, C(0, 0)), 0) = ][ = [2]
 * ωω = C(C(C(0, 0), 0), 0) = [] = [ω]
 * ε0 = C(Ω1, 0) = [Ω]
 * ε02 = C(C(Ω1, 0), C(Ω1, 0)) = [[Ω]&#93;[Ω] = [ε0]ε0
 * ε0ω = C(C(0, C(Ω1, 0)), C(Ω1, 0)) = [[][Ω]&#93;[Ω] = [ε0 + 1]ε0
 * ε1 = C(Ω1, C(Ω1, 0)) = [Ω][Ω] = [Ω]ε0
 * εω = C(C(0, Ω1), 0) = [[]Ω] = [Ω + 1]
 * εε 0 = C(C(C(Ω1, 0), Ω1), 0) = [ΩΩ] = [[ε0]Ω] = [Ω + ε0]
 * ζ0 = C(C(Ω1, Ω1), 0) = [[Ω]Ω] = [Ω2]
 * εζ 0 + 1 = C(Ω1, C(C(Ω1, Ω1), 0)) = [Ω][[Ω]Ω] = [Ω]ζ0
 * ζ0 = C(C(Ω1, Ω1), C(C(Ω1, Ω1), 0)) = [[Ω]Ω][[Ω]Ω] = [Ω2]ζ0
 * φ3(0) = C(C(Ω1, C(Ω1, Ω1)), 0) = [[Ω][Ω]Ω] = [[Ω]Ω2] = [Ω3]
 * φω(0) = C(C(C(0, Ω1), Ω1), 0) = [[[]Ω]Ω] = [[Ω + 1]Ω] = [Ωω]
 * φφ ω(0) (0) = C(C(C(C(C(C(0, Ω1), Ω1), 0), Ω1), Ω1), 0) = [[[[]Ω]ΩΩ]Ω] = [[[φω(0)]Ω]Ω] = [[Ω + φω(0)]Ω] = [Ωφω(0)]
 * Γ0 = C(C(C(Ω1, Ω1), Ω1), 0) = [[[Ω]Ω]Ω] = [[[Ω2]Ω] = [Ω2]
 * Γ1 = C(C(C(Ω1, Ω1), Ω1), C(C(C(Ω1, Ω1), Ω1), 0)) = [[[Ω]Ω]Ω][[[Ω]Ω]Ω] = [Ω2]Γ0
 * φ(1, 1, 0) = C(C(Ω1, C(C(Ω1, Ω1), Ω1)), 0) = [[Ω][[Ω]Ω]Ω] = [[Ω]Ω2] = [Ω2 + Ω]
 * φ(2, 0, 0) = C(C(C(Ω1, Ω1), C(C(Ω1, Ω1), Ω1)), 0) = [[[Ω]Ω][[Ω]Ω]Ω] = [[[Ω2]Ω2] = [Ω22]
 * φ(ω, 0, 0) = C(C(C(0, C(Ω1, Ω1)), Ω1), 0) = [[[][Ω]Ω]Ω] = [[Ω2 + 1]Ω] = [Ω2ω]
 * φ(1, 0, 0, 0) = C(C(C(Ω1, C(Ω1, Ω1)), Ω1), 0) = [[[Ω][Ω]Ω]Ω] = [[Ω3]Ω] = [Ω3]
 * SVO = C(C(C(C(0, Ω1), Ω1), Ω1), 0) = [[[[]Ω]Ω]Ω] = [[Ωω]Ω] = [Ωω]
 * LVO = C(C(C(C(Ω1, Ω1), Ω1), Ω1), 0) = [[[[Ω]Ω]Ω]Ω] = [[Ω2]Ω] = [ΩΩ]
 * BHO = C(C(Ω2, Ω1), 0) = [[L]Ω] = [Ω2]
 * TFB = C(C(Ω2, C(C(0, Ω2), 0)), 0) = L][[]L = L][L + 1 = [[L]Ωω] = [Ωω + 1]
 * ψ0(ΩΩ) = C(C(C(Ω1, Ω2), 0), 0) = [Ω]L = [[L + Ω]&#93; = [ΩΩ]
 * ψ0(ψI(0)) = C(C(C(C(Ω3, 0), Ω2), 0), 0) = [[[[R]&#93;L]&#93; = [[[L]L]&#93; = [[L2]&#93; = [I]
 * C(C(C(Ω3, Ω2), 0), 0) = [R]L = [[L2]&#93; = [M]