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As people are having trouble reading the source code of Bashicu Matrix calculator, which is the definition of BM itself, I wrote a document to guide for the reading the source code.
https://kyodaisuu.github.io/basmat/definition.html
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As I implemented ordinal conversion of pair sequence to BM calculator, next step is to go further. To implement it we have to find a rule to convert BM to some known ordinal expression.
As it was easy to convert pair sequence to Buchholz psi function, I want to know if the same approach can be applied to trio sequence to some extent. Although (0,0,0)(1,1,1) is normally written as ψ(Ω_ω), it would be easier to use the notation of ψ0(ψω(0)) = p0(pw(0)) to find a rule to convert. Once we can find a rule, we can implement it in the BM calculator.
 (0,0,0)(1,1,1) = p0(pw(0))
 (0,0,0)(1,1,1)(0,0,0) = p0(pw(0))+1
 (0,0,0)(1,1,1)(0,0,0)(1,1,1) = p0(pw(0)) + p0(pw(0))
 (0,0,0)(1,1,1)(1,0,0) = p0(pw(0))w = p0(pw(0)+p0(0))
 (0,0,0)(1,1,1)(1,0,0)(2,1,1) = w^w^(…

As I implemented a function to determine if Bashicu Matrix is standard in BM calculator, now I am trying to implement ordinal conversion.
Koteitan made a Pair sequence to Buchholz's Ψ function in four lines Javascript, and I read the code and translated. Actually I think koteitan's code only works for the standard form of pair sequence. In the BM calculator, standard form is checked and therefore I can just exclude nonstandard form.
The code converts the BM to omega and epsilon expression when ordinal is below zeta_0. It would be convenient because people are used to this kind of ordinal expression. Right now I just wrote a test code and paste the result here to verify before actually implementing it.
Update: I made some ordinal calculation …
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Robert Munafo classified natural numbers with classes. This method is useful for classifying large numbers of tetration level, but it is not useful for classifying numbers such as Graham's number.
Therefore in googology we classify functions which define large numbers. For example, in , an infinite set of functions \(E_i\) for some natural number \(i\) is introduced, and the nth hierarchy \(\mathcal{E}^n\) which containes certain classes of functions are defined. The set has hierarchy structure such that
\[ \mathcal{E}^0 \subseteq \mathcal{E}^1 \subseteq \mathcal{E}^2 \subseteq \cdots\]
By extending this definition to ordinal, fastgrowing hierarchy (FGH) is defined. FGH defined in googology wiki is
 \(f_0(n) = n + 1\)
 \(f_{\alpha+1}(n) = f^n_…

Bashicu matrix system is very strong, but it is hard to understand how it works. That is why I made Bashicu matrix calculator for showing the calculation process of Bashicu matrix.
After hyp cos found nonteminating sequence of Bashicu matrix, Bashcu updated his system in his blog post. Please note that it is different from the version that User:KurohaKafka posted on the talk page. KurohaKafka thinks (0,0,...,0)(1,1,...,1) terminates for Bashicu's new system, while KurohaKafka's version may not terminate.
Let us call the original version of Bashicu matrix as Bashicu Matrix version 1 (BM1), and the new version that Bashicu made as version 2 (BM2). The problem in BM2 is that it is too difficult to understand. I would suggest that we use BM1 fo…
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