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Kyodaisuu

aka Fish

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My favorite wikis
• I live in Japan
• Readers guide for the BM calculator source code

July 9, 2018 by Kyodaisuu

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

• Going beyond pair sequence in BM calculator

July 8, 2018 by Kyodaisuu

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^(…
• Converting pair sequence to ordinal

July 6, 2018 by Kyodaisuu

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 non-standard 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 …

• Classifying numbers with ordinals

April 15, 2018 by Kyodaisuu

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 n-th 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, fast-growing hierarchy (FGH) is defined. FGH defined in googology wiki is

• $$f_0(n) = n + 1$$
• \(f_{\alpha+1}(n) = f^n_…