User blog:Kyodaisuu/Going beyond pair sequence in BM calculator

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^(p0(pw(0))*2) = p0(pw(0)+p0(pw(0)))
 * (0,0,0)(1,1,1)(1,0,0)(2,1,1)(2,0,0)(3,1,1) = w^w^(p0(pw(0))*2) = p0(pw(0)+p0(pw(0))+p0(pw(0)))

up to here it can be confirmed with the current BM calculator.


 * (0,0,0)(1,1,1)(1,1,0)	= p0(pw(0)+p1(0))

This can be confirmed by the sequcne of BM calculator.

So the tentative rule up to here would be


 * for (a,b,0), if a increases write pb(. if a decreases or the third row decreases write 0, appropriate times of ), and +pb(
 * for (a,b,1), write pw(
 * To finish write 0 and appropriate times of )

Here "appropriate times of )" is complex because it is influenced by the numbers of pw that appeared.


 * (0,0,0)(1,1,1)(1,1,0)(2,1,0) = p0(pw(0)+p1(p1(0)))
 * (0,0,0)(1,1,1)(1,1,0)(2,2,0) = p0(pw(0)+p1(p2(0)))
 * (0,0,0)(1,1,1)(1,1,0)(2,2,0)(3,3,0) = p0(pw(0)+p1(p2(p3(0))))
 * (0,0,0)(1,1,1)(1,1,0)(2,2,0)(3,3,0)(4,4,0) = p0(pw(0)+p1(p2(p3(p4(0)))))
 * (0,0,0)(1,1,1)(1,1,0)(2,2,1) = p0(pw(0)+p1(pw(0))) = p0(p1(pw(0)+pw(0)))

It is getting complex and I am not very sure about this level...