User blog:PsiCubed2/My own version of BMS

It's basically a variation of BM1. We can call it BM1.2 or BM4.1, depending on your preference.

The rules are exactly the same as in BM1 (these can be found here) with a small modification of the way to find the bad node:

In BM1, the bad node Y is found by this algorithm:

(1) We call the rightmost element X

(2) We begin scanning leftwards

(3) Y is the right most element whose entries are all strictly smaller than the corresponding nonzero entries in X.

This rule-set lead to bad behavior as early as (0,0)(1,1)(2,1)(3,1)(2,0)(1,1)(2,1)(3,1), where the bad node in BM1 is (2,0) while a Buchhloz-hydra clone would require us to scan all the way to the left and pick (0,0) as our bad node.

So I suggest the following modification:

To find the bad node Y with my new algorithm, we do this:

(1) We call the rightmost element X

(2) We begin scanning leftwards

(3) Y is the right most element whose entries are all strictly smaller than the corresponding nonzero entries in X, and for which there's no element between Y and X whose first entry are strictly smaller than the first entry in Y.

Going back to (0,0)(1,1)(2,1)(3,1)(2,0)(1,1)(2,1)(3,1), the bad node will no longer be (2,0) because (1,1) is between that node and (3,1) and 1 is less than 2. So this modified algorithm will, indeed, go all the way to (0,0) as needed.

I believe (though I haven't yet proven) that in this version, the pair sequences work exactly as Buchholz Hydras (with my additional rule making sure that we don't mix-up nodes with different parents). It is also almost as simple as BM1 (unlike all the other versions floating around) which makes it especially attractive in my view.

Feel free to try and analyze this notation, find flaws, or suggest improvements.