I introduce my new hydra diagram based on **Upper-Branch-Ignoring Model**, which denotes the structure of Bashicu Matrix System (BMS) version BM4.

(For understanding the expantion rule of BMS, my mathematical definition may help you. For the mathematical notation in this article, see it)

## Before

Bubby3, Ecl1psed276 and Alemagno12 explained the mechanism of the bad-root-searching algorithm in their articles below in the easy way to understand:

- Bubby3 BM1 is weaker than expected. #How to fix it
- Ecl1psed276 and Alemagno12 A program-free definition for BM2?

They are same methods and Ecl1psed276 and Alemagno12 call it *sequence reduction method* and Bubby3 calls it *upper-branch removing method*, and so on.

Bubby3 showed an example of it with the matrix (0,0)(1,1)(2,2)(3,0)(2,1)(3,1)(4,1). I cite it here:

- (0,0)(1,1)(2,2)(3,0)(2,1)(3,1)(
**4**,1) - (0,0)(1,1)(2,2)(3,0)(2,1)(
**3**,1)(4,1) - (0,0)(1,1)(2,2)(3,0)(
**2**,1)(3,1)(4,1) - (0,0)(
**1**,1)~~(2,2)(3,0)~~(2,1)(3,1)(4,1) - (
**0**,0)(1,1)~~(2,2)(3,0)~~(2,1)(3,1)(4,1) - (0,0)(1,1)
~~(2,2)(3,0)~~(2,1)(3,1)(4,**1**) - (0,
**0**)~~(1,1)(2,2)(3,0)(2,1)(3,1)~~(4,1)

In this method, while searching the node which has the smaller level than the one of pivot node, the skipped columns are removed from the sequence.

The explanation is simple way, however, the matrix is changed by the reduction during the searching dynamically, so that many rewriting of the matrices occur when we decide the Ascension matrix \(A_{xy}\) and when we expand the next expansion.

## Proposal

I propose a static diagram which can express the dynamic behaviors of sequence reduction method or upper-branch removing method. I call it **Upper-Branch-Ignoring Model**.

Upper-Branch-Ignoring Model consists the same number hydras as the number of the matrix rows \(Y\). The level of the nodes of each hydra is the same as value of the elements \(S_{xy}\) in the matrix \(\mathrm{S}\). Each edge denotes the parent connection \(P_y(x)\) of the each element. Though its connections of the top hydra is normal, the lower hydras are not normal.

The connections of the lower hydras depend upper hydras.

This is how to draw the connections.

If we want to search the red node (we call it ** child ** now), at first see

**the node**which is on the upper hydra in the same column as the

**child**(Fig.3-1). Next, mark

**all ancestors of it**\(\forall a(P_{y-1})^a(x)\) (Fig.3-2). At last, seeing

**only nodes on the lower hydra in the same columns**of the

**upper ancestors**(Fig.3-3), search and connect the rightmost nodes which has the smaller level than the level of the

**child**(Fig.3-4), that is the

**parent**\(P_{y}(x)\).

The diagram is static and useful to find a bad root \(r\) and the ascension nodes \(A_{xy}\). Here is how to find them:

This is example for the matrix (0,0,0)(1,1,1)(2,0,0)(1,1,1). (that is the famous matrix because BM1 doesn't terminate with it.) The bad root of this matrix \(r\) is defined as the parent of the lowermost-nonzero element \(P_t(X-1)\) of the rightmost cut column \(X-1\) (\(\mathrm{S}_{X-1}=\)(1,1,**1**)).

The ascension nodes is **the all recursive children of them**(\(\{\forall x|\forall ar=P^{a}_y(r+x)\}\)). That's all. Simple. **pink node** are not the child of the bad root so that it doesn't ascent while copying and keep its level as it is.

In addition to this, we can also copy the the parent connection edges as it is. You don't need the re-analyzing of the parent connections when you expand the next.

In this way with Upper-Branch-Ignoring Model, We can get the bad roots, the ascension matrices and expansion easily and statically.

## To be continued?

With this diagram, I found another interesting characteristics in the structure of BMS. I'll tell you the story in the another article, next time...

## Examples

Here is the examples for the diagrams of Upper-Branch-Ignoring Model.

## References

- Bashicu_matrix_system, explanation in this wiki (note:rule is BM1)
- Bubby3, 'BM1 is weaker than expected. #How to fix it', Googology wiki blog, Mar. 25, 2018.
- My first post about Upper-Branch-Ignoring Model, 2, 3, twitter, June 17, 2018
- Ecl1psed276 and Alemagno12, 'A program-free definition for BM2?', Googology wiki blog, July 9, 2018.
- Bashicu, ' BASIC言語による巨大数のまとめ', 巨大数研究 wiki blog, Sept. 1, 2018.
- Koteitan, 'Purely mathematical definition of Bashicu Matrix', Googology wiki blog, Sept. 24, 2018.