User blog:Koteitan/The Hydra Diagram based on Upper-Branch-Ignoring Model for understanding BM4 structure

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 and Alemagno12 explained the mechanism of the bad-root-searching algorithm in their articles below: They are same methods and Alemagno12 calls it sequence reduction method and Bubby3 calls it upper-branch removing method, and so on.
 * Bubby3 Bubby3 BM1 is weaker than expected. #How to fix it
 * Alemagno12 Alemagno12 A program-free definition for BM2?

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)
 * 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 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)). So that the ascension nodes is the all recursive children of them(\(\{\forall x|\forall a\P^{a}_y(r+x)}\)). 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.

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

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