User blog:P進大好きbot/Summary on historical background of BMS

I summarise historical background of BMS (Bashicu Matrix System) for readers who want to study it. I appreciate any corrections with precise sources.

= What is BMS? =

BMS is a system to generate a large number, which was created by Bashicu to define several large numbers, e.g. Bashicu matrix number. The main article of BMS in googology wiki is here.

BMS has many variants created by Bashicu and other googologists, and they are also called BMS. Mainly, googologists focus on the second version of BMS called BM2.

= Why is no precise definition accessible? =

The original "definitions" of Bashicu matrix numbers are written by Bashicu here. He states that they are written in BASIC language, which is one of a progamming laguage, but they contain many compile-errors. Namely, they are never well-defined.

Bashicu are still updating the definitions many times in order to strengthen BMS, but seems not to be willing to fix the errors.

Therefore when we refer to BMS, we need to consider alternative definitions given by other googolgists or guess what Bashicu intends. At least, we can understand a partial behaviour of BMS by Bashicu's slide explaing an example of the expansion process of BMS in Japanese.

= Who knows the explicit definition? =

Maybe none. Since there is no definition written in an explicit way, the precise definition, if exists, is accessible by none other than Bashicu. Further, if Bashicu knew it, then he would write it.

Of course, the answer depends on to what extent a definition should be precise. For example, KurohaKafka said that the original "definition" should be dealt with as a valid definition, because there is no ambiguity at least for him. .

= How do googologists know the expansion rule of BMS? =

We can test the behaviour of BMS by Bashicu Matrix Calculator written in C language by Fish. It is one of the most trustworthy source of BMS, because Bashicu actually checked its behaviour.

Also, Bashicu himself implicitly asked Fish to translate his "definition", and hence the translation by Fish is reliable to Bashicu.

There are many trial to guess alternative definitions of BMS by many googologists. Here is a list of alternative definitions.
 * Explanation of Bashicu Matrix Calculator by Fish
 * List of definitions of BMS's by Fish
 * Classification of the "definitions" of BMS's by Koteitan
 * Alternative definition by rpakr
 * Alternative definition by Alemagno12.
 * ...and other definitions on Discord

= How can googologists analyse BMS? =

Since BMS is not completely defined, googologists "analyse" their own or someone's guess of the original definition. Moreover, since BMS is expected to be very strong as an ordinal notation, they need to use unknown ordinals or ordinal notations which have not been completely defined yet. Here is a list of the "analysis".
 * Analysis by KurohaKafka
 * Japanese slide explaining the expansion rule by rpakr
 * Analysis by Alemagno12
 * Analysis by Eclpsed276
 * ...and other analyses on Discord