User blog:Ecl1psed276/A list of all BMS versions and their differences

here you go

Version 1
Created: ???

Creator: BashicuHyudora

Description: This version is the first version of BMS. It uses an extremely simple method of finding the bad root of an expression, a method that ultimately didn't work and caused an infinite loop.

Terminates: No. An example of a non-terminating expression is (0,0)(1,1)(2,1)(3,1)(2,0)(1,1)(2,1)(3,1), and this one is believed to be the smallest one.

Version 2
Created: ???

Creator: BashicuHyudora

Description: The second version of BMS made by Bashicu. It was created after BM1 was shown to have an infinite loop. The method used to find the bad root in BM2 is more complex than BM1, but it is believed to be the most logical and consistent way to do it. If BM2 terminates, this is one of the most powerful notations known, easily surpassing notations like SAN. Even (0,0,0)(1,1,1)(2,2,2) is believed to be beyond pretty much any notation in existence, except for TON.

Terminates: Unknown. For a long time, BM2 was believed to terminate, but after some studying, Nish and Koteitan have determined that BM2 is likely to have an infinite loop somewhere past (0,0,0,0)(1,1,1,1). They are currently working to find an explicit counterexample.

Version 2.1
Created: ???

Creators: Koteitan

Description: idk

Terminates: No.

Version 2.2
Created: ???

Creators: Koteitan

Description: This version of BMS is the most unique of all the versions. It uses something called the "concestor method", a different method to find the bad root than BM2 uses. Analysis of this version looks identical to BM2 up to (0,0)(1,1)(2,0). However, analysis becomes extremely complicated from here on out, because sometimes adding a 1 to the ordinal doesn't actually correspond to adding a (0,0...0).

Terminates: Unknown.

Version 2.3
Created: July 18, 2018

Creators: Koteitan/Ecl1psed/Nish

Description: After discovering an inconsistency in the expansion of a BM2 expression, this version was created by Nish and Koteitan. I also independently discovered this version when I tried to make a program for BM2. However, it turned out that my program didn't exactly correspond with BM2, it instead corresponded with the newly defined BM2.3. It works exactly like BM2 all the way up to (0,0,0,0)(1,1,1,1), but it does not have the same inconsistency that BM2 has. It is accepted by most of the community that BM2.3 is better and more logical than BM2.

Terminates: Probably.

Version 2.4
Created: June 24, 2018

Creators: Yukito

Description: This version expands exactly like BM2.3, so we can treat the notations as equal for all intents and purposes. It's almost like Yukito independently discovered BM2.3, along with me and Nish/Koteitan. You just know that when 3 or 4 different people independently discover a notation at the same time that it is definitely a good notation.

Terminates: Probably.

Version 2.5
Created: July 20, 2018

Creators: Yukito

Description: idk

Terminates: idk

Version 2.6
Created: around July 22, 2018

Creators: Nish

Description: This version is based off of Nish's extension of primitive sequence system. It acts similarly to BM2 but is slightly more powerful. However, Nish thinks that BM2 catches up to this version at (0,0,0)(1,1,1)(2,2,1)(3,0,0).

Terminates: Unknown.

Version 3
Created: June 12, 2018

Creator: BashicuHyudora

Description: At the point of BM3's release, it was thought that BM2 was likely to terminate, so it was not known exactly why Bashicu decided to put this version out. I think the reason he did was to make a version of BM3 that has more consistent behaviour. Not entirely sure though.

Terminates: No. Just mere days after it's inclusion into Fish's calculator, Nish/Alemagno12 found an infinite loop with the expression (0,0,0)(1,1,1)(2,1,1)(1,1,1).

Version 3.1
Created: July 12, 2018

Creator: Nish

Description: This version acts like BM2, except in the calculation of the C vectors, if a vector does not equal (1,1,1...1,1) with only 1's, then we set it to (1,0,0,0...0,0). (sorry this is not the place to explain what a C vector is). This modification makes the analysis MUCH easier than BM2. For example, (1,1,1) now always adds \(\Omega_\omega\) inside the psi function, whereas with BM2, adding a (1,1,1) sometimes adds \(\Omega_\omega\), but other times it changes a \(\Omega\) into \(\Omega_\omega\). However, this also weakens the notation slightly. It is not known if the notation ever catches up to BM2. Some people, like myself, think that it catches up at (0,0,0)(1,1,1)(2,2,2), and others think it will never catch up at all.

Terminates: Unknown. The termination of BM3.1 is highly dependent on whether or not BM2 terminates. I can say with a fair amount of confidence that BM2 terminates if and only if BM3.1 terminates.

Version 3.1.1
Created: July 20, 2018

Creator: Ecl1psed (me)

Description: why does this version even exist lol In discord, I said my idea of a new version of BMS, based off BM3.1. However, within 5 minutes of sending that message, I realized that my idea would not work, because it would create expressions like (2,0)(4,0) which is obviously not allowed. But for some reason, everyone thought it would be a good idea to give my notation it's own version name, so here we are. Don't even get me started.

Terminates: without a doubt

Version 3.2
Created: July 23, 2018

Creator: Nish

Description: Remember how we got from BM2 to BM3.1? What if you instead start with BM2.3 instead of BM2? Then, you get this version, BM3.2. It is highly reminiscent of BM3.1.

Terminates: Probably. The termination of BM3.2 is highly dependent on whether or not BM2.3 terminates, which it probably does.

so yeah, there are quite a lot of versions of Bashicu Matrix System. Out of these, the ones that are most likely to terminate are BM2.3 and BM3.2. The analysis of 3.2 is much easier than 2.3, but 2.3 is more powerful. It is not known if they catch up.