This is the table of the rule sets of (almost) all sub versions of Bashicu Matrix invented in 20152018.
The aim of this entry is the comparison of the rule sets of them. The all sub versions can be categorized according to the difference of the three micro rules; Bad root searching rules, Bad part ascending rules and Ascension modification rules. In this article, the all sub versions are rewritten in the mathematical definition with the same format. So that the comparison become easier. Please use this for your proof of the termination or analysis.
Here is the Japanese version of this article.
(25 Feb., 2019 Note) Recently, Bashicu fixed the latest definition as 'BM4'. If you want to know about the definition in the mathematical equations, please see this mine. If you want the original definition, see here.
Contents
Rule sets
Rule sets Authors Dates 
Definitions  Bad root searching  Bad part ascending  Ascension modification  discussions 
BM1 Bashicu 21,August'15 
Left method  All branches enabled  No modify 


rTSS rpakr 25.March'18 
Left method  All branches enabled  No modify  same as trio sequence of BM1  
BM1.1^{}[1] Bashicu 18,March'16 
Upperbranchignoringmodel  All branches enabled  No modify  
BM2.1 koteitan 4.March'18 
Upperbranchignoringmodel  All branches enabled  No modify 


Bubby3's fix Bubby3 25,March'18 
Upperbranchignoringmodel  All branches enabled  No modify  same as BM1.1  
BM2 Bashicu 25,June'16 
Upperbranchignoringmodel  BM2 based  no modify  analysing link(Bashicu, Deedlit11, 84.229.92.191, 77.127.67.249, KurohaKafka, Alemagno12, Googleaarex, Bubby3, rpakr)  
BM4 Bashicu 1,Sep.'18 
Upperbranchignoringmodel  Upperbranchignoringmodel  no modify 


BM2.3 koteitan 18,Jun'18 
Upperbranchignoringmodel  Upperbranchignoringmodel  no modify 


BM3.1 Nish 18,July'18 

Upperbranchignoringmodel  BM2based  all 1 or (\(a'_{xy}\),0,…,0) 

BM3.2 Nish 23,July'18 

Upperbranchignoringmodel  Upperbranchignoringmodel  all 1 or (\(a'_{xy}\),…,0) 

BM3.1.1 Ecl1psed 20,July'18 

Upperbranchignoringmodel  BM2based  all 1 or all 0 

BM3.3 rpakr Ecl1psed 22,June'19 
Upperbranchignoringmodel  BM3.3  No modify  
BM2.2 koteitan 8,March'18 
Concestor method  All branches enabled  No modify 

Settings
 Bashicu matrix consist a matrix whose element is the natural number and a natural number named bracket.
 The matrix \(\begin{pmatrix}c_{11}&c_{21}\\c_{12}&c_{22}\end{pmatrix}\) in the Bashicu matrix is often described as \((c_{11},c_{12})(c_{21},c_{22})\). In this page it is describe so.
 \(f(n)\) is the function from a natural number into a natural number. in BM1, rTSS, BM1.1, Bubby3's fix, BM2 and BM4 \(f(n)\) is \(f(n)=n^2\), and in BM2.1, BM2.2, BM2.3 \(f(n)\) is \(f(n)=n+1\).
Common rules
A expansion procedure of a Bashicu matrix in a step is shown as below:
 Let the rightmost column \(C=(c_1, c_2, \cdots c_N)\).
 If the all elements of \(C\) are \(0\), The bad root is not found and the Jump to 5.
 Let the index of the lowermost nonzero element of the \(C\) be \(t\). (\(c_t\gt 0\land \forall k(k\gt t\rightarrow c_k=0)\))
 Decide the bad root \(R=(r_1, r_2, \cdots, r_N)\) by using the bad root searching rule of the rule set.
 remove the rightmost column \(C\).
 Replace \([n]\) into \([f(n)]\).
 If the bad root is not found, end the expansion procedure.
 let the bad root \(R\) and the partial matrix in the right side of it the bad part \(B=(b_{11},b_{12},\cdots,b_{1N})(b_{21},b_{22},\cdots,b_{1N})\cdots(b_{L1},b_{L2},\cdots,b_{LN})\).
 Before the bad root is copied,
\(\Delta=(\delta_{11}, \cdots, \delta_{1N})\cdots(\delta_{L1},\cdots,\delta_{LN})\)is defined as ‘’the ascension level’’ as: \(\forall x~\delta_{xy}=\begin{cases}c_yr_y&(y\lt t)\\0&(y \geq t)\end{cases}\).  As the flag if the each elements of the bad part ascend or not, decide a preascending matrix \(A'=(a'_{11}, a'_{12}, \cdots, a'_{1N})(a'_{21}, a'_{22}, \cdots, a'_{2N})\cdots(a'_{L1}, a'_{L2}, \cdots, a'_{LN})\) by using the bad part ascending rule of the rule set.
 By using the ascending modification rule of the rule set, decide an ascending matrix \(A\) from the preascending matrix \(A'\).
 By using the value of the bracket \(n\), combine the \(B_1, B_2, \cdots, B_n\) which is defined as below into the right of the matrix.
 \(B_1 = B + A \otimes \Delta\)
 \(B_{k+1} = B_k + A \otimes \Delta\)
 \(\otimes\) is the elementwise product as below:
\((a_{11},\cdots, a_{1N})\cdots(a_{L1},\cdots, a_{LN}) \otimes (b_{11},\cdots, b_{1N})\cdots(b_{L1},\cdots, b_{LN}) \)
\(= (a_{11}b_{11},\cdots, a_{1N}b_{1N})\cdots(a_{L1}b_{L1},\cdots, a_{LN}b_{LN})\)
That's all.
After the repetitions of the expansion above, the value of the bracket when the matrix is empty is the large number.
Bad root searching rules
Left method
 If there exists a column \(M=(m_1, \cdots, m_N)\) which meets \(m_k \lt c_k\) in the row \(t\) and any upper row \(k \leq t\), and which is in the left side of \(c\), let the right side column for \(M=(m_1, \cdots, m_N)\) be the bad root \(R\). If it is not found, the bad root is not found.
Upperbranchignoringmodel
 the ancestor of the \(c\) is defined as \(c\) itself or the ancestor of the parent of \(c\) recursively.
 the parent of the \(c\) is the rightmost element \(p\) which meets all of below recursively:
 \(p\) is in the same row of \(c\).
 \(p\) is in the left side of \(c\).
 \(p\) is smaller than \(c\).
 \(p\) is in the uppermost row, or \(p\) is in the same column as the ancestor of \(c'\). (\(c'\) is the element in the one row up of \(c\).)
Then, the bad root \(R\) is the column, in which there is the parent of the lowermost nonzero element of the rightmost column. If the parent is not found, the bad root is not found and end.
Concestor method
 the ancestor of the \(c\) is defined as \(c\) itself or the ancestor of the parent of \(c\) recursively.
 the parent of the \(c\) is the rightmost element \(p\) which meets all of below recursively:
 \(p\) is in the same row of \(c\).
 \(p\) is in the left side of \(c\).
 \(p\) is smaller than \(c\).
Then, the bad root \(R\) is the column, in which there is the parent of the lowermost nonzero element of the rightmost column. If the parent is not found, the bad root is not found and end.
bad part ascending rules
All branches enabled
 \(\forall x,y(a'_{xy}=1)\)
BM2based
The value of the ascension matrix \(a'_{xy}\) is defined as following.
\[p(x)=\max\left\{kk \lt x \land b_{k1} \lt b_{x1}\right\}\] \[\forall y \left(a'_{1y}=１\right)\] \[\forall x\gt 1,~y \left(a'_{xy}=\begin{cases}１ & (\mathrm{if}~a'_{p(x)y}=1 \land b_{1y} \lt b_{xy})\\ 0 & \mathrm{otherwize}\end{cases}\right) \]
\(p(x)\) means the parent row of the column \(x\). It depends on only 1st row.
The second line means that the bad root is ascent anyway.
The third line means that the node \(b_{xy}\) is ascent only if the node in parent row of it is ascent and the value of \(b_{xy}\) is larger than bad root \(b_{1y}\).
Upperbranchignoringmodel
 the ancestor of the \(c\) is defined as \(c\) itself or the ancestor of the parent of \(c\) recursively.
 the parent of the \(c\) is the rightmost element \(p\) which meets all of below recursively:
 \(p\) is in the same row of \(c\).
 \(p\) is in the left side of \(c\).
 \(p\) is smaller than \(c\).
 \(p\) is in the uppermost row, or \(p\) is in the same column as the ancestor of \(c'\). (\(c'\) is the element in the one row up of \(c\).)
For all \(x, y\) which meet \(r_y\) is the ancestor of \(b_{xy}\), \(a'_{xy}=1\), otherwise \(a'_{xy}=0\).
BM3.3
\begin{eqnarray} \mathrm{Ascension~matrix:}~a_{xy}&=&\left\{\begin{array}{ll} 1 &\Bigl(\mathrm{if}~ (y=t \land \exists i( r=(P_{y})^i(r+x))) \\ & ~~ ~\lor a_{xt}=1\\ & ~~ ~\lor (a_{(P_y(r+x)r)y}=1 \land P_y(r+x)>r)\Bigr) \\ 0 &(\mathrm{otherwise}) \end{array}\right.\\ \mathrm{parent~of}~S_{xy}:~P_{y}(x)&=&\left\{\begin{array}{ll} \max\{pp\lt x \land S_{py} \lt S_{xy} \land \exists i( p=(P_{y1})^i(x))\} & (\mathrm{if}~y\gt 1)\\ \max\{pp\lt x \land S_{py} \lt S_{xy} \} & (\mathrm{if}~y=1)\\ \end{array}\right.\\ \end{eqnarray} \(r \geq 1\) is the column index of the bad root. \(t \geq 1\) is the index of row in which there is the lowermost nonzero. \(S_{xy}\) is the element of the matrix in x th column and y th row. \((x,~y \geq 1)\)
Ascending modification rules
no modify
\(\forall xy (a_{xy}=a'_{xy})\)
All 1 or (\(a'_{x1}\),0,…,0)
For all \(x\), define \(a_{xy}\) from \(a'_{xy}\) as below:
 If \(\forall y\leq t(a'_{xy}=1)\) then \(\forall y(a_{xy}=1)\) otherwise \(a_{x1}=a'_{x1}\) and \(\forall y\gt 1 \left(a_{xy}=0\right)\).
All 1 or All 0
For all \(x\), define \(a_{xy}\) from \(a'_{xy}\) as below:
 if \(\forall y\leq t(a'_{xy}=1)\) then \(\forall x(a_{xy}=1)\), otherwise \(\forall y(a_{xy}=0)\).
Notes
 ↑ The rule doesn't have the formal name so that I named it as BM1.1 in this article.