Categorizing of the rule sets for all sub versions of bashicu matrix

This is the table of the rule sets of (almost) all sub versions of Bashicu Matrix invented in 2015-2018.

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 re-written 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.

Rule sets

Rule sets Authors Dates	Definitions	Bad root searching	Bad part ascending	Ascension modification	discussions
BM1 Bashicu 21,August'15	BASIC pseudo code	Left method	All branches enabled	No modify	explanation with equations non-terminated example(hyp/^,cos) non-terminated example(Bubby3) countermeasure(bashicu) It is said that (0,0)(1,1)(2,1)(3,1)(2,0)(1,1)(2,1)(3,1) is the smallest non-terminated matrix.
rTSS rpakr 25.March'18	definition	Left method	All branches enabled	No modify	same as trio sequence of BM1
BM1.1[1] Bashicu 18,March'16	BASIC pseudo code	Upper-branch-ignoring-model	All branches enabled	No modify	Non-terminated example(hyp/^,cos) Countermeasure(bashicu)
BM2.1 koteitan 4.March'18	C code	Upper-branch-ignoring-model	All branches enabled	No modify	same as BM1.1 non-terminated example(KurohaKafka) 12
Bubby3's fix Bubby3 25,March'18	Definition	Upper-branch-ignoring-model	All branches enabled	No modify	same as BM1.1
BM2 Bashicu 25,June'16	BASIC pseudo code	Upper-branch-ignoring-model	BM2 based	no modify	explain in SlideShare (in Japanese) explain by rpakr(in Japanese) non-terminated example analysing link(Bashicu, Deedlit11, 84.229.92.191, 77.127.67.249, KurohaKafka, Alemagno12, Googleaarex, Bubby3, rpakr)
BM4 Bashicu 1,Sep.'18	BASIC pseudo code C code	Upper-branch-ignoring-model	Upper-branch-ignoring-model	no modify	all differences between BM2 and BM4 BASIC code explanation BM4 is same as BM2.3.
BM2.3 koteitan Nish Ecl1psed 18,Jun'18	C code First explanation(in Japanese) equations	Upper-branch-ignoring-model	Upper-branch-ignoring-model	no modify	BM2.3 is same as BM4. example of which the expansion is different from BM2 I made it to be more logical because I feel the bad part ascending rule of BM2 is complicating. I heard Nish and Ecl1psed made same algorithm independently.
BM3.1 Nish 18,July'18	conversation on Discord introduction	Upper-branch-ignoring-model	BM2-based	all 1 or (\(a'_{xy}\),0,…,0)	Though its ascending rule is weaker than BM2, It is believed that it catch up BM2 and it is easy to analyze.
BM3.2 Nish 23,July'18	Conversation on Discord introduction	Upper-branch-ignoring-model	Upper-branch-ignoring-model	all 1 or (\(a'_{xy}\),…,0)	Though its ascending rule is weaker than BM4, it is easy to analyze.
BM3.1.1 Ecl1psed 20,July'18	Conversation on Discord introduction	Upper-branch-ignoring-model	BM2-based	all 1 or all 0	It can causes dangerous sequence as (2,0)(4,0) so that it is believed to be non-terminated.
BM3.3 rpakr Ecl1psed 22,June'19	original	Upper-branch-ignoring-model	BM3.3	No modify	definition by equations
BM2.2 koteitan 8,March'18	Definition C code	Concestor method	All branches enabled	No modify	Analysis by Bubby3 Nish said \((0,0,0)(1,1,1)\) is weaker than \(\zeta_0\). I made it by the changing bad root searching rule because I feel BM2's rule is complicating I should change the bad part ascending rule.

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:

1. Let the rightmost column \(C=(c_1, c_2, \cdots c_N)\).
2. If the all elements of \(C\) are \(0\), The bad root is not found and the Jump to 5.
3. Let the index of the lowermost non-zero element of the \(C\) be \(t\). (\(c_t\gt 0\land \forall k(k\gt t\rightarrow c_k=0)\))
4. Decide the bad root \(R=(r_1, r_2, \cdots, r_N)\) by using the bad root searching rule of the rule set.
5. remove the rightmost column \(C\).
6. Replace \([n]\) into \([f(n)]\).
7. If the bad root is not found, end the expansion procedure.
8. let the bad root \(R\) and the partial matrix in the right side of it the bad part \(B=(b_{11},b_{12},\cdots,b_{1N})(b_{21},b_{22},\cdots,b_{1N})\cdots(b_{L1},b_{L2},\cdots,b_{LN})\).
9. 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: \(\forall x~\delta_{xy}=\begin{cases}c_y-r_y&(y\lt t)\\0&(y \geq t)\end{cases}\).
10. As the flag if the each elements of the bad part ascend or not, decide a pre-ascending 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.
11. By using the ascending modification rule of the rule set, decide an ascending matrix \(A\) from the pre-ascending matrix \(A'\).
12. 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 element-wise 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

1. 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.

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Upper-branch-ignoring-model

• 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:

1. \(p\) is in the same row of \(c\).
2. \(p\) is in the left side of \(c\).
3. \(p\) is smaller than \(c\).
4. \(p\) 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 non-zero element of the rightmost column. If the parent is not found, the bad root is not found and end.

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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:

1. \(p\) is in the same row of \(c\).
2. \(p\) is in the left side of \(c\).
3. \(p\) is smaller than \(c\).

Then, the bad root \(R\) is the column, in which there is the parent of the lowermost non-zero element of the rightmost column. If the parent is not found, the bad root is not found and end.

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bad part ascending rules

All branches enabled

• \(\forall x,y(a'_{xy}=1)\)

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BM2-based

The value of the ascension matrix \(a'_{xy}\) is defined as following.

\[p(x)=\max\left\{k|k \lt x \land b_{k1} \lt b_{x1}\right\}\] \[\forall y \left(a'_{1y}=１\right)\] \[\forall x\gt 1,~y \left(a'_{xy}=\begin{cases}１ & (\mathrm{if}~a'_{p(x)y}=1 \land b_{1y} \lt b_{xy})\\ 0 & \mathrm{otherwize}\end{cases}\right) \]

\(p(x)\) 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 \(b_{1y}\).

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Upper-branch-ignoring-model

• 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:

1. \(p\) is in the same row of \(c\).
2. \(p\) is in the left side of \(c\).
3. \(p\) is smaller than \(c\).
4. \(p\) 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 \(a'_{xy}=0\).

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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\{p|p\lt x \land S_{py} \lt S_{xy} \land \exists i( p=(P_{y-1})^i(x))\} & (\mathrm{if}~y\gt 1)\\ \max\{p|p\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. \((x,~y \geq 1)\)

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Ascending modification rules

no modify

\(\forall xy (a_{xy}=a'_{xy})\)

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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 \(\forall y\gt 1 \left(a_{xy}=0\right)\).

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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 \(\forall y(a_{xy}=0)\).