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I wrote another definition of Bashicu_matrix_system (BMS) with BM4 rule in mathematical notation.

Japanese version is here.

## Bashicu Matrix System (version BM2.3=BM4)

K is the large number which is originally defined by Bashicu in BASIC pseudo code. $$\mathrm{expand}$$ is the function to expand a BMS with the rule BM2.3=BM4. The size of this number is unknown. (BM2.3 and BM4 are the same rule.)

\begin{eqnarray*} \mathrm{Number:}~K&=&\mathrm{Bm}^{10}(9)\\ \mathrm{Function:}~\mathrm{Bm}(n)&=&\mathrm{expand}((\underbrace{0,0,\cdots,0}_{n+1})(\underbrace{1,1,\cdots,1}_{n+1})[n])\\ \mathrm{Rule:}~\mathrm{expand}([n])&=&n\\ \mathrm{expand}({\boldsymbol S}[n])&=&\left\{\begin{array}{ll} \mathrm{expand}({\boldsymbol S}_0\cdots{\boldsymbol S}_{X-2}[f(n)])&(\mathrm{if}~\forall y~S_{(X-1)y}=0)\\ \mathrm{expand}({\boldsymbol G}{\boldsymbol B}^{(0)}{\boldsymbol B}^{(1)}{\boldsymbol B}^{(2)} \cdots {\boldsymbol B}^{(f(n))}[f(n)])&(\mathrm{otherwise})\\ \end{array}\right.\\ \mathrm{Activation~function:}~f(n)&=&n^2\\ \mathrm{Matrix:}~{\boldsymbol S}&=&{\boldsymbol S}_0{\boldsymbol S}_1\cdots{\boldsymbol S}_{X-1}\\ \mathrm{Vector:}~{\boldsymbol S}_x&=&(S_{x0},S_{x1},\cdots,S_{x(Y-1)})\\ \mathrm{Good~part:}~{\boldsymbol G}&=&{\boldsymbol S}_0{\boldsymbol S}_1\cdots{\boldsymbol S}_{r-1}\\ \mathrm{Bad~part:}~{\boldsymbol B}^{(a)}&=&{\boldsymbol B}_0^{(a)}{\boldsymbol B}_1^{(a)}\cdots{\boldsymbol B}_{X-2-r}^{(a)}\\ {\boldsymbol B}_x^{(a)}&=&(B_{x0}^{(a)},B_{x1}^{(a)},\cdots,B_{x(Y-1)}^{(a)})\\ B_{xy}^{(a)}&=&S_{(r+x)y}+a\Delta_{y}A_{xy}\\ \mathrm{Ascension~offset:}~\Delta_{y}&=&\left\{\begin{array}{ll} S_{(X-1)y}-S_{ry}&(\mathrm{if}~y\lt t)\\ 0 &(\mathrm{if}~y\geq t) \end{array}\right.\\ \mathrm{Ascension~matrix:}~A_{xy}&=&\left\{\begin{array}{ll} 1 &(\mathrm{if}~ \exists a( r=(P_{y})^a(r+x)))\\ 0 &(\mathrm{otherwise}) \end{array}\right.\\ \mathrm{Lowermost~nonzero:}~t&=&\max\{y|S_{(X-1)y}\gt 0\}\\ \mathrm{Bad~root:}~r &=& P_t(X-1)\\ \mathrm{parent~of}~S_{xy}:~P_{y}(x)&=&\left\{\begin{array}{ll} \max\{p|S_{py} \lt S_{xy} \land \exists a\gt 0( p=(P_{y-1})^a(x))\} & (\mathrm{if}~y\gt 0)\\ \max\{p|S_{py} \lt S_{xy} \land p\lt x \} & (\mathrm{if}~y=0)\\ \end{array}\right.\\ \end{eqnarray*}

## Pair Sequence System

The following K is the definition of the Pair_sequence_number and $$\mathrm{expand}$$ is the expansion rule of Pair Sequence System which is the special type of Bashicu Matrix System. The size of this number is $$f_{\psi(\Omega_\omega)+1}(10)$$ in FGH.

\begin{eqnarray*} \mathrm{Number:}~K&=&\mathrm{Pair}^{10}(9)\\ \mathrm{Function:}~\mathrm{Pair}(n)&=&\mathrm{expand}\left((0,0)(1,1)\cdots (n+1,n+1)[n]\right)\\ \mathrm{Rule:}~\mathrm{expand}([n])&=&n\\ \mathrm{expand}({\boldsymbol S}[n])&=&\left\{\begin{array}{ll} \mathrm{expand}((S_{00},S_{01})\cdots(S_{(X-2)0},S_{(X-2)1})[f(n)])&(\mathrm{if}~\forall yS_{(X-1)y}=0) \\ \mathrm{expand}({\boldsymbol G}{\boldsymbol B}^{(0)}{\boldsymbol B}^{(1)}{\boldsymbol B}^{(2)} \cdots {\boldsymbol B}^{(f(n))}[f(n)])&(\mathrm{otherwise}) \end{array}\right.\\ \mathrm{Activation~function:}~f(n)&=&n^2\\ \mathrm{Matrix:}~{\boldsymbol S}&=&(S_{00},S_{01})(S_{10},S_{11})\cdots (S_{(X-1)0},S_{(X-1)1})\\ \mathrm{Good~part:}~{\boldsymbol G}&=&(S_{00},S_{01})(S_{10},S_{11})\cdots (S_{(r-1)0},S_{(r-1)1})\\ \mathrm{Bad~part:}~{\boldsymbol B}^{(a)}&=&(B_{00}^{(a)},B_{01}^{(a)})(B_{10}^{(a)},B_{11}^{(a)})\cdots (B_{(X-2-r)0}^{(a)},B_{(X-2-r)1}^{(a)})\\ B_{x0}^{(a)}&=&\left\{\begin{array}{ll} S_{(r+x)0}+a(S_{(X-1)0}-S_{r0})&~(S_{(X-1)1}\gt 0)\\ S_{(r+x)0} &~(\mathrm{otherwise})\\ \end{array}\right.\\ B_{x1}^{(a)}&=&S_{(r+x)1}\\ \mathrm{Bad~root:}~r &=& \left\{\begin{array}{ll} P_1(X-1) & (S_{(X-1)1} \neq 0)\\ P_0(X-1) & (\mathrm{otherwise}) \end{array}\right.\\ \mathrm{parent~of}~S_{x1}:~P_1(x)&=&\max\{p|S_{p1} \lt S_{x1} \land \exists a\gt 0( p=(P_0)^a(x))\}\\ \mathrm{parent~of}~S_{x0}:~P_0(x)&=&\max\{p|S_{p0} \lt S_{x0} \land p\lt x\}\\ \end{eqnarray*}

## Primitive Sequence System

The following K is the definition of the Primitive Sequence Number and $$\mathrm{expand}$$ is the expansion rule of Pair Sequence System which is the further special type of Pair Sequence System. The size of this number is $$f_{\psi(\epsilon_0)+1}(10)$$ in FGH.

\begin{eqnarray*} \mathrm{Number:}~K&=&\mathrm{Primivive}^{10}(9)\\ \mathrm{Function:}~\mathrm{Primivive}(n)&=&\mathrm{expand}\left((0,1,\cdots,n+1)[n]\right)\\ \mathrm{Rule:}~\mathrm{expand}([n])&=&n\\ \mathrm{expand}({\boldsymbol S}[n])&=&\left\{\begin{array}{ll} \mathrm{expand}((S_0,S_1,\cdots,S_{X-2})[f(n)])&(\mathrm{if}~S_{X-1}=0) \\ \mathrm{expand}({\boldsymbol G}{\boldsymbol B}\underbrace{{\boldsymbol B}{\boldsymbol B} \cdots {\boldsymbol B}}_{f(n)~\mathrm{times}}[f(n)])&(\mathrm{otherwise}) \end{array}\right.\\ \mathrm{Activation~function:}~f(n)&=&n^2\\ \mathrm{Sequence:}~{\boldsymbol S}&=&(S_0, S_1, \cdots, S_{X-1})\\ \mathrm{Good~part:}~{\boldsymbol G}&=&(S_0, S_1, \cdots, S_{r-1})\\ \mathrm{Bad~part:}~{\boldsymbol B}&=&(S_r, S_{r+1}, \cdots, S_{X-2})\\ \mathrm{Bad~root:}~r &=& \max\{p|S_p \lt S_{X-1} \land p \lt X-1\}\\ \end{eqnarray*}

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