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Bashicu matrix system
Typematrix
Based on\(n^2\)
Growth rate\(f_{>\psi(\psi_{I_\omega}(0))}(n)\)

Bashicu matrix system is a notation designed to produce large numbers.[1] It was invented by Bashicu in 2014.[2] By approximating with FGH, 1-row matrix (primitive sequence system) is level \(\varepsilon_0\) and 2-row matrix (pair sequence system) is level \(\vartheta(\Omega_\omega)\). 3-row matrix \begin{pmatrix}
 0 & 1 & 2 & 3 \\
 0 & 1 & 2 & 2 \\
 0 & 1 & 1 & 1
\end{pmatrix} is level \(\psi(\psi_{I_\omega}(0))\). FGH ordinal corresponding to n-row matrix \begin{pmatrix}
 0 & 1 \\
 0 & 1 \\
 \vdots & \vdots \\
 0 & 1
\end{pmatrix} is not known. The corresponding function is possibly weaker than Loader's function.

Bashicu matrix has 2 versions.[3] The first version, BM1 is equivalent to pair sequence system. Calculation of BM1 does not always stop in 3-row matrix system, and therefore BM2 should be used in high order system. As BM2 is difficult to understand, BM1 is still useful for understanding the basic behaviour of Bashicu matrix system.

Definition Edit

Notation Edit

Bashicu matrix is a matrix such as

\begin{pmatrix}
 a_{11} & a_{12} & a_{13}\\
 a_{21} & a_{22} & a_{23}
\end{pmatrix}

where all elements are nonnegative integers. The matrix can be also written as the sequence of transposes of columns (sequence expression of Bashicu matrix);

\((a_{11},a_{21})(a_{12},a_{22})(a_{13},a_{23})\)

Bashicu matrix \(BM\) works as a function from a natural number \(n\) to a natural number \(BM[n]\), and is written as \(S[n]\), where \(S\) is either matrix or sequence expression. When the function is approximated with a function from Hardy hierarchy with ordinal index \(\alpha\), we say the matrix \(BM\) represents \(\alpha\). For example:

\begin{pmatrix}
 0 & 1 & 2 & 3 & 3\\
 0 & 1 & 2 & 3 & 2
\end{pmatrix}
= (0,0)(1,1)(2,2)(3,3)(3,2) = \psi(\psi_1(\Omega_2))

Transformation rules Edit

The rules below are described for sequence expression of a matrix.

Terminology Edit

  • The length of a sequence is the number of pairs of brackets.
  • A sequence of length 1 (i.e. has one pair of brackets) is called an element of the sequence.
  • \(S\) denotes any sequence.
  • \(Z\) denotes \((0,0,...,0)\), which has one or more zeros.
  • \(f(n) = n^2\). (in variants of the notation this can be modified)
  • \(A\frown B\) is concatenation of sequence. For example, \((0,0)(1,1)\frown (2,2)(3,3)=(0,0)(1,1)(2,2)(3,3)\)

Element comparisonEdit

From here description is about BM1.

Denote \(A = (a_0,a_1,...,a_m), B = (b_0,b_1,...,b_m), D=\{ 0,1,...,m \}\). We make the following definition:

\[A < B \Leftrightarrow \forall i \in D ((a_i<b_i) \vee \exists j \in D (b_j=0 \wedge j \le i))\]

In simple words: \(A<B\) only if either \(B\) has no zeroes and all elements of \(B\) are larger than corresponding elements of \(A\), or \(B\) has some zeroes and all numbers in \(B\) to the left of leftmost zero are larger than corresponding elements of \(A\).

Calculation rules Edit

Rule 1. \([n]=n\)

Rule 2. \(S Z[n]=S[f(n)]\)

Rule 3. If neither Rule 1 or 2 is applicable, apply the rules below in order.

Rule 3-1. Denote the i-th element from the left in \(S\) by \(S_i\) and the length of \(S\) by \(n\). That is, \(S=S_1S_2\cdots S_n\). Note that \(S_n\) is not equal to \(Z\) since Rule 2 is not applicable.

  • If there is no element that is smaller than \(S_n\), replace \(S_n\) by \(Z\); \(S = S_1 S_2 \ldots S_{n-1} Z\) (and then apply Rule 2).
  • If there is at least one element that is smaller than \(S_n\), let the rightmost one be \(S_i\). We define the good part of the sequence to be \(G = S_1 \ldots S_{i-1}\), the bad part of the sequence to be \(B = S_i \ldots S_{n-1}\), \(L = S_i\), and \(N = S_n\). Note that now \(S=G\frown B\frown N\).

Rule 3-2. From \(L = (L_0, L_1, \ldots ,L_m)\) and \(N = (N_0,N_1, \ldots ,N_m)\), \(\Delta=(\Delta_0,\Delta_1,\cdots\Delta_m)\) (difference) is calculated as

  • \(\Delta_i=0\) if there is a zero among \(N_0\) to \(N_{i+1}\) (letting \(N_{m+1}=0\))
  • \(\Delta_i=N_i-L_i\) otherwise

As \(L < N\) by choice of \(L\), \(\Delta_i \ge 0\).

Rule 3-3: \(B(i)\) is defined as follows.

  • \(B(0) = B\)
  • \(B(i+1)\) is \(B(i) + \Delta\); here \(+\) means coordinate-wise addition of \(\Delta\) to every element of \(B(i)\). For example, when \(B(0) = (1,1,1)(2,2,2)(3,3,3)\) and \(\Delta = (3,1,0)\), we have \(B(1) = (4,2,1)(5,3,2)(6,4,3)\) and \(B(2) = (7,3,1)(8,4,2)(9,5,3)\).

Rule 3-4: \(S[n] = \{ G \frown B(0) \frown B(1) \frown \ldots\ \frown B(f(n))\}[f(n)] \)

Definition using a program Edit

Bashicu originally defined the system with BASIC program language. The program by Bashicu was not intended to be actually run, and because of that Fish wrote a program, Bashicu matrix calculator, intended for demonstrating the calculation process (the demonstration program was verified by Bashicu). Therefore, official definition of Bashicu matrix can be found in the source code of this program.[4] It can calculate both BM1 and BM2.[5] Bashicu matrix calculator also has a web interface. The program has 4 options of "n increment". The original definition of Bashicu matrix is the option of "n=n * n", and other options are variants. By using the option of "Simulate Hardy function", the calculation exactly matches Hardy function with Wainer hierarchy for ordinals below \(\epsilon_0\).

Example Edit

\((0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,2,0)[2]\) is calculated as follows:

  • By Rule 3-1: In the sequence \(S = (0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,2,0)\), we look for the rightmost element that is smaller than \(S_4 = (4,2,0)\), which is \(S_1 = (1,1,1)\). Therefore \(G = (0,0,0), B = (1,1,1)(2,2,2)(3,3,3), N = (4,2,0)\)
  • By Rule 3-2: As \(L = (1,1,1)\) and \(N = (4,2,0)\), \(\Delta = (3,0,0)\)
  • By Rule 3-3: \(B(0) = (1,1,1)(2,2,2)(3,3,3), B(1) = (4,1,1)(5,2,2)(6,3,3), B(2) = (7,1,1)(8,2,2)(9,3,3), B(3) = (10,1,1)(11,2,2)(12,3,3), B(4) = (13,1,1)(14,2,2)(15,3,3)\)
  • By rule 3-4: \(S [2] = {G \frown B(0) \frown B(1) \frown B(2) \frown B(3) \frown B(4)} [4]\)

Therefore \((0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,2,0)[2] = (0,0,0)(1,1,1)(2,2,2)(3,3,3)(4,1,1)(5,2,2)(6,3,3)(7,1,1)(8,2,2)(9,3,3)(10,1,1)(11,2,2)(12,3,3)(13,1,1)(14,2,2)(15,3,3)[4]\) By writing in matrix form, \[\begin{pmatrix} 0 & 1 & 2 & 3 & 4\\ 0 & 1 & 2 & 3 & 2\\ 0 & 1 & 2 & 3 & 0\\ \end{pmatrix}[2] = \begin{pmatrix} 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15\\ 0 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 \\ \end{pmatrix}[4]\]

This calculation matches calculation result of Bashicu matrix calculator.

Analysis Edit

The first question was whether the calculation always terminates. It was not answered until User:KurohaKafka posted a proof of this fact on Japanese BBS 2ch.net in 2016.[6] However, User:Hyp cos disproved it in the talk page of this article by showing a non-teminating sequence. After that, Bashicu updated the system, making Bashicu Matrix verion 2 (BM2), showing the algorithm by a BASIC program,[7] hoping that calculation of \begin{pmatrix}
 0 & 1 \\
 0 & 1 \\
 \vdots & \vdots \\
 0 & 1
\end{pmatrix} terminates (but not proven yet).

The growth rate of 1-row matrix (primitive sequence system) is \(f_{\varepsilon_0}(n)\)[8] and the growth rate of 2-row matrix (pair sequence system) is \(f_{\vartheta(\Omega_\omega)}(n)\) in BM1.

Complete analysis of the growth rate of 3-row matrix (trio sequence system) with FGH is difficult because the system is so strong. People are analyzing the system to some extent.[9][10][11][12][13]

Sources Edit

  1. Bashicu Matrix Calculator
  2. Bashicu's GWiki account
  3. User blog:Kyodaisuu/Update of Bashicu Matrix Calculator
  4. Source code of Bashicu matrix calculator
  5. User blog:Kyodaisuu/Update of Bashicu Matrix Calculator
  6. Proof that calculation of "standard form" Bashicu matrix terminates and copy
  7. BASIC program of Bashicu Matrix system
  8. A blog post introducing primitive sequence system
  9. Analysis of trio sequence system by Bashicu with his own ordinal notations
  10. Bashicu's calculation in more detail
  11. Analysis by KurohaKafka
  12. Analysis by Bubby3
  13. Analysis by Googleaarex

See also Edit

Googology in Asia

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Pair sequence number · Bashicu matrix system
Indian counting system: Lakh · Crore
Chinese and Japanese counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Buddhist text: Tallakshana · Dvajagravati · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra
Other: Taro's multivariable Ackermann function · Sushi Kokuuhen

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