Bashicu matrix system: Difference between revisions

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'''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an [[expansion system]] with the base of the standard form being \( \{((\underbrace{0,0,...,0,0}_n),(\underbrace{1,1,...,1,1}_n)) : n\in\mathbb{N}\} \) and the expansion A[n] of an array A at a natural number n being defined in the following way:
'''Bashicu matrix system''' ('''BMS''') is an [[ordinal notation system]] invented by [[BashicuHyudora]]. It is a [[sequence system]], with the sequences in question being two-dimensional arrays of natural numbers (i.e. sequences of columns, where columns are sequences of natural numbers and have the same length). It is also an [[expansion system]] with the base of the standard form being \( \{((\underbrace{0,0,...,0,0}_n),(\underbrace{1,1,...,1,1}_n)) : n\in\mathbb{N}\} \) and the expansion \( A[n] \) of an array \( A \) at a natural number \( n \) being defined in the following way:


- The parent of an entry x (a natural number in the array) is the last entry y before it in the same row, such that the entry directly above y (if it exists) is an ancestor of the entry above x, and y<x. The ancestors of an entry x are defined recursively as the parent of x and the ancestors of the parent of x.
- The parent of an entry \( x \) (an entry is a natural number in the array) is the last entry \( y \) before it in the same row, such that the entry directly above \( y \) (if it exists) is an ancestor of the entry above \( x \), and \( y<x \). The ancestors of an entry \( x \) are defined recursively as the parent of \( x \) and the ancestors of the parent of \( x \).


- If A is empty, then A[n]=A for all natural numbers n. Otherwise let C be the last column of A, and let m<sub>0</sub> be maximal such that the m<sub>0</sub>-th element of C has a parent if such an m<sub>0</sub> exists, otherwise m<sub>0</sub> is undefined. Let G and B<sub>0</sub> be arrays such that A=G+B<sub>0</sub>+(C), where + is concatenation, and the first column in B<sub>0</sub> contains the parent of the m<sub>0</sub>-th element of C if m<sub>0</sub> is defined, otherwise B<sub>0</sub> is empty.
- If \( A \) is empty, then \( A[n]=A \) for all natural numbers \( n \). Otherwise let \( C \) be the last column of \( A \), and let \( m_0 \) be maximal such that the \( m_0 \)-th element of \( C \) has a parent if such an \( m_0 \) exists, otherwise \( m_0 \) is undefined. Let \( G \) and \( B_0 \) be arrays such that \( A=G+B_0+(C) \), where \( + \) is concatenation, and the first column in \( B_0 \) contains the parent of the \( m_0 \)-th element of \( C \) if \( m_0 \) is defined, otherwise \( B_0 \) is empty.


- Say that an entry in B<sub>0</sub> "ascends" if it is in the first column of B<sub>0</sub> or has an ancestor in the first column of B<sub>0</sub>. Define B<sub>1</sub>,B<sub>2</sub>,...,B<sub>n</sub> as copies of B<sub>0</sub>, but in each B<sub>i</sub>, each ascending entry x is increased by i times the difference between the entry in C in the same row as x and the entry in the first column of B<sub>0</sub> in the same row as x.
- Say that an entry in \( B_0 \) "ascends" if it is in the first column of \( B_0 \) or has an ancestor in the first column of \( B_0 \). Define \( B_1,B_2,...,B_n \) as copies of \( B_0 \), but in each \( B_i \), each ascending entry \( x \) is increased by \( i \) times the difference between the entry in \( C \) in the same row as \( x \) and the entry in the first column of \( B_0 \) in the same row as \( x \).


- A[n]=G+B<sub>0</sub>+B<sub>1</sub>+...+B<sub>n</sub>, where + is again concatenation.
- \( A[n]=G+B_0+B_1+...+B_n \), where \( + \) is again concatenation.


For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref>[https://arxiv.org/abs/2307.04606 Proof of well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension.
For a long time, the problem of finding a proof of its well-orderedness was a famous problem in apeirology, but now there is at least a claimed proof.<ref>[https://arxiv.org/abs/2307.04606 Proof of well-foundedness of BMS]</ref> The proof utilizes [[stability]], so the problem of finding a self-contained proof that BMS is well-ordered remains open for now. A related open problem is the well-orderedness of Y sequence, which is similar enough to BMS (below the limit of BMS) that it can be considered an extension.
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