basmat - Bashicu Matrix Calculator

This program calculates Bashicu Matrix. Bashicu matrix system is a notation designed to produce large numbers. 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 written in the form of $(a_{11},a_{21})(a_{12},a_{22})(a_{13},a_{23})$ ; sequence of transpose of each column. With an algorithm invented by Bashicu in 2014 and updated afterwards, Bashicu matrix BM works as a function from a natural number n to a natural number BM[n] (provided that the calculation ends), and written as (0,0)(1,1)(1,1)[3]. It is known that 2-row matrix, pair sequence, can be approximated with Hardy function. When the function is approximated with Hardy function, the matrix itself represents the ordinal of the Hardy function, and therefore can be written as:

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

See Definition of Bashicu Matrix if you want to understand the definition of Bashicu Matrix by reading the source code.

Installation

See Installation.

Usage

basmat [-v ver] [-o opt] [-s seq] [-t stp] [-dh] ini

• ini - Initial variable in the form of BM[n],
  • BM sequence expression of Bashicu matrix.
  • n: natural number. n=2 if not given.
  • ex. “(0,0)(1,1)[3]” (Quote to escape brackets in shell)
• ver - Version of Bashicu matrix system. Default = 4.
  • Available versions: 1, 2, 2.1, 2.2, 2.3, 3, 3.1, 3.2, 3.3, 4
  • See basmat version for detail
• opt - Calculation option.
  • opt = 1: n is constant. (Default)
  • opt = 2: n = n+1 for each loop.
  • opt = 3: n = n*n for each loop.
  • opt = 4: Simulate Hardy. n=n+1 for successor, and copy n-1.
• seq - Maximum length of sequence. Default = 20.

• stp - Maximum step of calculation. Default = 0 (no limit).

• -d Show detailed process of calculation.
• -T Test for development.
• -h Show this help and exit.

Example:

 basmat -d "(0)(1)(2)[3]"
 basmat -o 2 -s 100000 "(0,0)(1,1)[3]"
 basmat -o 4 -s 100000 "(0,0)(1,1)[3]"
 basmat -o 1 -s 13000 -t 100000 "(0,0)(1,1)[3]"

Version History

See ChangeLog.

Web interface

Web interface is available.

The online version of basmat has some limit on the length of initial variable, the maximum length <= 10000, and maximum step of calculation = 1000.

Source code of the web interface is also available.

Authors

• Bashicu invented versions 1,2,3 and wrote BASIC program
• Fish wrote this C program using Bashicu’s algorithm
• koteitan invented 2.1, 2.2 and 2.3 and wrote C program of versions 2.1, 2.2, 2.3, 3.1 and 3.2
• Nish invented version 3.1 and 3.2
• rpakr invented version 3.3
• Ecl1psed invented version 3.3