Primitive sequence number

The primitive sequence number is a large number made by a Japanese Googology Wiki user BashicuUser:BashicuHyudora. It was posted to the large number thread Part.10Although 2channel is banned in China and it bans the access from United States, the archive can be seen here. of website 2channel as the program (message No. 109) and its examples (No. 135-140)the article explaining itGoogology in Japan - exploring large numbers. The algorithm calculating primitive sequence number is called primitive sequence system; it is like Beklemishev's worm and it has the strength of \(f_{\varepsilon_0}(n)\). The size of primitive sequence number is about \(f_{\varepsilon_0+1}(10)\).
Pair sequence system, which is the expansion of primitive sequence system into two rows by Bashicu, has strength approximated to \(f_{\psi_0(\Omega_{\omega})}(n)\) with respect to Buchholz's function. The expansion into three rows by Bashicu is called as Trio sequence system. The generalization of them by Bashicu are called as Bashicu matrix system. It has \(n\) rows. Their investigation is currently in progress.
Hyper primitive sequence system, which is an extension of primitive sequence system using difference sequence system by a Japanese Googology Wiki user Yukito has the strength of \(f_{\psi(\Omega_{\omega})}(n)\) with respect to Buchholz's function.
== Definition ==
The first definition of the primitive sequence number was posted into the communication website 2channel large number thread Part.10its archive as the pseudo code in BASIC language. After that, the definitions are maintained on the article Summary of the large number in BASIC Language in Wikia user blog.
== Mathematical definition ==
Although the original definition is written in the pseudo code of BASIC language, it can be described in the mathematical definition as following:
=== Large function ===
Primitive sequence is a list of non negative integer \(S = (S_0, S_1, \ldots, S_k)\).
Primitive sequence has the behavior as the function from a non negative integer \(n\) into a non negative integer, the value of \(S[n]\) is defined as below:
* \( ()[n]=n \).
* good part \(g\) and bad part \(b\) of the primitive sequence are defined as following: (\(r\) is the largest non negative integer which gives \(r<k\) and \(S_r<S_k\) in below.)
** When \(r\) exists, let \(g\) and \(b\) be \(g=(S_0,\ldots,S_{r-1}),~b= (S_{r},\ldots,S_{k-1})\).
** When \(r\) does not exists, let them be \(g=(S_0, \ldots, S_{k−1}),~b=() \).
* Let \(S[n]\) be \(S[n] = (g \frown b \underbrace{\frown b \frown \cdots \frown b}_{f(n) \mathrm{'s}~\frown b})[f(n)]\). Where, \(f(n)\) is \(f(n)=n^2\).
That's the definition of \(S[n]\).
\(\frown\) is the concatenation of the sequence, for example, \((0, 3, 2) \frown (1, 4, 5) = (0, 3, 2, 1, 4, 5)\).
That is the definition modeled after Beklemishev's worm. Note that the position which make division of \(g\) and \(b\) is different from the one of Beklemishev's worm.
=== Large numbers ===
Primitive sequence number is defined as \(P^{10}(9)~(P(n)=(0,\cdots,n)[n])\) with respect to the primitive sequence system above.
In addition, a Taiwanese Googologist Mi created and drew a Gijinka (anthropomorphization) of \((0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)[10]\) (19 0s), and a Japanese Googologist 銀杏 coined the name 久界 (kyuukai) for it.銀杏, tweet. Retrieved December 4th 2019. A Japenese Googlogy Wiki user koteitan created a dancing video of 久界, a Japanese Googology Wiki user fish, and a Japanee Googologist 巨大数大好きbot (cf. #External Links).
== Explanation of definition ==
The easy explanation of finding \(r\) is, searching the number \(S_k-1\) from the right side of the sequence, the index of the one is found as \(r\). For example, in the case of \((0, 1, 2, 3, 3, 1, 2, 3, 2, 3, 2, \underbrace{2}_{=S_k})[2]\) , the rightmost number \(S_k\) is \(2\), search \(1\) to the left side from there, the first found \(1\) is \(S_5=1\) so that \(r=5\). In this case \(b\) and \(g\) become \((\underbrace{0, 1, 2, 3, 3}_{=g}, \underbrace{1, 2, 3, 2, 3, 2}_{=b}, 2)[2]\). The bracket is \([2]\) and \(f(n)=n^2\), they give
\begin{eqnarray*}
S[2]&=&g\frown b~\underbrace{\frown b\frown b\frown b\frown b}_{2^2~{\rm ’s}}\\
&=&(\underbrace{0, 1, 2, 3, 3}_{=g}, \underbrace{1, 2, 3, 2, 3, 2}_{=b}, \underbrace{1, 2, 3, 2, 3, 2}_{=b}, \underbrace{1, 2, 3, 2, 3, 2}_{=b}, \underbrace{1, 2, 3, 2, 3, 2}_{=b}, \underbrace{1, 2, 3, 2, 3, 2}_{=b})[4].
\end{eqnarray*}
This \(S_r\) is often called as bad root.
== Others ==
Primitive sequence is often written in the format like \(S = (S_0 S_1 \ldots S_k) = (S_0)(S_1) \ldots (S_k)\) as the special case of the Bashicu matrix system.
The calculation of the Primitive sequence system can be done with Bashicu Matrix Calculator. For example, the calculation result of (0)(1)(2)(3)[2] becomes like this. (It grows if you give the bigger 'maximum length'.) Although it might look to be continue to grow, this sequence always achieve to the empty sequence at the last and it leave \([n]\). Then the calculation is terminated after the output of \([n]=n\). If you give (0)(1)(2)[2] with \(f(n)=n\), you can see the process until the termination.
In addition that, it can be modified to match to Hardy hierarchy and it can calculate \(H_{\omega^\omega}(2) = 8\) as like this.
Considering the function \(P(n) = (0,1,2, \ldots, n)[n]\), The grow rate of the \(P(n)\) is about \(f_{\varepsilon_0}(n)\). Primitive sequence number is \(P^{10}(9)\) and its size is about \(f_{\varepsilon_0+1}(10)\).
=== Correspondence with ordinals ===
There is a correspondence assigning to each primitive sequence an ordinal less than \(\varepsilon_0\) so that the corresponding ordinals are always reduced in the calculation of primitive sequences. Since there is no infinite descending chain of ordinals, the calculation of the primitive sequences necessarily reach end. Googology Wiki user 12AbBa gave an introductory lecture on how to prove the termination of primitive sequence in a series of lectures on ordinal collapsing functions in Ross Mathematics ProgramZongshu Wu, Ordinal Collapsing Functions, lecture note in Ross Mathematics Program, Google drive, 2023-07-09 (updated and retrieved on 2024-06-16), and Japanese Googology Wiki user Mizudora independently proved the termination of primitive sequence.Mizudora (みずどら) Termination proof of primitive sequence (in Japanese) 2024-07-30.
Indeed, there is a program which displays the ordinals corresponding to primitive sequences (cf. #Correspondence with ordinals 2). The inverse correspondence from ordinals to primitive sequences of standard form is demonstrated in the following way:
# Write the hydra tree corresponding to a given ordinal \(\alpha\). The figure above represents \(\omega^{\omega^\omega+\omega^3}+\omega^{\omega^2+1}\) as the paper by Kirby and Paris (1982).Kirby, L.; Paris, J. (1982), "Accessible independence results for Peano arithmetic"
# Start below from root node.
# Add a new element 0 into the last of the sequence when you go up in 1 above from the root node.
# Add a new element "(the last element) + 1" into the last of the sequence when you go up in 1 above from the other node.
# When you reach the end of the branch and go down to the branching node (common anscestor) and go on the another branch, add "(the branching element) + 1".
# so that the number in the node becomes the value "(height)-1".
# Here is the example to corresponding with the figure like this. The part of \(\omega^{\omega^\omega}\) is shown as (0,1,2,3). after that, go down 3 nodes, go to the next branch (segment), get (1,2,2,2), it is added into the sequence. After that, (0,1,2,2,1) is added as the same. Finally, the hydra tree is corresponded with the primitive sequence (0,1,2,3,1,2,2,2,0,1,2,2,1).
#\(\omega^{\omega^\omega+\omega^3}+\omega^{\omega^2+1} = (0,1,2,3,1,2,2,2,0,1,2,2,1)\).
=== Examples ===
Here are examples of the correspondence between ordinals below \(\varepsilon_0\) and primitive sequences of standard form.
\begin{eqnarray*}
1 &=& (0) \\
2 &=& (0,0) \\
3 &=& (0,0,0) \\
\omega &=& (0,1) \\
\omega+1 &=& (0,1,0) \\
\omega \cdot 2 &=& (0,1,0,1) \\
\omega^2 &=& (0,1,1) \\
\omega^2+\omega &=& (0,1,1,0,1) \\
\omega^3 &=& (0,1,1,1) \\
\omega^\omega &=& (0,1,2) \\
\omega^{\omega^\omega} &=& (0,1,2,3) \\
\omega^{\omega^{\omega^\omega}} &=& (0,1,2,3,4) \\
\omega^{\omega^{(\omega^\omega+1)}} &=& (0,1,2,3,4,2) \\ 
\omega^{\omega^{\omega^{\omega^\omega}}} &=& (0,1,2,3,4,5) \\
\end{eqnarray*}
=== Programs ===
==== Expansion ====
The python code which performs Primitive sequence system is shown in the site The Py_1 Function, which is the online JAM to make the large number with short codes.
The code is 72 bytes and it calculates (0)(1)(2)(3)(4)(5)(6)(7)(8)(9)[9] in \(f(n)=n^2\).
x,*m=range(9,-1,-1)
while m:
 q,*m=m;x*=x
 if q:m=m[:m.index(q-1)+1]*x+m
Here are a few codes:
* Expansion calculators:
** in C by fish: Bashicu matrix calculator
** in Javascript by koteitan: yaBMS
** in Javascript by koteitan: prss.html (See also a blog post with detailed explanations in Japanese or its English translation.)
* Ordinal converters:
** in Python 2 by fish: Primitive sequence analyzer 原始数列解析 (See also a blog post with detailed explanations in Japanese.)
** in Javascript by koteitan: p2o
* Hydra notation converters:
** in Javascript by koteitan: hydraviewer(for Pair sequence system)
** in Javascript by naruyoko: BMS Hydra viewer(Multi-row supported)
==== Correspondence with ordinals ====
The assignment in #Correspondence with ordinals is also implemented into programs:
* fish, Primitive sequence analyzer 原始数列解析
* koteitan, Primitive sequence to ordinal in four lines Javascript
== References ==
== External Links ==
* koteitan, 久界が踊ります. (A link to a dancing video of 久界, fish, and 巨大数大好きbot.)
==See also==
* Pair sequence system
* Bashicu matrix system
* Hyper primitive sequence system
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