YY sequence or Yabai Y sequence is a difference sequence system introduced by a Japanese googologist Yukito on 17/07/2020.[1][2] It is intended to be a notation stronger than Y sequence. Although it has not been formalised yet, the expression \((1,2,5)\) in YY sequence is expected to correspond to the limit of Bashicu matrix system version 2.3 and the expression \((1,3)\) in Y sequence, and the expression \((1,3)\) in YY sequence corresponds to the limit of Y sequence.
Convention[]
In order to distinguish a sequence in YY sequence and other sequence notations, people frequently put a shorthand of the notation in front of the expression. For example, (1,3) in Y sequence is denoted by Y(1,3), (1,3) in YY sequence is denoted by YY(1,3), and so on.
As Y sequence allows ω as an entry only when we consider the limit expression Y(1,ω), the limit expression of YY sequence is denoted by YY(1,ω). It does not mean that Y sequence and YY sequence are extended to be systems of sequences of transfinite entries.
Extensions[]
Yukito also indicated the existence of higher extensions on 19/07/2020: YYY sequence, YYYY sequence, and Yω sequence.[2] According to Yukito, there is a sequence "Y1 sequence, Y2 sequence, Y3 sequence, …" such that Y1 sequence is Y sequence, Y2 sequence is YY sequence, Y3 sequence is YYY sequence, Y4 sequence is YYYY sequence, and the limit of Yn(1,ω) with n<ω "coincides" with Yω(1,3).
The last equality probably means that there exists a correspondence from standard expressions in Yn sequence to ordinals for any ordinals n≦ω such for which expansions give fundamental sequences and the limit of ordinals corresponding to Yn(1,ω) coincides with the ordinal corresponding to Yω(1,3). Although none of the systems has been formalised, it means that the whose system is expected to be much stronger than Y sequence, which is expected to be much stronger than BMS version 2.3.
Although there was a table of analysis among BMS, Y sequence, YY sequence, YYY sequence, and YYYY sequence which tells us how higher extension outgrows YY sequence, it has been deleted since 19/07/2020. According to Yukito, he found a bug in Yn sequence for all n≧3.
Nayuta mentioned that Yn sequence employs a similar idea to an unborn version of N primitive called NΩ.0, and some weird property of YYY(1,3,3) implies that NΩ.0 might not work as intended. Therefore NΩ.0 is canceled.
Relation to Bell numbers[]
A Japanese googologist koteitan[3] discovered that the sequence YY(1,2,5,15,52,203) appearing in the expansion of YY(1,3) is an initial segment of the sequence of Bell numbers.[4][5][6] Yukito agreed that the expansion of YY(1,3) coincides with the sequence of Bell numbers. According to Yukito, he actually knew Bell numbers but the equality is a coincidence.
Sources[]
- ↑ The user page of Yukito in the Japanese Googology Wiki.
- ↑ 2.0 2.1 Yukito, YYvsY, Google Spreadsheet.[dead link]
- ↑ The user page of koteitan in Googology Wiki.
- ↑ koteitan, Relation between YY sequence and Bell numbers.
- ↑ Bell numbers, A0000110 - OEIS.
- ↑ E. T. Bell, Exponential polynomials], Annals of Mathematics, Second Series, Vol. 35, No. 2 (1934), pp. 258--277.
See also[]
By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By aster: White-aster notation · White-aster
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's function
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4 · Googology Wiki can have an article with any gibberish if it's assigned to a number
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea
By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system
By ふぃっしゅ (Fish): Ackermann function
By koteitan: Ackermann function · Beklemishev's worms · KumaKuma ψ function
By Mitsuki1729: Ackermann function · Graham's number · Conway's Tetratri · Fish number 1 · Fish number 2 · Laver table
By みずどら: White-aster notation
By Naruyoko Naruyo: p進大好きbot's Translation map for pair sequence system and Buchholz's ordinal notation · KumaKuma ψ function · Naruyoko is the great
By 猫山にゃん太 (Nekoyama Nyanta): Flan number 4 version 3 · Fish number 5 · Laver table
By Okkuu: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 5 · Fish number 6
By rpakr: p進大好きbot's ordinal notation associated to Extended Weak Buchholz's function · Standardness decision algorithm for Taranovsky's ordinal notation
By ふぃっしゅ (Fish): Computing last 100000 digits of mega · Approximation method for FGH using Arrow notation · Translation map for primitive sequence system and Cantor normal form
By Kihara: Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's \(\mathcal{O}\)
By koteitan: Translation map for primitive sequence system and Cantor normal form
By Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
By Nayuta Ito: Comparison of Steinhaus-Moser Notation and Ampersand Notation
By Okkuu: Verification of みずどら's computation program of White-aster notation
By p進大好きbot: Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the \(\Sigma_1\)-soundness and the pointwise well-definedness of \(\textrm{TR}(T,n)\) for the case where \(T\) is the formalisation of the base theory
By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud
See also: Template:Googology in Asia