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Beklemishev's worms are a construction described by Lev D. Beklemishev (Russian: Беклемишев Лев Дмитриевич[1][2]) that result in a one-player game that takes a long time to terminate.[3]

It is closely related to Kirby-Paris hydras.

Description[]

A worm is simply a list of nonnegative integers \([W_0, W_1, \ldots, W_n]\). In a game Beklemishev calls "the Worm battle," our hero Cedric is presented with an arbitrary worm \(W\), and his task is to reduce it to an empty list. On turn \(m\) of the game, the worm is altered by the function \(\text{next}(W, m)\):

  • If \(W_n = 0\), then \(\text{next}(W, m) = [W_0, W_1, \ldots, W_{n-1}]\). (That is, Cedric chops off the worm's head.)
  • Otherwise, define \(k = \max_{i < n} W_i < W_n\). We define the good part of the sequence to be \(g = [W_0, \ldots, W_k]\) and the bad part to be \(b = [W_{k+1}, \ldots, W_{n-1}, W_n - 1]\). (Note that \(W_n\) is decremented by 1.) If \(k\) is nonexistent, define \(g\) to be an empty list and \(b = [W_0, \ldots, W_{n-1}, W_n - 1]\). We then define \(\text{next}(W, m) = g + b + b + \cdots + b + b\) with \(m+1\) copies of \(b\). (Here + means sequence concatenation, so for example [0, 3, 2] + [1, 4, 5] = [0, 3, 2, 1, 4, 5].)

Beklemishev proved, in a theorem he calls the Worm principle, that Cedric can always defeat the worm regardless of the initial value of \(W\). He further showed that this fact is unprovable in Peano arithmetic.

The Worm's expansion rule is almost identical to Primitive Sequence System's expansion system.

From this, we can create a specific fast-growing function. Define \(\text{Worm}(n)\) to be the number of steps required to defeat a worm starting with \(W = [n]\). Then \(\text{Worm}(n)\) is a function that dominates all functions provably recursive in Peano arithmetic, giving the function a growth rate comparable to \(f_{\varepsilon_0}(n)\).

Examples[]

A Japanese Googology Wiki user koteitan created a simulator for Beklemishev's worms (cf. #Computer programs). Here are examples of small inputs:

  • Start: [1]
  • Step 1: [0,0]
  • Step 2: [0]
  • Step 3: []

So \(\text{Worm}(1) = 3\).

  • Start: [2]
  • Step 1: [1,1]
  • Step 2: [1,0,1,0,1,0]
  • Step 3: [1,0,1,0,1]
  • Step 4: [1,0,1,0,0,0,0,0,0]
  • Step 10: [1,0,1]
  • Step 11: [\(1,0^{13}\)]
  • Step 24: [1]
  • Step 25: [\(0^{26}\)]
  • Step 51: []

So \(\text{Worm}(2) = 51\).

Computer programs[]

Variants[]

Sources[]

  1. [1]
  2. [2]
  3. Beklemishev, L. (2006). The Worm principle. In Z. Chatzidakis, P. Koepke, & W. Pohlers (Eds.), Logic Colloquium '02 (Lecture Notes in Logic, pp. 75-95). Cambridge: Cambridge University Press. doi:10.1017/9781316755723.005

See also[]

Original numbers, functions, notations, and notions

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


Methodology

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


Implementation of existing works into programs

Proofs, translation maps for analysis schema, and other mathematical contributions

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


Entertainments

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
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