FANDOM


Pair sequence number is a the output of a program made by BashicuHyudora and posted at a googology-related thread of the Japanese site BBS 2ch.net in 2014.[1][2][3][4] The algorithm of the program is called the pair sequence system, a weak version of Bashicu's matrix system by the same creator. It is supposed to calculate number comparable to \(f_{\vartheta(\Omega_\omega)+1}(10)\). It is an extension of a system named the primitive sequence system, still by the same author, which generates a number approaching \(f_{\varepsilon_0+1}(10)\).[5]

A pair sequence is a finite sequence of pairs of nonnegative integers, for example (0,0)(1,1)(2,2)(3,3)(3,2). A pair sequence P works as a function from natural numbers to natural numbers, (though we write P[n] rather than P(n)), for example \(n \mapsto (0,0)(1,1)(2,2)(3,3)(3,2)[n]\) is a function. The function P[n] is usually approximated with a function of the form \(H_\alpha\) from the Hardy hierarchy (we note \(P = \alpha\)). For example, \((0,0)(1,1)(2,2)(3,3)(3,2) = \psi(\psi_1(\Omega_2))\).

Original BASIC Code Edit

In the following program, in the loop starting from "for D=0 to 9" and ending at "next", a number close of \(f_{\vartheta(\Omega_\omega)}(C)\) is generated. The program repeats this loop 10 times and finally output a number close of \(f_{\vartheta(\Omega_\omega)+1}(10)\).

dim A[∞],B[∞]:C=9
for D=0 to 9
 for E=0 to C
  A[E]=E:B[E]=E
 next
 for F=C to 0 step -1
  C=C*C
  for G=0 to C
   if A[F]=0 | (A[F-G]<A[F] & (B[F]=0 | B[F-G]<B[F])) then H=G:G=C
  next
  if B[F]=0 then I=0 else I=A[F]-A[F-H]
  for J=1 to C*H
   A[F]=A[F-H]+I:B[F]=B[F-H]:F=F+1
  next
 next
next
print C

Equivalent C code Edit

This code is optimized to take as few place as possible, so that it matches the rules of the Bignum Bakeoff contest. The pair sequence number is returned from main(). The code has 278 characters without whitespaces.

#define A a[f]
#define B b[f]
#define M = malloc(9),
#define W while (

main(f) {
   int *a M *b M c = 9, d = 10, h, i, k;
   W d--) {
      f = h = c + 1;
      W h--) a[h] = b[h] = h;
      W f--) {
         c *= c;
         h = f + 1;
         W h--)
            (a[h] < A && (b[h] < B || !B) || A + B == 0)?
            (k = B ? A - a[h]: 0, i = f - h, h = 0): 0;
         h = f + c * i; a = realloc(a, h); b = realloc(b, h);
         W f < h) A = a[f-i] + k, B = b[f-i], f++;
      }
   }
   return c;
}

Verification code Edit

The Bashicu matrix calculator[6] shows the calculation process of pair sequence system.

Here are some examples of the calculation of some pair sequences. The algorithm is modified so that it always take n=2.

Corresponding ordinals Edit

Up to \(\varepsilon_0\) Edit

When all the values of the second row are 0, it is the same as the primitive sequence system. We have:

\begin{array}{ll} (0,0) &=& 1 \\ (0,0)(0,0) &=& 2 \\ (0,0)(0,0)(0,0) &=& 3 \\ (0,0)(1,0) &=& \omega \\ (0,0)(1,0)(0,0)(0,0) &=& \omega+2 \\ (0,0)(1,0)(0,0)(1,0) &=& \omega \cdot 2 \\ (0,0)(1,0)(1,0) &=& \omega^2 \\ (0,0)(1,0)(1,0)(0,0)(1,0) &=& \omega^2+\omega \\ (0,0)(1,0)(2,0) &=& \omega^\omega \\ (0,0)(1,0)(2,0)(3,0) &=& \omega^{\omega^\omega} \\ (0,0)(1,0)(2,0)(3,0)(4,0) &=& \omega^{\omega^{\omega^\omega}} \\ \end{array} (0,0)(1,1) has fundamental sequence as follows. Here, n is not changed.

\begin{array}{ll} (0,0)(1,1)[1] &=& (0,0)(1,0)[1] \\ (0,0)(1,1)[2] &=& (0,0)(1,0)(2,0)[2] \\ (0,0)(1,1)[3] &=& (0,0)(1,0)(2,0)(3,0)[3] \\ (0,0)(1,1)[4] &=& (0,0)(1,0)(2,0)(3,0)(4,0)[4] \\ \end{array} Therefore, \(\{\omega, \omega^\omega, \omega^{\omega^\omega}, \omega^{\omega^{\omega^\omega}}, \ldots\}\) and \begin{array}{ll} (0,0)(1,1) &=& \varepsilon_0 \\ \end{array}

Up to \(\varepsilon_1\) Edit

As for (0,0)(1,1)(1,0), \[(0,0)(1,1)(1,0)[4] = (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)[4]\] and the fundamental sequence is \begin{array}{ll} (0,0)(1,1) &=& \varepsilon_0 \\ (0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 \cdot 2 \\ (0,0)(1,1)(0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 \cdot 3 \\ (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 \cdot 4 \\ (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 \cdot 5 \\ \end{array} Therefore, \[(0,0)(1,1)(1,0) = \varepsilon_0 \cdot \omega\]

\[(0,0)(1,1)(1,0)(1,0)[2] = (0,0)(1,1)(1,0)(0,0)(1,1)(1,0)(0,0)(1,1)(1,0)[2]\] has fundamental sequence of \begin{array}{ll} (0,0)(1,1)(1,0) &=& \varepsilon_0 \cdot \omega \\ (0,0)(1,1)(1,0)(0,0)(1,1)(1,0) &=& \varepsilon_0 \cdot \omega \cdot 2 \\ (0,0)(1,1)(1,0)(0,0)(1,1)(1,0)(0,0)(1,1)(1,0) &=& \varepsilon_0 \cdot \omega \cdot 3 \\ \end{array}

Therefore, \[(0,0)(1,1)(1,0)(1,0) = \varepsilon_0 \cdot \omega^2 \] In this way, adding (1,0) to the end of the sequence makes the ordinal \(\omega\) times. Adding (1,0)(2,0) to the end of the sequence \[(0,0)(1,1)(1,0)(2,0)[4] = (0,0)(1,1)(1,0)(1,0)(1,0)(1,0)(1,0)[4]\] corresponds to multiplying \(\omega^\omega\) to the ordinal, and therefore \[(0,0)(1,1)(1,0)(2,0) = \varepsilon_0 \cdot \omega^\omega\]

As for (0,0)(1,1)(1,1), \[(0,0)(1,1)(1,1)[4] = (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1)[4]\] and the following fundamental sequence is obtained. \begin{array}{ll} (0,0)(1,1) &=& \varepsilon_0 \\ (0,0)(1,1)(1,0)(2,1) &=& \varepsilon_0^2 \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1) &=& \varepsilon_0^{\varepsilon_0} \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1) &=& \varepsilon_0^{\varepsilon_0^2} \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1) &=& \varepsilon_0^{\varepsilon_0^{\varepsilon_0}} \\ \end{array} Therefore, \[(0,0)(1,1)(1,1) = \varepsilon_1 = \psi(1) \]

Up to Feferman–Schütte ordinal = \(\Gamma_0\) Edit

Similar calculation results in:

\begin{eqnarray*} (0,0)(1,1)(2,0) &=& \varepsilon_{\omega} = \psi(\omega) \\ (0,0)(1,1)(2,0)(2,0) &=& \varepsilon_{\omega^2} = \psi(\omega^2) \\ (0,0)(1,1)(2,0)(3,0) &=& \varepsilon_{\omega^\omega} = \psi(\omega^\omega) \\ (0,0)(1,1)(2,0)(3,1) &=& \varepsilon_{\varepsilon_0} = \psi(\psi(0)) \\ (0,0)(1,1)(2,0)(3,1)(4,0)(5,1) &=& \varepsilon_{\varepsilon_{\varepsilon_0}} = \psi(\psi(\psi(0))) \\ (0,0)(1,1)(2,1) &=& \zeta_0 = \phi(2,0) = \psi(\Omega) \\ (0,0)(1,1)(2,1)(3,1) &=& \Gamma_0 = \phi(1,0,0) = \psi(\Omega^\Omega) \\ \end{eqnarray*}

Up to Large Veblen ordinal = \(\psi(\Omega^{\Omega^\Omega})\) Edit

\begin{eqnarray*} (0,0)(1,1)(2,1)(3,1)(4,0) &=& ψ(\Omega^{\Omega \cdot \omega}) \\ (0,0)(1,1)(2,1)(3,1)(4,1) &=& ψ(\Omega^{\Omega \cdot \Omega}) = ψ(\Omega^{\Omega^2}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(4,0) &=& ψ(\Omega^{\Omega^2 \cdot \omega}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(4,1) &=& ψ(\Omega^{\Omega^2 \cdot \Omega}) = ψ(\Omega^{\Omega^3}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,0) &=& ψ(\Omega^{\Omega^\omega}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1) &=& ψ(\Omega^{\Omega^\Omega}) \\ \end{eqnarray*}

Up to \(\vartheta(\Omega_\omega)\) Edit

\begin{eqnarray*} (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1) &=& ψ(\Omega^{\Omega^{\Omega^2}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)(7,1) &=& ψ(\Omega^{\Omega^{\Omega^\Omega}}) \\ (0,0)(1,1)(2,2) &=& \psi(\varepsilon_{\Omega+1}) = \psi(\psi_1(0)) \\ (0,0)(1,1)(2,2)(0,0) &=& \psi(\psi_1(0))+1 \\ (0,0)(1,1)(2,2)(1,0) &=& \psi(\psi_1(0)) \omega \\ (0,0)(1,1)(2,2)(2,0) &=& \psi(\psi_1(0) \omega) \\ (0,0)(1,1)(2,2)(3,0) &=& \psi(\psi_1(\omega)) \\ (0,0)(1,1)(2,2)(3,0)(4,0) &=& \psi(\psi_1(\omega^\omega)) \\ (0,0)(1,1)(2,2)(3,0)(4,1) &=& \psi(\psi_1(\psi(0)))=\psi(\psi_1(\varepsilon_0)) \\ (0,0)(1,1)(2,2)(3,1) &=& \psi(\psi_1(\Omega)) \\ (0,0)(1,1)(2,2)(3,2) &=& \psi(\psi_1(\Omega_2)) \\ (0,0)(1,1)(2,2)(3,3) &=& \psi(\psi_1(\psi_2(0))) \\ (0,0)(1,1)(2,2)(3,3)(4,4) &=& \psi(\psi_1(\psi_2(\psi_3(0)))) \\ (0,0)(1,1)(2,2)(3,3)...(9,9) &=& \psi(\psi_1(\psi_2(\psi_3(\psi_4(\psi_5(\psi_6(\psi_7(\psi_8(0))))))))) \\ \end{eqnarray*}

By defining \(Pair(n) = (0,0)(1,1) \ldots (n,n)[n]\), one has \[Pair(n) \approx f_{\vartheta(\Omega_\omega)}(n)\]

Sources Edit

  1. First version of BASIC program
  2. Modified version of BASIC program
  3. Name of the system and the number
  4. Blog post which introduces pair sequence system
  5. BASIC program of \(\varepsilon_0\) level and a blog post which introduces it
  6. Bashicu matrix calculator

See also Edit

Googology in Asia

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Pair sequence number · Bashicu matrix system
Indian counting system: Lakh · Crore
Chinese and Japanese counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Buddhist text: Tallakshana · Dvajagravati · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra
Other: Taro's multivariable Ackermann function · Sushi Kokuuhen

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