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Loader's number is the output of loader.c, a C program by Ralph Loader that came in first place for the Bignum Bakeoff contest, whose objective was to write a C program (in 512 characters or less) that generates the largest possible output on a theoretical machine with infinite memory. It is among the largest computable numbers ever devised.

The program diagonalizes over the Huet-Coquand calculus of constructions. Its output, affectionately nicknamed Loader's number, is defined as \(D^5(99)=D(D(D(D(D(99)))))\), where \(D(k)\) is the sum of all possible bit strings described by the first k expressions of the calculus of constructions (encoding everything as binary numbers).

David Moews has shown that \(D(99)\) is larger than \(2↑↑30,419\) (where ↑↑ is tetration), and that even \(D^2(99)\) would be much larger than \(f_{\varepsilon_0+\omega^3}(1,000,000)\) in the fast-growing hierarchy, using Cantor's definition of fundamental sequences. \(D^2(99)\) thus is obviously much larger than the output of Marxen.c, which is upper bounded at the aforementioned \(f_{\varepsilon_0+\omega^3}(1,000,000)\).

The final output of \(D^5(99)\) is much larger than TREE(3), SCG(13), and, say, BH(100). It is probably overpowered by finite promise games and greedy clique sequences. Loader's function is computable, so \(\Sigma(n) > D^5(99)\) for relatively small n, say, n = 100.

Code Edit

#define R { return
#define P P (
#define L L (
#define T S (v, y, c,
#define C ),
#define X x)
#define F );}

int r, a;
P y, X
   R y - ~y << x;
}
Z (X
   R r = x % 2 ? 0 : 1 + Z (x / 2 F
L X
   R x / 2 >> Z (x F
#define U = S(4,13,-4,
T  t)
{
   int
      f = L t C         
      x = r;
   R
         f - 2 ?
         f > 2 ?
         f - v ? t - (f > v) * c : y :
         P f, P T  L X  C 
                          S (v+2, t  U y C  c, Z (X )))
         :
         A (T  L X  C 
                T  Z (X ) F
}
A (y, X
   R L y) - 1
      ? 5 << P y, X 
      : S (4, x, 4, Z (r) F
#define B (x /= 2) % 2 && (
D (X 
{
   int
      f,
      d,
      c = 0,
      t = 7,
      u = 14;
   while (x && D (x - 1 C  B 1))
      d = L L D (X ) C
         f = L r C
         x = L r C
         c - r || (
            L u) || L r) - f ||
            B u = S (4, d, 4, r C 
                   t = A (t, d) C
            f / 2 & B  c = P d, c C 
                              t  U t C 
                              u  U u) )
             C
         c && B
            t = P
               ~u & 2 | B
                  u = 1 << P L c C  u) C 
               P L c C  t) C
            c = r  C
         u / 2 & B 
            c = P t, c C 
            u  U t C 
            t = 9 );
   R a = P P t, P u, P x, c)) C 
                                a F
}
main ()
   R D (D (D (D (D (99)))) F

See alsoEdit


External links Edit

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