There are numbers vastly beyond the range of human comprehension that can
be compressed to simple programs that output them in unary. For example,
define the hyperoperation function over positive integers:
h(a, n, b) {
if n=1 and b=1 output bit 0, return a+1;
else if n=1 output bit 1, return h(a+1, 1, b-1);
else if b=1 return a;
else return h(a, n-1, h(a, n, b-1));
}
This performs operation n on a and b, where n=1 means add, 2 means
multiply, 3 means exponentiate, and higher numbers mean n-2 Knuth up
arrows. In other words, h(a, n, b) means iterate operation n-1 on b copies
of a, evaluating right to left.
For example, h(3, 6, 3) = 3^^^^3 = 3^^^(3^^^3), which is huge beyond
comprehension. If you think of numbers with more digits than the number of
ways you could rearrange all the atoms in the universe, you aren't even
close.
Then you have Graham's number, which is mind bendingly enormous in a whole
other dimension. https://en.m.wikipedia.org/wiki/Graham%27s_number
Yet I can compress it easily. It is g(64), given:
g(x) {
if x=1 return h(3, 6, 3);
else return h(3, g(x-1)+2, 3);
}
That is, g(64) is 3^^^^...3 with g(63) up arrows.
This doesn't mean you can compress all huge strings. Quite the opposite. If
I gave you a string of g(64) 1 bits, or even g(1), or even just 3^^^3, it
would be impossible to count them. The vast majority of strings do not have
any description that is shorter than the string itself, and you would have
no way to know.
On Fri, Nov 1, 2019, 4:21 PM <[email protected]> wrote:
> you can compress 1,000,000,000,000,000,000,000,000 bits to 80 bits.
> and they can be in all sorts of wonderous pattern shapes.
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