HN Reader

If it's not loading for anyone else [0]

[0] https://web.archive.org/web/20251027173129/https://benbrubak...

The best current lower bound for BB(6) is 2↑↑2↑↑2↑↑9 (google "Knuth Up Arrow" if this makes no sense), a number so inconceivably large it gives me the willies.

In particular, this means that in going from BB(5) to BB(6), you have already crossed the line where the actual busy beaver TM can no longer be simulated step by step in the lifetime of our universe (or a googol lifetimes of our universe for that matter).

It really is mind bending how fast this function grows.

TLDR; As BB(n) gets larger, they can encode more random walk style problems that have a stop condition related to the position of the random walk. Proving that such a condition is unlikely may be easy, but proving it never occurs is very difficult.

Despite all the reporting on BB(5) I had never seen anyone convey that equivalent high-level formulation from 1993, that's very cool!

EDIT: For fun I converted it to Rust and expected to see it spew a few million numbers into my terminal, but no, this equivalent loop actually terminates after 15 steps, which is fascinating given that the Turing machine takes 47 million steps:

    let mut x = 0;
    loop {
        x = match x % 3 {
            0 => 5 * x + 18,
            1 => 5 * x + 22,
            _ => break
        } / 3;
    }

OEIS link: https://oeis.org/A386909

EDIT 2: Of course the article mentions this in the next paragraph, which is what I get for being immediately nerd-sniped.

That was a really well written article. I think even somebody who had never heard of a Turing Machine could probably have gotten a pretty reasonable quick understanding of roughly what BB(5) and BB(6) are and how the Antihydra works and its greater mathematical/historical context. That's hard to do, good job!

Why we care about Busy Beaver numbers, from “Who Can Name the Bigger Number?” by Scott Aaronson:

Now, suppose we knew the Nth Busy Beaver number, which we’ll call BB(N). Then we could decide whether any Turing machine with N rules halts on a blank tape. We’d just have to run the machine: if it halts, fine; but if it doesn’t halt within BB(N) steps, then we know it never will halt, since BB(N) is the maximum number of steps it could make before halting. Similarly, if you knew that all mortals died before age 200, then if Sally lived to be 200, you could conclude that Sally was immortal. So no Turing machine can list the Busy Beaver numbers—for if it could, it could solve the Halting Problem, which we already know is impossible.

But here’s a curious fact. Suppose we could name a number greater than the Nth Busy Beaver number BB(N). Call this number D for dam, since like a beaver dam, it’s a roof for the Busy Beaver below. With D in hand, computing BB(N) itself becomes easy: we just need to simulate all the Turing machines with N rules. The ones that haven’t halted within D steps—the ones that bash through the dam’s roof—never will halt. So we can list exactly which machines halt, and among these, the maximum number of steps that any machine takes before it halts is BB(N).

Conclusion? The sequence of Busy Beaver numbers, BB(1), BB(2), and so on, grows faster than any computable sequence. Faster than exponentials, stacked exponentials, the Ackermann sequence, you name it. Because if a Turing machine could compute a sequence that grows faster than Busy Beaver, then it could use that sequence to obtain the D‘s—the beaver dams. And with those D’s, it could list the Busy Beaver numbers, which (sound familiar?) we already know is impossible. The Busy Beaver sequence is non-computable, solely because it grows stupendously fast—too fast for any computer to keep up with it, even in principle.

https://www.scottaaronson.com/writings/bignumbers.html

The Antihydra will halt if:

The sequence is (truly/fairly) random in its distribution of mods 1/2.

Even fair coins flipped infinitely would - on occasion - have arbitrary long results of heads or tails.

So the question becomes, is the anti-hydra sequence 'sufficiently' random?

Is this like SETI@home, Bitcoin, and Ai code generation?

In the old days we used to just chop wood, and burn it to keep warm. Then sit down and watch the sportsball game on TV to waste time.