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Comment by lukevp | Hacker News Reader

I think the problem is that some of the Turing machines are infinite, so getting 47 million steps in is the _entrypoint_ and you must then run until it halts (or you determine it’s not going to halt somehow). So the search space is unbounded, but it seems like you could snapshot the state of each machine at the 47 million steps and then incrementally search a couple thousand steps forward on each one looking for halting? I may be totally off though, I won’t pretend to fully grok this.

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If you get < 47M steps out of all possible 5 state machines, they you have proven the conjecture. A 5 state requires at most 47M states. I don't know if that's an upper bound or a precise estimate, but if they all halt, it's true.

The tape under the turing machine is infinite, so it's certainly not limited to 47M states. The conjecture states that all machines that halt will do so within 47M states.

Steps rather than states.

The 47 million thing is based on a specific machine, which was identified as the current 5-state champion in 1990 and is known to halt.

https://bbchallenge.org/story#goal

Since nobody has found any busier 5-state beaver than that since 1990, Scott Aaronson conjectured that this might be the actual maximum for halting 5-state machines.

In "Story" website section:

"This conjecture says that 47,176,870 is the maximum number of steps that a 5-state Turing machine can run before halting (starting from all-0 memory tape)."

So, we can run up to this number and either the machine will stop - and we'll go to the next one until exhaust the whole set - or the machine won't stop, and we'll refute the conjecture.

No, you seem to be misunderstanding the conjecture. It's about the maximum number of steps executed by a Turing machine that eventually halts.

If you simulate a Turing machine up to that many steps, and find it hasn't halted yet, then the conjecture is refuted if you know the machine's execution will eventually terminate -- but famously, there's no algorithm that can reliably tell you whether or not a given TM halts.

If it was just about the maximum number of steps that could be executed by a 5-state Turing machine, it would be trivially false, because it's easy to construct a Turing machine that runs forever.

Yeah, I was surprised the statement is constructed this way. Adding condition of eventual halting - which is what makes it the busy beaver - makes the difference :) .

I think that this wording relies on carefully interpreting "before halting". A TM which runs forever can run more than 47 million steps, but it does not run more than 47 million steps steps before halting, since it never halts.

I have changed the wording on the website in order to make it hopefully more straightforward:

"This conjecture says that if a 5-state Turing machine runs for more than 47,176,870 steps without halting then it will never halt (starting from all-0 memory tape)."

> So, we can run up to this number and either the machine will stop - and we'll go to the next one until exhaust the whole set - or the machine won't stop, and we'll refute the conjecture

No, you do not refute the conjecture if you run up to this number and a machine does not stop. Finding a machine that runs for more than 47,176,870 steps is easy - there are plenty of machines that run forever. The trick is that it needs to stop.

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