I’m a little out of my depth, but I’d guess a lot of them would probably fall into one of two categories: Something we believe should go on forever (and not halt) if the math problem is resolved the way we expect, but theoretically could suddenly halt after some absurdly long number of steps. Or something where it halts for a given input after some number of steps unless something some counter example exists where it goes on forever.

In the first, you can’t really do anything but just keep watching it not halt but it isn’t telling you anything about the infinity to go. (Say a program that spits out twin primes, we expect an infinite number but we don’t really know)

And in the second case we’d just have to keep trying larger and larger inputs making this just an extension of the first category if we wrote a program to do that for us. And if we did find an example where it goes on forever without repeating states, how would you even know? It’d be like the first situation again.

Once we have scalable quantum computers, fusion power, time travel and an indestructable material, I figure we can bundle all that together with instructions to send a particle back after T+1 on termination. Some problems will stay unsolved as they go on to the heat-death of the universe but maybe one or a few comes back with a useful result!

Certainly with the right investments we'll get there within the next 5 years if you ask Musk and Altman. While a time machine might sound uncertain in that timefram, I'm sure AI will figure it out for us.

Ah that makes a lot of sense!