It's a good question. Yes and no. If the "God of Busy Beaver" told us a value of BB(n) (for large enough n) then that would reduce some math problems to "simply" running a TM for that many steps and seeing if it halted. However, there are (at least) two issues with this: (1) As we can see, the number of steps will be unbelievably large (even just for 6-states, 2-symbols!), so it won't really be practical to run a TM that long and (2) The only known way to prove BB(n) numbers is by directly proving the halting / non-halting of every single TM of that size ... so, in other words, we will not know BB(n) until we have already proven all the interesting math that can be encoded in n-state TMs!
No comments yet.