What subset of TMs are there where the halting problem is decidable? Halting problem is not decidable in general, but are there interesting/useful subsets where it IS decidable? Bonus points if that subset is actually helpful.
Example:
If we know BB(x), then the halting problem for TMs with x states is decidable. The problem is BB(x) is not a computable function, as it basically requires solving the halting problem in the first place, so this doesn't help construct a general halting problem solver for arbitrary x. We do however, know BB(x) for x &lt; 5 (which isn't very helpful I imagine, but still...).
Still, I guess that means technically the halting problem is decidable for subsets which contain all TMs with the same finite amount of states, but we still can't construct a TM that decides it without first solving the halting problem.
Similarly, any finite subset of TMs is trivially decidable by some TM that just happens to correctly map the input TMs to truth values. Again, we can't construct this TM. Not very interesting or helpful.
If you don't look at turing machines in particularly: Sometimes you can make an argument &quot;if it halts at all, then it must halt within x steps&quot; where x can be calculated, so you decide the halting problem by calculating x and performing x steps.
One particular case: A program with nested loops, where an upper limit for the number of iterations of each loop can be calculated before the loop is executed, will halt. And if you take such a program and insert identifiable &quot;non-halting&quot; constructs, like &quot;for (;;) {}&quot; in a C program, then the halting problem is decidable.