there may still be a useful semi-/decidable subset.
Yes, but note that the entire language HALT is semi-decidable (or Turing-recognisable if you use Sipser’s terminology); in this case it’s only the (strictly) decidable subset that is useful.
So is there any result on "how much" of the halting problem is solvable? E.g. is there a result that says there is a program which can decide whether the input halts for "most" inputs, in some sense of "most"?
Comments
Yes, but note that the entire language HALT is semi-decidable (or Turing-recognisable if you use Sipser’s terminology); in this case it’s only the (strictly) decidable subset that is useful.
So is there any result on "how much" of the halting problem is solvable? E.g. is there a result that says there is a program which can decide whether the input halts for "most" inputs, in some sense of "most"?