Re:Did Turing prove the undecidability of the halting problem?
by Tainnor on Tuesday July 02, 2024 @10:34AM
(#40857100)
> Figuring out whether a single given program halts is decidable
It's maybe nitpicky, but by definition "decidable" applies to a language, i.e. a set of programs, not a single program. Of course, there is a trivial TM (either the one that always accepts or the one that always rejects) that, for any given single program, gives the correct answer for exactly that program, but that is rather uninteresting. So this framing doesn't make a lot of sense.
What is actually decidable is the language consisting of pairs of a TM (+ input) and a valid proof of its termination, and that's why we can prove (some) programs to be terminating (or any other more interesting property).
It's maybe nitpicky, but by definition "decidable" applies to a language, i.e. a set of programs, not a single program. Of course, there is a trivial TM (either the one that always accepts or the one that always rejects) that, for any given single program, gives the correct answer for exactly that program, but that is rather uninteresting. So this framing doesn't make a lot of sense.
What is actually decidable is the language consisting of pairs of a TM (+ input) and a valid proof of its termination, and that's why we can prove (some) programs to be terminating (or any other more interesting property).