# Showing that the set of DTMs that run forever is not Turing-recognizable

The language A, that is all DTMS that run forever on input.

Would this not just be the HALT problem?

Therefore no reduction or proof is necessary, other then stating that?

ANSWER FOUND: I think i have it, if it was Turing Recognizable, then it would be possible to determine when HALT should reject and therefore HALT would be decidable. Since it is not-decidable, then A must be not Turing Recognizable.

• Yes, it is just the complement of HALT_TM but without the descriptions that do not describe valid TMs. So why do you think would it not be Turing-recognizable? – Ryan Jul 27 '15 at 18:30
• Since you found your answer, you should not leave it in a comment but either add it as another answer, or include it as an edit in the question. IMHO. – babou Jul 28 '15 at 12:20

This is not the halting problem, the halting problem is the set of TMs that do halt, the language $A$ is the set of TMs that don't halt. It's the complement of the halting problem.