Can a problem be undecidable but yet Turing recognizable?

Question

Can a promblem be undecidable but yet be fully Turing recognizable?

For example - if language is Turing recognizable OR it's complement is Turing recognizable - so it still undecidable?

And another question - does Atm is np or np hard?

Does a list of all the configuration that accepts the word w can be verifier?

Thanks!

Answer 1

Re: your first question, yes. Recognizable languages are also called computably (or recursively) enumerable sets, and they can indeed be non-decidable. For example, the set of $e$ such that the $e$th Turing machine (in some standard enumeration) halts on input $e$ is recognizable but not decidable - and is called the halting problem.

Your second and third questions, though, don't make sense to me:

Does Atm is np or np hard? Does a list of all the configuration that accepts the word w can be verifier?

What is Atm? And, the set of all Turing machines that accept the word $w$ is indeed recognizable but not decidable (in fact, equivalent to the halting problem in a precise sense), but I'm not sure that's what you're asking . . .