Prove a language in not in coNP

Question

Can someone tell me how to prove a particular problem/language does not belong to coNP?

Intuitively, we should show that the complement of the problem is not in NP, but I am not sure this is the way to do this.

Thanks.

Answer 1

Generally speaking, this is quite difficult. For example, right now we know that

P ⊆ NP ⊆ PSPACE ⊆ EXPTIME

but don't know whether any individual one of those "subset of" relationships are strict. This means that no one knows of a problem in EXPTIME or PSPACE, for example, that is not in NP. As a result, even problems that we strongly suspect are not in NP, such as TBQF, as of now can't be proven not to be in NP.