# Is there an oracle that separates two complexity classes known to be equal?

We know that there exist two oracles $A$ and $B$ such that $P^A=NP^A$ and $P^B\neq NP^B$, this implies the obstacle of proving $P\neq NP$ using diagonalization. I just wonder if there exist two complexity classes, say $A$ and $B$ and oracle $O$ such that $A=B$ but $A^O\neq B^O$?

I am not sure if it is a research-level problem. So don't hesitate to close it if not.

Thanks.

• Then it means $P\ne NP$, otherwise, how can an oracle separate them? – pyao May 18 '11 at 12:26
• OK, here "the same" means they are shown to be same, but may use completely different definitions, like $IP=PSPACE$. – pyao May 18 '11 at 12:27
• I suggest that you change the question to reflect this. – Dave Clarke May 18 '11 at 13:21
• related: 1 – Kaveh May 19 '11 at 6:07