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Is there a known example of two complexity classes, $A$ and $B$, such that:

  1. $A \neq B$;

  2. there is an oracle $O$ for which $A^O = B^O$?

I strongly believe that there are such examples, as otherwise $P = PSPACE$ (note that $P^{PSPACE} = PSPACE^{PSPACE}$), but I was looking for an example of this.

Guilherme Rito
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2 Answers2

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Let $\mathsf{C}$ be a complexity class of your choice, let $\mathsf{O}$ be an oracle of your choice, and define $\mathsf{D} = \mathsf{C}^\mathsf{O}$. Then $\mathsf{C}^\mathsf{O} = \mathsf{D}^\mathsf{O}$, but it could happen that $\mathsf{C} \neq \mathsf{D}$. For example, you can take $\mathsf{C} = \mathsf{P}$ and $\mathsf{O} = \mathsf{HALT}$, the halting problem.

This might seem like cheating, but it's exactly the same answer as the one by Willard Zhan.

Yuval Filmus
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See Ryan Williams' answer here. An easy example is that $\mathsf{AC^0}\neq \mathsf{AC^0[2]}$, but they are equal relative to a parity oracle.

Wei Zhan
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