Baker-Gill-Solovay theorem

Theorem

There exists oracles $A,B$ such that $\mathbf{P}^A = \mathbf{NP}^A$ and $\mathbf{P}^B \neq \mathbf{NP}^B$.

(where $\mathbf{P}^O$ is the set containing every language that can be decided by polynomial time deterministic Turing machine with oracle access to $O$, and $\mathbf{NP}^O$ that which can be decided by polynomial time non-deterministic Turing Machine with oracle access to $O$, these are relativized complexity classes)

References

1. S. Arora, B. Barak. Computational Complexity: A Modern Approach, Cambridge University Press, 2009, pp. 74-75.
2. T. Baker, J. Gill, and R. Solovay, “Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question,” SIAM J. Comput., vol. 4, no. 4, pp. 431–442, Dec. 1975, doi: 10.1137/0204037.
3. https://cstheory.stackexchange.com/questions/21663/baker-gill-solovay-pb-ne-npb-relativization-what-class-is-b-in
4. https://courses.cs.cornell.edu/cs6810/2023fa/lec6.pdf
5. https://faculty.cc.gatech.edu/~ladha/toc/L21.pdf