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$P^A$ and $NP^A$ are shorthand notation to be used with care. What we mean by $\mathcal C^A$ depends heavily on the syntactical definition of the class $\mathcal C$ and is not well-defined if it's not clear which syntactical definition of $\mathcal C$ we are referring to. $\mathcal C^A$ is more an invariant of $\mathcal C$-machines than of the set of languages decided by $\mathcal C$-machines.

While $P=NP$ means that deterministic polynomial-time machines $P$-machines can decide exactly the same languages as non-deterministic polynomial-time machines, $NP$-machines, $P^B\neq NP^B$ for some oracle $B$ as proven by Baker, Gill, and Solovay (1975) means that deterministic polynomial-time machines $P$-machines with additional access to the oracle $B$ can decide strictly fewer languages than non-deterministic ones $NP$-machines with access to $B$. They Baker, Gill, and Solovay also give an oracle $A$ with respect to which these two types of machines can decide exactly the same languages.
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$P^A$ and $NP^A$ are shorthand notation to be used with care. What we mean by $\mathcal C^A$ depends heavily on the syntactical definition of the class $\mathcal C$ and is not well-defined if it's not clear how to syntactically define which syntactical definition of $\mathcal C$ we are referring to. It $\mathcal C^A$ is more a property an invariant of the machines $\mathcal C$-machines than of the set of languages decided by $\mathcal C$-machines.

While $P=NP$ means that deterministic polynomial-time machines can decide exactly the same languages as non-deterministic polynomial-time machines, it was $P^B\neq NP^B$ for some oracle $B$ as proven by Baker, Gill, and Solovay (1975) means that deterministic polynomial-time machines with access to some oracle $B$ can decide strictly fewer languages than non-deterministic ones with access to the same oracle $B$. They also give an oracle $A$ with respect to which these two types of machines can decide the same languages.
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$P^A$ and $NP^A$ are shorthand notation to be used with care. What we mean by $\mathcal C^A$ depends heavily on the syntactical definition of the class $\mathcal C$ and is not well-defined if it's not clear how to syntactically define $\mathcal C$. It is more a property of the machines than of the languages. While $P=NP$ means that deterministic polynomial-time machines can decide exactly the same languages as non-deterministic polynomial-time machines, it was proven by Baker, Gill, and Solovay (1975) that deterministic polynomial-time machines with access to some oracle $B$ can decide strictly fewer languages than non-deterministic ones with access to the same oracle $B$. They also give an oracle $A$ with respect to which these two types of machines can decide the same languages.