Yes. The usual collapse relativizes because every machine involved receives the same <oracle machine>[oracle] $A$. Induct on the levels of the <polynomial hierarchy>. The hypothesis gives
$$
\Sigma_1^{\mathbf P,A}=\mathbf{NP}^A=\mathbf P^A.
$$
If the preceding level is contained in $\mathbf P^A$, then its oracle queries can be simulated in polynomial time with oracle $A$. A nondeterministic machine for the next existential level is therefore only an $\mathbf{NP}^A$ machine, hence a $\mathbf P^A$ machine by hypothesis. Complements give the corresponding universal level. The induction yields
$$
\boxed{\mathbf{PH}^A=\mathbf P^A.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 124 / 3 / iii / Solution

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An oracle machine may query membership in a fixed language in one computation step.


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