# Natural relativized worlds

The oracles that are used in relativized collapses or separations of complexity classes rarely represent $natural$ algorithmic problems. They are typically constructed "artificially" with techniques like diagonalization, for the sole purpose of achieving the relativized collapse or separation. A notable exception is that for any ${\bf PSPACE}$-complete set $L$ it holds that ${\bf P}^L={\bf NP}^L$. This indeed leads to "natural relativized worlds," since there are ${\bf PSPACE}$-complete languages that correspond to natural algorithmic tasks. Generally, I am looking for such examples, where the oracle represents a natural problem. (And, of course, it is not trivially implied by the above example).

In particular, is there any such known "natural relativized world" in which ${\bf P}$ and ${\bf NP}$ are separated?

$Note:$ A random oracle here does not count as natural, because it does not represent a natural algorithmic problem. It rather represents that a statement holds for almost all oracles, regardless to their "naturality."

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As far as I know, diagonalization is the only method we know of that can separate non-small complexity classes, so no surprise the oracles separating $\mathsf{NP}$ from $\mathsf{P}$ are such sets.