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Is it known whether $Logspace$ is strictly contained in the polynomial time hierarchy ?

Are there oracles relative to which these classes are equal / distinct ?

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    $\begingroup$ No, this is not known. For all we know, PH may collapse to uniform $\mathrm{AC}^0[6]$. $\endgroup$ – Emil Jeřábek Feb 18 at 11:30
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This is equivalent to $LOGSPACE≠NP$ (which is obviously open). The proof of that equivalence relativizes (at least under the usual oracle models).

And there are oracles making $LOGSPACE = NP$ (the PSPACE-complete TQBF works) and making them not equal (the oracle separating P from NP works).

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