On Czerwinski's "${\rm P} \neq {\rm NP}$ relative to a ${\rm P}$-complete oracle"
(2312.04395)Abstract
In this paper, we take a closer look at Czerwinski's "${\rm P}\neq{\rm NP}$ relative to a ${\rm P}$-complete oracle" [Cze23]. There are (uncountably) infinitely-many relativized worlds where ${\rm P}$ and ${\rm NP}$ differ, and it is well-known that for any ${\rm P}$-complete problem $A$, ${\rm P}A \neq {\rm NP}A \iff {\rm P}\neq {\rm NP}$. The paper defines two sets ${\rm D}{\rm P}$ and ${\rm D}{\rm NP}$ and builds the purported proof of their main theorem on the claim that an oracle Turing machine with ${\rm D}{\rm NP}$ as its oracle and that accepts ${\rm D}{\rm P}$ must make $\Theta(2n)$ queries to the oracle. We invalidate the latter by proving that there is an oracle Turing machine with ${\rm D}{\rm NP}$ as its oracle that accepts ${\rm D}{\rm P}$ and yet only makes one query to the oracle. We thus conclude that Czerwinski's paper [Cze23] fails to establish that ${\rm P} \neq {\rm NP}$.
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