On Czerwinski's "${\rm P} \neq {\rm NP}$ relative to a ${\rm P}$-complete oracle"

Michael C. Chavrimootoo; Tran Duy Anh Le; Michael P. Reidy; Eliot J. Smith

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arXiv · 2312.04395

On Czerwinski's "${\rm P} \neq {\rm NP}$ relative to a ${\rm P}$-complete oracle"

Michael C. Chavrimootoo·Tran Duy Anh Le·Michael P. Reidy·Eliot J. Smith

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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 $\Theta(2^n)$ 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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Michael C. Chavrimootoo, Tran Duy Anh Le, Michael P. Reidy, Eliot J. Smith. 2023-12-07. On Czerwinski's "${\rm P} \neq {\rm NP}$ relative to a ${\rm P}$-complete oracle". https://arxiv.org/abs/2312.04395

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