A Critique of Sopin's "${\rm PH} = {\rm PSPACE}$"

TL;DR

This paper critically examines Valerii Sopin's claim that the polynomial hierarchy equals PSPACE, identifying flaws in the original proof and arguing that the claim is not substantiated.

Contribution

The paper provides a detailed critique of Sopin's proof, highlighting its errors and clarifying that the polynomial hierarchy's equality with PSPACE remains unproven.

Findings

Sopin's proof contains significant flaws.

The claim ${ m PH} = { m PSPACE}$ is not supported by the critique.

The paper clarifies the current understanding of the hierarchy.

Abstract

We critique Valerii Sopin's paper "${\rm PH} = {\rm PSPACE}$" [Sop14]. The paper claims to resolve one of the major open problems of theoretical computer science by leveraging the Skolemization of existential quantifiers of quantified boolean formulas to show that ${\rm QBF}$ (a well-known ${\rm PSPACE}$-complete problem) is in $\Pi_4^p$, and thus ${\rm PH} = {\rm PSPACE}$. In this critique, we highlight problems in that paper and conclude that it fails to establish that ${\rm PH} = {\rm PSPACE}$.