A Critique of Lin's "On $\text{NP}$ versus $\text{coNP}$ and Frege Systems"

TL;DR

This paper critically analyzes Lin's attempt to prove NP ≠ coNP, identifying flaws in his construction and demonstrating that his main claim is unsubstantiated, thereby challenging his conclusions.

Contribution

The paper provides a detailed critique of Lin's proof attempt, exposing errors and clarifying the invalidity of his claims regarding NP versus coNP.

Findings

Lin's construction of language L_d is flawed

L_d cannot be recognized in polynomial time as claimed

Lin's main proof that NP ≠ coNP is invalid

Abstract

In this paper, we examine Lin's "On NP versus coNP and Frege Systems" [Lin25]. Lin claims to prove that $\text{NP} \neq \text{coNP}$ by constructing a language $L_d$ such that $L_d \in \text{NP}$ but $L_d \notin \text{coNP}$. We present a flaw in Lin's construction of $D$ (a nondeterministic Turing machine that supposedly recognizes $L_d$ in polynomial time). We also provide a proof that $L_d \not\in \text{NP}$. In doing so, we demonstrate that Lin's claim that $\text{NP} \neq \text{coNP}$ is not established by his paper. In addition, we note that a number of further results that Lin claims are not validly established by his paper.