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

Nicholas DeJesse, Spencer Lyudovyk, Dhruv Pai, Michael Reidy

TL;DR

This paper critically reexamines Lin's claim that $NP\neq coNP$ by analyzing the language $L_d$ and the nondeterministic machine $D$ that Lin uses to place $L_d$ in $NP$. The authors show a flaw in $D$'s construction and provide a proof that $L_d\notin NP$, thereby undermining the main result. They further argue that the relativization barrier, the existence of ${\rm coNP}$-intermediate languages, and the implications for Frege proof systems do not follow given the corrected view. The critique highlights the fragility of diagonalization-based arguments for separating ${\rm NP}$ and ${\rm coNP}$ and stresses the need for rigorous validation of such constructions.

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.

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

TL;DR

This paper critically reexamines Lin's claim that by analyzing the language and the nondeterministic machine that Lin uses to place in . The authors show a flaw in 's construction and provide a proof that , thereby undermining the main result. They further argue that the relativization barrier, the existence of -intermediate languages, and the implications for Frege proof systems do not follow given the corrected view. The critique highlights the fragility of diagonalization-based arguments for separating and and stresses the need for rigorous validation of such constructions.

Abstract

In this paper, we examine Lin's "On NP versus coNP and Frege Systems" [Lin25]. Lin claims to prove that by constructing a language such that but . We present a flaw in Lin's construction of (a nondeterministic Turing machine that supposedly recognizes in polynomial time). We also provide a proof that . In doing so, we demonstrate that Lin's claim that 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.
Paper Structure (15 sections, 9 theorems, 6 equations)

This paper contains 15 sections, 9 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The "For All" Accepting Criterion and coNP
  4. Universal Turing Machines
  5. Analysis of Lin's Arguments
  6. Introduction to Lin's Arguments
  7. Analysis of Lin's Theorem 5.3
  8. $N_3$ = "On input $w$:
  9. Analysis of Lin's Further Claims
  10. Analysis of Lin's "Breaking the Relativization Barrier"
  11. Analysis of Lin's "Rich Structure of coNP"
  12. Analysis of Lin's "Frege Systems"
  13. Conclusion
  14. Acknowledgments
  15. Additional Proofs

Key Result

Theorem 1

The language $L_d$ is in ${\rm NP}$, where $L_d$ is accepted by the universal nondeterministic Turing machine $D$ that for some $k \in {\mathbb{N}^+}$ runs within time $O(n^k)$.

Theorems & Definitions (15)

  • Definition 1: lin2024np
  • Definition 2: lin2024np
  • Theorem 1: lin2024np
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • proof
  • Definition 3: lin2024np
  • Theorem 5: lin2024np
  • Theorem 6: lin2024np
  • ...and 5 more