Refuting Tianrong Lin's arXiv:2110.05942 "Resolution of The Linear-Bounded Automata Question"

TL;DR

This paper critically examines Tianrong Lin's claimed proof resolving the longstanding LBA question, demonstrating its incompleteness and errors, and clarifies that the proof cannot be fixed.

Contribution

It provides a detailed refutation of Lin's proof, showing that his approach is flawed and cannot establish the claimed separation of complexity classes.

Findings

Lin's proof is incomplete and contains errors.

The strategy used in Lin's proof cannot be repaired.

The claimed resolution of the LBA question is invalid.

Abstract

In the preprint mentioned in the title Mr. Tianrong claims to prove $\textrm{NSPACE}[n]\neq\textrm{DSPACE}[n]$, resolving a longstanding open problem in automata theory called the LBA question. He claims to achieve this by showing more generally $\textrm{NSPACE}[S(n)]\neq\textrm{DSPACE}[S(n)]$ for suitable $S(n)$. We demonstrate that his proof is incomplete, even wrong, and his strategy cannot be repaired. Update to include recent developments of Mr. Tianrong's preprint.