Resolution of The Linear-Bounded Automata Question
Tianrong Lin
TL;DR
The paper resolves the linear-bounded automata question by proving a general separation ${\rm DSPACE}[S(n)]\subsetneqq {\rm NSPACE}[S(n)]$ for every space-constructible $S(n)\ge n$, implying in particular ${\rm DSPACE}[n]\subsetneqq {\rm NSPACE}[n]$. It introduces a diagonalization method against deterministic $S(n)$-space machines using a universal nondeterministic TM, producing a language $L_d$ that lies in ${\rm NSPACE}[S(n)]$ but not in ${\rm DSPACE}[S(n)]$, and further shows $L\subsetneqq NL$ via padding, yielding $L\subsetneqq P$. As a corollary, no deterministic $O(\log n)$-space TM can decide STCON. The work advances the understanding of nondeterminism in space-bounded computation, with implications for context-sensitive languages and the boundaries between complexity classes, while also outlining open questions for intermediate space scales between $\log n$ and $n$.
Abstract
This paper resolves a famous and longstanding open question in automata theory, i.e., the {\it linear-bounded automata question} (or shortly, LBA question), which can also be phrased succinctly in the language of computational complexity theory as $$ {\rm NSPACE}[n]\overset{?}{=}{\rm DSPACE}[n]. $$ In fact, we prove a more general result that $$ {\rm DSPACE}[S(n)]\subsetneqq {\rm NSPACE}[S(n)] $$ where $S(n)\geq n$ is a space-constructible function. Our proof technique is based on diagonalization against deterministic $S(n)$ space-bounded Turing machines with a universal nondeterministic Turing machine and on other novel and interesting new techniques. Our proof also implies the following consequences, which resolve some famous open questions in complexity theory: (1). ${\rm DSPACE}[n]\subsetneqq {\rm NSPACE}[n]$; (2). $L\subsetneqq NL$; (3). $L\subsetneqq P$; (4). There exists no deterministic Turing machine working in $O(\log n)$ space deciding the $st$-connectivity question (STCON).