Resolution of The Linear-Bounded Automata Question
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 $$ \text{NSPACE}[n]\overset{?}{=}\text{DSPACE}[n]. $$ In fact, we prove a more general result that $$ \text{DSPACE}[S(n)]\subsetneqq \text{NSPACE}[S(n)] $$ 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 by a universal nondeterministic Turing machine, and on other novel and interesting new technique. Our proof also implies the following consequences, which resolve some famous open questions in complexity theory: (1). $\text{DSPACE}[n]\subsetneqq \text{NSPACE}[n]$; (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);
- Publication:
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arXiv e-prints
- Pub Date:
- October 2021
- DOI:
- 10.48550/arXiv.2110.05942
- arXiv:
- arXiv:2110.05942
- Bibcode:
- 2021arXiv211005942L
- Keywords:
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- Computer Science - Computational Complexity;
- Computer Science - Formal Languages and Automata Theory;
- 68Q15;
- 68Q17;
- 03D15
- E-Print:
- [v16] fixed a blemish