Resolution of the Linear-Bounded Automata Question

10/11/2021

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by   Tianrong Lin, et al.

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This work resolve a longstanding open question in automata theory, i.e. the linear-bounded automata question ( shortly, LBA question), which can also be phrased succinctly in the language of computational complexity theory as NSPACE[n]?=DSPACE[n]. We prove that NSPACE[n]≠ DSPACE[n]. Our proof technique is based on diagonalization against all deterministic Turing machines working in O(n) space by an universal nondeterministic Turing machine running in O(n) space. Our proof also implies the following consequences: (1) There exists no deterministic Turing machine working in O(log n) space deciding the st-connectivity question (STCON); (2) L≠ NL; (3) L≠ P.