The Separation of N P and PSPACE

There is a important and interesting open question on NP ? = PSPACE in computational complexity. It is a widespread belief that NP (cid:54) = PSPACE . In this paper, we confirm this conjecture by showing that there is a language L d accepted by no polynomial-time nondeterministic Turing machines but accepted by a nondeterministic Turing machine running within space O ( n k ) for all k ∈ N 1 , by virtue of the premise of NTIME[ S ( n )] ⊆ DSPACE[ S ( n )] , and then by diagonalization against all polynomial-time nondeterministic Turing machines via a universal nondeterministic Turing machine M 0 running in space O ( n k ) for all k ∈ N 1 . We further show that L d ∈ PSPACE , which leads to the conclusion NP (cid:54) = PSPACE . Our approach is based on standard diagonalization similar to [Lin21a, Lin21b] with many novel auxiliary methods.

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