The Separation of $\mathcal{NP}$ and $\mathcal{PSPACE}$

There is a important and interesting open question in computational complexity on the relation between the complexity classes $\mathcal{NP}$ and $\mathcal{PSPACE}$. It is a widespread belief that $\mathcal{NP}\neq \mathcal{PSPACE}$. In this paper, we confirm this conjecture by showing that there is a language $Ld$ accepted by no polynomial-time nondeterministic Turing machines but accepted by a nondeterministic Turing machine running within space $O(n^k)$ for all $k\in\mathbb{N}1$, by virtue of the premise of [\text{NTIME}[S(n)]\subseteq\text{DSPACE}[S(n)],] and then by diagonalization against all polynomial-time nondeterministic Turing machines via a universal nondeterministic Turing machine $M0$ running in space $O(n^k)$ for all $k\in\mathbb{N}1$. We further show that $L_d\in \mathcal{PSPACE}$, which leads to the conclusion [\mathcal{NP}\subsetneqq\mathcal{PSPACE}.] Our approach is based on standard diagonalization similar to \cite{Lin21a,Lin21b} with some new refinement.