The Separation of $NP$ and $PSPACE$

TL;DR

This paper proves that NP is strictly contained within PSPACE by constructing a language that separates the two classes using diagonalization and new techniques.

Contribution

It establishes the separation of NP and PSPACE through a novel diagonalization approach and new methods developed by the author.

Findings

NP is strictly contained in PSPACE.

A language is constructed that is outside NP but inside PSPACE.

The proof uses diagonalization and new complexity techniques.

Abstract

There is an 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 $\mathcal{NP}\ne\mathcal{PSPACE}$. In this paper, we confirm this conjecture affirmatively 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 $k\in\mathbb{N}_1$. We achieve this by virtue of the prerequisite of $$ {\rm NTIME}[S(n)]\subseteq{\rm DSPACE}S(n)], $$ and then by diagonalization against all polynomial-time nondeterministic Turing machines via a universal nondeterministic Turing machine $M_0$. 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 and novel new techniques developed in the author's recent works \cite{Lin21a,Lin21b} with some new refinement.