The Separation of $NP$ and $PSPACE$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917972593868800 |
|---|---|
| author | Lin, Tianrong |
| author_facet | Lin, Tianrong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_11886 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Separation of $NP$ and $PSPACE$ Lin, Tianrong Computational Complexity 68Q15, 68Q17 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. |
| title | The Separation of $NP$ and $PSPACE$ |
| topic | Computational Complexity 68Q15, 68Q17 |
| url | https://arxiv.org/abs/2106.11886 |