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The Separation of $NP$ and $PSPACE$

Tianrong Lin

TL;DR

This paper addresses the longstanding question of whether $\mathcal{NP}$ is distinct from $\mathcal{PSPACE}$ by constructing a language $L_d$ that is not decidable by any polynomial-time nondeterministic TM, yet is decidable within polynomial space. The approach relies on the standard diagonalization principle together with a diagonalization against an enumeration of all polynomial-time NTMs, realized via a universal nondeterministic TM $M_0$ that decodes encodings and simulates candidate machines within space bounds tied to their time parameters. The key results show $L_d\in\mathcal{PSPACE}$ while $L_d\notin\mathcal{NP}$, and thus $\mathcal{NP}\subsetneqq\mathcal{PSPACE}$ (via Savitch’s theorem $\mathcal{PSPACE}=\mathcal{NSPACE}$). The work extends the standard diagonalization repertoire with refinements from the author’s prior research and asserts a clear separation between these central complexity classes, with discussion of open problems and broader implications.

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.

The Separation of $NP$ and $PSPACE$

TL;DR

This paper addresses the longstanding question of whether is distinct from by constructing a language that is not decidable by any polynomial-time nondeterministic TM, yet is decidable within polynomial space. The approach relies on the standard diagonalization principle together with a diagonalization against an enumeration of all polynomial-time NTMs, realized via a universal nondeterministic TM that decodes encodings and simulates candidate machines within space bounds tied to their time parameters. The key results show while , and thus (via Savitch’s theorem ). The work extends the standard diagonalization repertoire with refinements from the author’s prior research and asserts a clear separation between these central complexity classes, with discussion of open problems and broader implications.

Abstract

There is an important and interesting open question in computational complexity on the relation between the complexity classes and . It is a widespread belief that . In this paper, we confirm this conjecture affirmatively by showing that there is a language accepted by no polynomial-time nondeterministic Turing machines but accepted by a nondeterministic Turing machine running within space for all . We achieve this by virtue of the prerequisite of and then by diagonalization against all polynomial-time nondeterministic Turing machines via a universal nondeterministic Turing machine . We further show that , which leads to the conclusion 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.

Paper Structure

This paper contains 7 sections, 9 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organization
  3. Preliminaries
  4. Enumeration of Polynomial-Time Nondeterministic Turing Machines
  5. $\mathcal{NP}$ Differs from $\mathcal{PSPACE}$
  6. Proof of Theorem \ref{['theorem1']}
  7. Conclusions and Open Problems

Key Result

Theorem 1

$\mathcal{NP}\neq \mathcal{PSPACE}$. In other words, $\mathcal{NP}\subsetneqq \mathcal{PSPACE}$.

Figures (1)

  • Figure 1: Cantor pairing function

Theorems & Definitions (23)

  • Theorem 1
  • Corollary 2
  • Definition 2.1: $k$-tape nondeterministic Turing machine, AHU74
  • Definition 2.2: cf. polynomial-time deterministic Turing machines in Coo00
  • Remark 2.1
  • Lemma 2.1: Theorem 7.4 (b) in Pap94, p. 147
  • proof
  • Lemma 2.2: Lemma 10.1 in AHU74
  • Lemma 2.3: Corollary 2 in AHU74, p. 372
  • Definition 3.1: Rud76, p. 27, Definition 2.7
  • ...and 13 more