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L is different from NP

J. Andres Montoya

TL;DR

This work develops a framework to separate LOGSPACE from NP by combining syntactic-morphism reductions, real-time and pebble automata theory, and an entropy-based highness program. It shows that the LOGSPACE class \mathcal{L} equals the union of the pebble levels \mathcal{REG}_k and that the nondeterministic real-time class \mathcal{NRT} is high within \mathcal{L}, preventing Greibach's hardest quasi-real-time language L_R from lying in \mathcal{L}. Consequently, the paper derives the separation \mathcal{L} ≠ \mathcal{NP} and discusses conditional implications for related separations such as \mathcal{NL} ≠ \mathcal{P} and \mathcal{L} ≠ \mathcal{NL}. The results rest on proving the existence of high-entropy, polylogarithmically complex constructions (S_k) and translating these into lower bounds on the entropy of pebble-automaton computations, thereby ruling out containment of certain languages in low pebble levels.

Abstract

We prove that the class LOGSPACE (L, for short) is different from the class NP.

L is different from NP

TL;DR

This work develops a framework to separate LOGSPACE from NP by combining syntactic-morphism reductions, real-time and pebble automata theory, and an entropy-based highness program. It shows that the LOGSPACE class \mathcal{L} equals the union of the pebble levels \mathcal{REG}_k and that the nondeterministic real-time class \mathcal{NRT} is high within \mathcal{L}, preventing Greibach's hardest quasi-real-time language L_R from lying in \mathcal{L}. Consequently, the paper derives the separation \mathcal{L} ≠ \mathcal{NP} and discusses conditional implications for related separations such as \mathcal{NL} ≠ \mathcal{P} and \mathcal{L} ≠ \mathcal{NL}. The results rest on proving the existence of high-entropy, polylogarithmically complex constructions (S_k) and translating these into lower bounds on the entropy of pebble-automaton computations, thereby ruling out containment of certain languages in low pebble levels.

Abstract

We prove that the class LOGSPACE (L, for short) is different from the class NP.
Paper Structure (22 sections, 40 theorems, 159 equations)

This paper contains 22 sections, 40 theorems, 159 equations.

Table of Contents

  1. Syntactic Morphisms and Hard Problems
  2. Real-Time Classes
  3. Languages in $\mathcal{RT}$
  4. The Pebble Hierarchy
  5. $\mathcal{L}$ is different from $\mathcal{NP}$
  6. Proving that Sequence $\left\{ L_{n}\right\} _{n\geq 1}$ is High
  7. Entropy of Pebble Automata
  8. Sets of High Entropy
  9. Appendix 1: Sections 2, 3 and 4
  10. Appendix 2: Section 5
  11. Polylogarithmic Factors
  12. $\mathcal{S}_{k}$ Is a H-Set for $L_{k}$
  13. The Random Variables
  14. Promise Automata for the Pair $\left( L_{k+1}^{\ast },\mathcal{S}_{k+1}^{\ast }\right)$
  15. $\mathcal{S}_{k}$ Is a High Entropy Set for $L_{k}:$ Six Inequalities
  16. ...and 7 more sections

Key Result

theorem 1

Greibach's Argument Let $\mathcal{C}$ and $\mathcal{D}$ be two complexity classes. Suppose $\mathcal{C}=\dbigcup\limits_{i\geq 1}\mathcal{C}_{i}$, suppose that $\dbigcup\limits_{i\geq 1}\mathcal{C}_{i}$ is an invariant hierarchy, suppose that $\mathcal{D}$ is principal, and suppose that $\mathcal{D}

Theorems & Definitions (138)

  • remark 1
  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • theorem 1
  • proof
  • remark 2
  • definition 5
  • remark 3
  • ...and 128 more