Download PDFOpen PDF in browserNP on Logarithmic SpaceEasyChair Preprint 9555, version 216 pages•Date: November 3, 2023Abstract$P$ versus $NP$ is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is $P$ equal to $NP$? It was essentially mentioned in 1955 from a letter written by John Nash to the United States National Security Agency. However, a precise statement of the $P$ versus $NP$ problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity classes are $L$ and $NL$. Whether $L = NL$ is another fundamental question that it is as important as it is unresolved. We prove that $NP \subseteq NSPACE(\log^{2} n)$ just using logarithmic space reductions. Keyphrases: completeness, complexity classes, logarithmic space, polynomial time, reduction
