NP on Logarithmic Space: Mediterranean International Conference of Pure & Applied Mathematics and Related Area (MICOPAM 2023)

The $P$ versus $NP$ problem 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 $DSPACE(S(n))$ and $NSPACE(S(n))$ for every space-constructible function $S(n)$. We prove that $NP \subseteq NSPACE(\log^{2} n)$ just using logarithmic space reductions.