Hard 3-CNF-SAT problems are in $P$ -- A first step in proving $NP=P$ (2001.00760v1)

Abstract: The relationship between the complexity classes $P$ and $NP$ is an unsolved question in the field of theoretical computer science. In the first part of this paper, a lattice framework is proposed to handle the 3-CNF-SAT problems, known to be in $NP$. In the second section, we define a multi-linear descriptor function ${\cal H}\varphi$ for any 3-CNF-SAT problem $\varphi$ of size $n$, in the sense that ${\cal H}\varphi : {0,1}^n \rightarrow {0,1}^n$ is such that $Im \; {\cal H}\varphi$ is the set of all the solutions of $\varphi$. A new merge operation ${\cal H}\varphi \bigwedge {\cal H}{\psi}$ is defined, where $\psi$ is a single 3-CNF clause. Given ${\cal H}\varphi$ [but this can be of exponential complexity], the complexity needed for the computation of $Im \; {\cal H}\varphi$, the set of all solutions, is shown to be polynomial for hard 3-CNF-SAT problems, i.e. the one with few ($\leq 2^k$) or no solutions. The third part uses the relation between ${\cal H}\varphi$ and the indicator function $\mathbb{1}{{\cal S}\varphi}$ for the set of solutions, to develop a greedy polynomial algorithm to solve hard 3-CNF-SAT problems.