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

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 ${\cal H}_φ$ for any 3-CNF-SAT problem $φ$ of size $n$, in the sense that ${\cal H}_φ: \{0,1\}^n \rightarrow \{0,1\}^n$ is such that $Im \; {\cal H}_φ$ is the set of all the solutions of $φ$. A new merge operation ${\cal H}_φ\bigwedge {\cal H}_ψ$ is defined, where $ψ$ is a single 3-CNF clause. Given ${\cal H}_φ$ [but this can be of exponential complexity], the complexity needed for the computation of $Im \; {\cal H}_φ$, 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 ${\cal H}_φ$ and the indicator function $\mathbb{1}_{{\cal S}_φ}$ for the set of solutions, to develop a greedy polynomial algorithm to solve hard 3-CNF-SAT problems.

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