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The relationship between the complexity classes $P$ and $NP$ is an unsolvedquestion in the field of theoretical computer science. In the first part ofthis 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 descriptorfunction ${\cal H}_\varphi$ for any 3-CNF-SAT problem $\varphi$ of size $n$, inthe sense that ${\cal H}_\varphi : \{0,1\}^n \rightarrow \{0,1\}^n$ is suchthat $Im \; {\cal H}_\varphi$ is the set of all the solutions of $\varphi$. Anew 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 canbe of exponential complexity], the complexity needed for the computation of $Im\; {\cal H}_\varphi$, the set of all solutions, is shown to be polynomial forhard 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 indicatorfunction $\mathbb{1}_{{\cal S}_\varphi}$ for the set of solutions, to develop agreedy polynomial algorithm to solve hard 3-CNF-SAT problems.

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