Hard 3-CNF-SAT problems are in P– A first step in proving NP=P
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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 H_φ for any 3-CNF-SAT problem φ of size n, in the sense that H_φ : {0,1}^n →{0,1}^n is such that Im H_φ is the set of all the solutions of φ. A new merge operation H_φ H_ψ is defined, where ψ is a single 3-CNF clause. Given H_φ [but this can be of exponential complexity], the complexity needed for the computation of Im H_φ, the set of all solutions, is shown to be polynomial for hard 3-CNF-SAT problems, i.e. the one with few (≤ 2^k) or no solutions. The third part uses the relation between H_φ and the indicator function 1_ S_φ for the set of solutions, to develop a greedy polynomial algorithm to solve hard 3-CNF-SAT problems.
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