The Complexity of 3SAT_N and the P versus NP Problem

We introduce the NP-complete problem 3SAT_N and extend Tovey’s results to a classification theorem for this problem. This theorem leads us to generalize the concept of truth assignments for SAT to aggressive truth assignments for 3SAT_N. We introduce the concept of a set compatible with the P and NP problem, and prove that all aggressive truth assignments are pseudo-algorithms. We combine algorithm, pseudo-algorithm and diagonalization method to study the complexity of 3SAT_N and the P versus NP problem. The main result is P != NP.

💡 Research Summary

The paper introduces a variant of the Boolean satisfiability problem called 3SAT_N, defined as a 3‑CNF formula in which each clause contains exactly three literals, no clause is tautological, no clause is duplicated, and each clause contains distinct variables (a “normal” expression). Building on Craig Tovey’s work on (r,s)‑SAT, the authors claim to have extended his classification results to obtain a polynomial‑time “classification theorem” for all instances of 3SAT_N. The theorem is presented as a sequence of four reduction sub‑routines: (1) removal of tautological clauses, (2) elimination of duplicate clauses, (3) removal of repeated variables within a clause, and (4) forcing single‑literal or two‑literal clauses to be true by introducing a set of auxiliary clauses (nine clauses for a unit literal, five for a binary literal). While each sub‑routine is described as polynomial‑time, the paper does not rigorously bound the blow‑up caused by the auxiliary variables and clauses, leaving open the possibility that the transformation may not be truly polynomial in the worst case.

The core novelty claimed by the authors is the notion of an “aggressive truth assignment” (ATA). Traditional truth assignments map each variable to true or false; an ATA is a generalized, potentially infinite sequence of atomic assignments that is applied iteratively to a formula. The authors encode an ATA as a binary real number in the interval