On the Tractability of Un/Satisfiability

TL;DR

This paper demonstrates that X3SAT can effectively prove P = NP by simplifying satisfiability checks through scope reductions and exact-1 disjunctions, making unsatisfiability verification straightforward.

Contribution

It introduces a novel approach using scope reductions and exact-1 disjunctions to efficiently determine satisfiability and unsatisfiability in X3SAT formulas.

Findings

X3SAT simplifies unsatisfiability checks

Scope reductions lead to efficient satisfiability verification

The method proves P = NP under certain conditions

Abstract

This paper shows effectiveness of X3SAT in proving P = NP. This is due to the fact that it is easy to check unsatisfiability of a particular truth assignment. A truth assignment leads to some reductions of clauses by means of "exactly-1 disjunction". The reductions result in a conjunction of literals, called the scope of the assignment. It is proved that this particular truth assignment makes the formula unsatisfiable iff the scope_s is unsatisfiable for some s, which is trivial to check. Then, each literal such that its scope is unsatisfiable is removed from the formula. Therefore, the formula is satisfiable iff the scope of every literal is satisfied.