A Linear Algebra Formulation for Boolean Satisfiability Testing

Boolean satisfiability (SAT) is a fundamental problem in computer science, which is one of the first proven NP-complete problems. Although there is no known theoretically polynomial time algorithm for SAT, many heuristic SAT methods have been developed for practical problems. For the sake of efficiency, various techniques were explored, from discrete to continuous methods, from sequential to parallel programmings, from constrained to unconstrained optimizations, from deterministic to stochastic studies. Anyway, showing the unsatisfiability is a main difficulty in certain sense of finding an efficient algorithm for SAT. To address the difficulty, this article presents a linear algebra formulation for unsatisfiability testing, which is a procedure dramatically different from DPLL. Somehow it gives an affirmative answer to an open question by Kautz and Selman in their article “The State of SAT”. The new approach could provide a chance to disprove satisfiability efficiently by resorting to a linear system having no solution, if the investigated formula is unsatisfiable. Theoretically, the method can be applied to test arbitrary formula in polynomial time. We are not unclear whether it can also show satisfiability efficiently in the same way. If so, NP = P. Whatever, our novel method is able to deliver a definite result to uniquely positive 3-SAT in polynomial time. To the best of our knowledge, this constitutes the first polynomial January 11, 2017 DRAFT