An apparatus and methods for the production of satisfiability reports are provided. In an exemplary embodiment, a method of producing a report is provided. The method includes generating a complete assignment for a CNF formula, deriving first second sets of clauses that are unsatisfied by the reference point, making decision assignments, performing BCP then recomputing the second set of clauses. One feature of this embodiment is that it provides for efficient solutions for SAT problems. Other embodiments provide apparatus and software products that implement the disclosed methods. This Abstract is provided for the sole purpose of complying with the Abstract requirement rules that allow a reader to quickly ascertain the subject matter of the disclosure contained herein. This Abstract is submitted with the explicit understanding that it will not be used to interpret or to limit the scope or the meaning of the claims.