Finding Models for Blocked 3-SAT Problems in Linear Time by Systematical Refinement of a Sub-model

We report a polynomial time SAT problem instance, the Blocked SAT problem. A blocked clause set, an instance of the Blocked SAT problem, contains only blocked clauses. A close is blocked (for resolution) if it has a literal on which no resolution is possible in the clause set. We know from work of O. Kullmann that a blocked clause can be added or deleted from a clause set without changing its satisfiability. Hence, any blocked clause set is satisfiable, but it is not clear how to find a satisfying assignment for it. We introduce the Blocked SAT Solver algorithm, which provides a model for Blocked SAT problems in linear time, if we know at least one blocked literal per clause. To collect these information polynomial time is needed in general. We show that in case of 3-SAT we can collect these information in linear time. This means that the Blocked 3-SAT problem is a linear time problem. We also discuss how to use blocked clauses if the whole clause set is not blocked.