Complete Algorithms

This chapter is concerned with sound and complete algorithms for testing satisfiability, i.e., algorithms that are guaranteed to terminate with a correct decision on the satisfiability/unsatisfiability of the given CNF. One can distinguish between a few approaches on which complete satisfiability algorithms have been based. The first approach is based on existential quantification, where one successively eliminates variables from the CNF without changing the status of its satisfiability. When all variables have been eliminated, the satisfiability test is then reduced into a simple test on a trivial CNF. The second approach appeals to sound and complete inference rules, applying them successively until either a contradiction is found (unsatisfiable CNF) or until the CNF is closed under these rules without finding a contradiction (satisfiable CNF). The third approach is based on systematic search in the space of truth assignments, and is marked by its modest space requirements. The last approach we will discuss is based on combining search and inference, leading to algorithms that currently underly most modern complete SAT solvers. We start in the next section by establishing some technical preliminaries that will be used throughout the chapter. We will follow by a treatment of algorithms that are based on existential quantification in Section 3.3 and then algorithms based on inference rules in Section 3.4. Algorithms based on search are treated in Section 3.5, while those based on the combination of search and inference are treated in Section 3.6. Note that some of the algorithms presented here could fall into more than one class, depending on the viewpoint used. Hence, the classification presented in Sections 3.3-3.6 is only one of the many possibilities.