TR-2007008: The Satisfiability Problem—From the Theory of NP-Completeness to State-of-the-Art SAT Solvers

Given a formula of the propositional calculus, determining whether there exists a variable assignment such that the formula evaluates to true under the usual rules of interpretation is called the Boolean Satisfiability Problem, commonly abbreviated as SAT. SAT has a central role in the theory of computational complexity as it was the first computational task shown to be NP-complete. Subsequent problems have been shown to belong to the same family by proving they are at least as hard as SAT. Roughly, a task is NPcomplete if a good algorithm for it would entail a good algorithm for SAT. Thus, despite its appealing simplicity, one can think of SAT as the “core” problem in this family of hard problems. SAT is of special concern to AI because of its direct connection to reasoning and theorem-proving. Deductive reasoning is simply the complement of satisfiability, and researches first became interested in SAT precisely for this reason. Nonetheless, in recent years there has been an explosion of interest in SAT because more and more practical problems in AI that utilize other forms of reasoning such as constraint-based reasoning with applications in planning, configuration, diagnosis, resource allocation, scheduling, and electronic design automation, make direct appeal to satisfiability, i.e., can easily be represented as SAT problems. The first SAT solver is traditionally attributed to Davis and Putnam in 1960. Since then a wealth of algorithms have been developed and several approaches have been proposed, including variations of backtrack search, local search, continuous formulations and algebraic manipulation. In this report we introduce SAT through the theory of NP-completeness, and follow it with a broad overview of the different techniques and algorithms used to solve the SAT problem, noting that given vast amount of algorithms that have been proposed only the classical and most prominent are discussed. Evaluating the performance of SAT algorithms has become a problem in itself, which has yielded interesting insights. We conclude this report with an overview of the different methods used to evaluate the performance of SAT algorithms.