On Finding All Minimally Unsatisfiable Subformulas

Much attention has been given in recent years to the problem of finding Minimally Unsatisfiable Subformulas (MUSes) of Boolean formulas. In this paper, we present a new view of the problem, strongly linking it to the maximal satisfiability problem. From this relationship, we have developed a novel technique for extracting all MUSes of a CNF formula, tightly integrating our implementation with a modern SAT solver. We also present another algorithm for finding all MUSes, developed independently but based on the same relationship. Experimental comparisons show that our approach is consistently faster than the other, and we discuss ways in which ideas from both could be combined to improve further. Many computational problems in a wide range of fields are posed as constraint satisfaction problems, often in the form of Boolean CNF formulas analyzed with satisfiability (SAT) solvers. While SAT solvers can return a short proof in the form of a satisfying assignment when a formula is satisfiable, typically no proof or explanation is given when a formula is found to be unsatisfiable. Explanations of infeasibility are often valuable, and techniques for finding them have been developed for use in these problems. Some techniques have focused on reducing the original set of constraints to produce a minimal, unsatisfiable core representing a cause of infeasibility. In this paper, we present a new approach to finding these cores, focusing on a complete method for finding all unsatisfiable cores of any given formula. Consider an unsatisfiable CNF formula . A Minimally Unsatisfiable Subformula (MUS) of is a subset of 's clauses that is both unsatisfiable and minimal in the sense that all of its proper subsets are satisfiable. An MUS can be seen as an irreducible cause of the infeasibility of the original formula. could have multiple reasons for its infeasibility. In this case would contain multiple MUSes, and fixing any single MUS may not make satisfiable. As long as any MUS is present in the formula, it will remain infeasible. In many applications, it is valuable to find the set of all MUSes, because diagnosing infeasibility is hard, if not impossible, without a complete view of its causes. Additionally, an algorithm that finds all MUSes provides a basis for approximations and techniques that find multiple, though not all, MUSes. Many methods for finding MUSes have been developed in recent years, both for Boolean satisfiability problems and for other types of constraints. Most techniques find φ φ φ φ φ φ On Finding All Minimally Unsatisfiable Subformulas Mark H. Liffiton and Karem A. Sakallah Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor 48109-2122 {liffiton, karem}@eecs.umich.edu