Hashing-based approximate counting of minimal unsatisfiable subsets

In many areas of computer science, we are given an unsatisfiable Boolean formula F in CNF, i.e. a set of clauses, with the goal to analyse the unsatisfiability. Examination of minimal unsatisfiable subsets (MUSes) of F is a kind of such analysis. While researchers in the past two decades focused mainly on techniques for explicit identification of MUSes, there have recently emerged various applications that do not require the explicit identification of MUSes. For instance, in the domain of diagnosis, it is often sufficient to count the number of MUSes. While in theory, one can simply count all MUSes by explicitly enumerating them, in practice, the complete explicit enumeration is often not possible for instances with a reasonably large number of MUSes. In this work, we describe our approximate MUS counting procedure called AMUSIC. Our approach avoids exhaustive MUS enumeration by combining the classical technique of universal hashing with advances in QBF solvers along with usage of union and intersection of MUSes to achieve runtime efficiency. Our prototype implementation of AMUSIC is shown to scale to instances that were clearly beyond the realm of enumeration-based approaches.