In various areas of computer science, we deal with a set of constraints to be satisfied. If the constraints cannot be satisfied simultaneously, it is desirable to identify the core problems among them. Such cores are called minimal unsatisfiable subsets (MUSes). The more MUSes are identified, the more information about the conflicts among the constraints is obtained. However, a full enumeration of all MUSes is in general intractable due to the large number (even exponential) of possible conflicts. Moreover, to identify MUSes, algorithms have to test sets of constraints for their simultaneous satisfiability. The type of the test depends on the application domains. The complexity of the tests can be extremely high especially for domains like temporal logics, model checking, or SMT. In this paper, we propose a recursive algorithm that identifies MUSes in an online manner (i.e., one by one) and can be terminated at any time. The key feature of our algorithm is that it minimises the number of satisfiability tests and thus speeds up the computation. The algorithm is applicable to an arbitrary constraint domain and its effectiveness demonstrates itself especially in domains with expensive satisfiability checks. We benchmark our algorithm against the state-of-the-art algorithm Marco on the Boolean and SMT constraint domains and demonstrate that our algorithm really requires less satisfiability tests and consequently finds more MUSes in the given time limits.
Given an unsatisfiable formula F in CNF, i.e. a set of clauses, the problem of Minimal Unsatisfiable Subset (MUS) seeks to identify a minimal subset of clauses such that N is unsatisfiable. The emerging viewpoint of MUSes as the root causes of unsatisfiability has led MUSes to find applications in a wide variety of diagnostic approaches. Recent advances in identification and enumeration of MUSes have motivated researchers to discover applications that can benefit from rich information about the set of MUSes. One such extension is that of counting the number of MUSes. The current best approach for MUS counting is to employ a MUS enumeration algorithm, which often does not scale for the cases with a reasonably large number of MUSes. Motivated by the success of hashing-based techniques in the context of model counting, we design the first approximate MUS counting procedure with guarantees, called . Our approach avoids exhaustive MUS enumeration by combining the classical technique of universal hashing with advances in QBF solvers along with a novel usage of union and intersection of MUSes to achieve runtime efficiency. Our prototype implementation of is shown to scale to instances that were clearly beyond the realm of enumeration-based approaches.
In many areas of computer science, we are given an unsatisfiable set of constraints with the goal to provide an insight into the unsatisfiability. One of common approaches is to identify minimal unsatisfiable subsets (MUSes) of the constraint set. The more MUSes are identified, the better insight is obtained. However, since there can be up to exponentially many MUSes, their complete enumeration might be intractable. Therefore, we focus on algorithms that enumerate MUSes online, i.e. one by one, and thus can find at least some MUSes even in the intractable cases. Since MUSes find applications in different constraint domains and new applications still arise, there have been proposed several domain agnostic algorithms. Such algorithms can be applied in any constraint domain and thus theoretically serve as ready-to-use solutions for all the emerging applications. However, there are almost no domain agnostic tools, i.e. tools that both implement domain agnostic algorithms and can be easily extended to support any constraint domain. In this work, we close this gap by introducing a domain agnostic tool called MUST. Our tool outperforms other existing domain agnostic tools and moreover, it is even competitive to fully domain specific solutions.
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