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** Approximate Counting of Minimal Unsatisfiable Subsets**.

Jaroslav Bendik, and Kuldeep S. Meel.

In * Proceedings of International Conference on Computer-Aided Verification (CAV)*, July 2020.

** Invited to FMSD issue dedicated to the best papers from CAV 2020**

Given an unsatisfiable formula F in CNF, i.e. a set of clauses, the problem of Minimal Unsatisfiable Subset (MUS) seeks to identify the minimal subset of clauses N \subseteq F 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 finding 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, which has shown to describe the inconsistency metrics for general propositional knowledge bases. 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 counting procedure with(epsilon,delta) guarantees, called AMUSIC. Our approach avoids exhaustive MUS enumeration by combining the classical technique of universal hashing withadvances in QBF solvers along with a novel 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 reach of enumeration-based approaches.

@inproceedings{bm20, title={Approximate Counting of Minimal Unsatisfiable Subsets}, author={Bendik, Jaroslav and Meel, Kuldeep S.}, month=jul, booktitle=CAV, year={2020}, bib2html_rescat={Counting}, bib2html_dl_pdf={../Papers/cav20-bm.pdf}, note={Invited to FMSD issue dedicated to the best papers from CAV 2020}, bib2html_pubtype={Refereed Conference,Award Winner}, abstract={Given an unsatisfiable formula F in CNF, i.e. a set of clauses, the problem of Minimal Unsatisfiable Subset (MUS) seeks to identify the minimal subset of clauses N \subseteq F 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 finding 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, which has shown to describe the inconsistency metrics for general propositional knowledge bases. 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 counting procedure with (epsilon,delta) guarantees, called AMUSIC. 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 AMUSIC is shown to scale to instances that were clearly beyond the reach of enumeration-based approaches. }, }

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