A branch and bound algorithm for extracting smallest minimal unsatisfiable subformulas
 Mark Liffiton,
 Maher Mneimneh,
 Inês Lynce,
 Zaher Andraus,
 João MarquesSilva,
 Karem Sakallah
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Explaining the causes of infeasibility of Boolean formulas has practical applications in numerous fields, such as artificial intelligence (repairing inconsistent knowledge bases), formal verification (abstraction refinement and unbounded model checking), and electronic design (diagnosing and correcting infeasibility). Minimal unsatisfiable subformulas (MUSes) provide useful insights into the causes of infeasibility. An unsatisfiable formula often has many MUSes. Based on the application domain, however, MUSes with specific properties might be of interest. In this paper, we tackle the problem of finding a smallestcardinality MUS (SMUS) of a given formula. An SMUS provides a succinct explanation of infeasibility and is valuable for applications that are heavily affected by the size of the explanation. We present (1) a baseline algorithm for finding an SMUS, founded on earlier work for finding all MUSes, and (2) a new branchandbound algorithm called Digger that computes a strong lower bound on the size of an SMUS and splits the problem into more tractable subformulas in a recursive search tree. Using two benchmark suites, we experimentally compare Digger to the baseline algorithm and to an existing incomplete genetic algorithm approach. Digger is shown to be faster in nearly all cases. It is also able to solve far more instances within a given runtime limit than either of the other approaches.
 Title
 A branch and bound algorithm for extracting smallest minimal unsatisfiable subformulas
 Journal

Constraints
Volume 14, Issue 4 , pp 415442
 Cover Date
 20091201
 DOI
 10.1007/s1060100890588
 Print ISSN
 13837133
 Online ISSN
 15729354
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Boolean satisfiability
 SAT
 Infeasibility
 Minimal unsatisfiable subformula
 MUS
 Smallest minimal unsatisfiable subformula
 SMUS
 Authors

 Mark Liffiton ^{(1)}
 Maher Mneimneh ^{(1)}
 Inês Lynce ^{(2)}
 Zaher Andraus ^{(1)}
 João MarquesSilva ^{(3)}
 Karem Sakallah ^{(1)}
 Author Affiliations

 1. University of Michigan, Ann Arbor, MI, USA
 2. Technical University of Lisbon, Lisbon, Portugal
 3. University of Southampton, Southampton, UK