This paper tackles the problem of deciding whether a given clause belongs to some minimally unsatisfiable subset (MUS) of a formula, where the formula is in conjunctive normal form (CNF) and unsatisfiable. Deciding MUS-membership helps the understanding of why a formula is unsatisfiable. If a clause does not belong to any MUS, then removing it will certainly not contribute to restoring the formula’s consistency. Unsatisfiable formulas and consistency restoration in particular have a number of practical applications in areas such as software verification or product configuration. The MUS-membership problem is known to be in the second level of polynomial hierarchy, more precisely it is \(\Sigma{^p_2}\) -complete. Hence, quantified Boolean formulas (QBFs) represent a possible avenue for tackling the problem. This paper develops a number of novel QBF formulations of the MUS-membership problem and evaluates their practicality using modern off-the-shelf solvers.


Minimal Model Conjunctive Normal Form Truth Assignment Boolean Formula Membership Problem 
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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Mikoláš Janota
    • 1
  • Joao Marques-Silva
    • 1
    • 2
  1. 1.INESC-IDLisbonPortugal
  2. 2.University College DublinIreland

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