On exact selection of minimally unsatisfiable subformulae

A minimally unsatisfiable subformula (MUS) is a subset of clauses of a given CNF formula which is unsatisfiable but becomes satisfiable as soon as any of its clauses is removed. The selection of a MUS is of great relevance in many practical applications. This expecially holds when the propositional formula encoding the application is required to have a well-defined satisfiability property (either to be satisfiable or to be unsatisfiable). While selection of a MUS is a hard problem in general, we show classes of formulae where this problem can be solved efficiently. This is done by using a variant of Farkas' lemma and solving a linear programming problem. Successful results on real-world contradiction detection problems are presented.