Minimal Sets over Monotone Predicates in Boolean Formulae

The importance and impact of the Boolean satisfiability (SAT) problem in many practical settings is well-known. Besides SAT, a number of computational problems related with Boolean formulas find a wide range of practical applications. Concrete examples for CNF formulas include computing prime implicates (PIs), minimal models (MMs), minimal unsatisfiable subsets (MUSes), minimal equivalent subsets (MESes) and minimal correction subsets (MCSes), among several others. This paper builds on earlier work by Bradley and Manna and shows that all these computational problems can be viewed as computing a minimal set subject to a monotone predicate, i.e. the MSMP problem. Thus, if cast as instances of the MSMP problem, these computational problems can be solved with the same algorithms. More importantly, the insights provided by this result allow developing a new algorithm for the general MSMP problem, that is asymptotically optimal. Moreover, in contrast with other asymptotically optimal algorithms, the new algorithm performs competitively in practice. The paper carries out a comprehensive experimental evaluation of the new algorithm on the MUS problem, and demonstrates that it outperforms state of the art MUS extraction algorithms.