Computing Minimal Sets on Propositional Formulae I: Problems & Reductions

Boolean Satisfiability (SAT) is arguably the archetypical NP-complete decision problem. Progress in SAT solving algorithms has motivated an ever increasing number of practical applications in recent years. However, many practical uses of SAT involve solving function as opposed to decision problems. Concrete examples include computing minimal unsatisfiable subsets, minimal correction subsets, prime implicates and implicants, minimal models, backbone literals, and autarkies, among several others. In most cases, solving a function problem requires a number of adaptive or non-adaptive calls to a SAT solver. Given the computational complexity of SAT, it is therefore important to develop algorithms that either require the smallest possible number of calls to the SAT solver, or that involve simpler instances. This paper addresses a number of representative function problems defined on Boolean formulas, and shows that all these function problems can be reduced to a generic problem of computing a minimal set subject to a monotone predicate. This problem is referred to as the Minimal Set over Monotone Predicate (MSMP) problem. This exercise provides new ways for solving well-known function problems, including prime implicates, minimal correction subsets, backbone literals, independent variables and autarkies, among several others. Moreover, this exercise motivates the development of more efficient algorithms for the MSMP problem. Finally the paper outlines a number of areas of future research related with extensions of the MSMP problem.