Deciding the Satisfiability of Propositional Formulas in Finitely-Valued Signed Logics

Signed logic is a way of expressing the semantics of many-valued connectives and quantifiers in a formalism that is well-suited for automated reasoning. In this paper we consider propositional, finitely-valued formulas in clausal normal form. We show that checking the satisfiability of formulas with three or more literals per clause is either NP-complete or trivial, depending on whether the intersection of all signs is empty or not. The satisfiability of bijunctive formulas, i.e., formulas with at most two literals per clause, is decidable in linear time if the signs form a Helly family, and is NP-complete otherwise. We present a polynomial-time algorithm for deciding whether a given set of signs satisfies the Helly property. Our results unify and extend previous results obtained for particular sets of signs.