The SAT Problem of Signed CNF Formulas

Abstract

Signed conjunctive normal form (signed CNF) is a classical conjunctive clause form using a generalised notion of literal, called signed literal.A signed literal is an expression of the form S:p, where p is a classical atom and S, its sign, is a subset of a domain N.The informal meaning is “p takes one of the values in S”.Signed formulas are a logical language for knowledge representation that lies in the intersection of the areas constraint programming (CP) many-valued logic (MVL), and annotated logic programming (ALP). This central rôle of signed CNF justifies a detailed study of its subclasses including algorithms for and complexities of associated satisfiability problems (SAT problems). Although signed logic is used since the 1960s, there are only few systematic investigations of its properties. In contrast to work done in ALP and MVL, our present work is a more fine-grained study for the case of propositional CNF. We highlight the most interesting lines of current research: (i) signed versions of some main proponents of classical deduction systems including non-trivial refinements having no classical counterpart; (ii) incomplete local search methods for satisfiability checking of signed formulas; (iii) phase transition phenomena as known, for example, from classical SAT and the influence of the cardinality of N on the crossover point; (iv) the complexity of the SAT problem for signed CNF and its subclasses.