Logical Foundations and Complexity of 4QL, a Query Language with Unrestricted Negation

Abstract: The paper discusses properties of a DATALOG$^{\neg\neg}$-like query language 4QL, originally outlined in [MS10]. Negated literals in heads of rules naturally lead to inconsistencies. On the other hand, rules do not have to attach meaning to some literals. Therefore 4QL is founded on a four-valued semantics, employing the logic introduced in [MSV08, VMS09] with truth values: 'true', 'false', 'inconsistent' and 'unknown'. 4QL allows one to use rules with negation in heads and bodies of rules, it is based on a simple and intuitive semantics and provides uniform tools for "lightweight" versions of known forms of nonmonotonic reasoning. In addition, 4QL is tractable as regards its data complexity and captures PTIME queries. Even if DATALOG$^{\neg\neg}$ is known as a concept for the last 30 years, to our best knowledge no existing approach enjoys these properties. In the current paper we: - investigate properties of well-supported models of 4QL - prove the correctness of the algorithm for computing well-supported models - show that 4QL has PTIME data complexity and captures PTIME.