  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.
