Non-Deterministic Approximation Fixpoint Theory and Its Application in Disjunctive Logic Programming

Semantics of various formalisms for knowledge representation can often be described by fixpoints of corresponding operators. For example, in many logics theories of a set of formulas can be seen as fixpoints of the underlying consequence operator [52]. Likewise, in logic programming, default logic or formal argumentation, all the major semantics can be formulated as different types of fixpoints of the same operator (see [22]). Such operators are usually non-monotonic, and so one cannot always be sure whether their fixpoints exist, and how they can be constructed. In order to deal with this'illusive nature' of the fixpoints, Denecker, Marek and Truszczyński [22] introduced a method for approximating each value z of the underlying operator by a pair of elements (x, y). These elements intuitively represent lower and upper bounds on z, and so a corresponding approximation operator for the original, non-monotonic operator, is constructed. If the approximating operator that is obtained is precision-monotonic, intuitively meaning that more precise inputs of the operator give rise to more precise outputs, then by Tarski and Knaster's Fixpoint Theorem the approximating operator has fixpoints that can be constructively computed, and which in turn approximate the fixpoints of the approximated operator, if such fixpoints exist. The usefulness of the algebraic theory that underlies the computation process described above was demonstrated on several knowledge representation formalisms, such as propositional logic programming [20], default logic [23], autoepistemic logic [23], abstract argumentation and abstract dialectical frameworks [50], hybrid MKNF [39], the graph description language SCHACL [11], and active integrity constraints [10], each one of which was shown to be an instantiation of this abstract theory of approximation.