Constraint Logic Programming (CLP) is a logic programming formalism used to solve problems requiring the consideration of constraints, like resource allocation and automated planning and scheduling. It has previously been extended in various directions, for example to support fuzzy constraint satisfaction, uncertainty, or negation, with different notions of semiring being used as a unifying abstraction for these generalizations. None of these extensions have studied clauses with negation allowed in the body. We investigate an extension of CLP which unifies many of these extensions and allows negation in the body. We provide semantics for such programs, using the framework of approximation fixpoint theory, and give a detailed overview of the impacts of properties of the semirings on the resulting semantics. As such, we provide a unifying framework that captures existing approaches and allows extending them with a more expressive language.

Choice constructs are an important part of the language of logic programming, yet the study of their semantics has been a challenging task. So far, only two-valued semantics have been studied, and the different proposals for such semantics have not been compared in a principled way. In this paper, an operator-based framework allow for the definition and comparison of different semantics in a principled way is proposed.

A wide variety of nonmonotonic semantics can be expressed as approximators defined under AFT (Approximation Fixpoint Theory). Using traditional AFT theory, it is not possible to define approximators that rely on information computed in previous iterations of stable revision. However, this information is rich for semantics that incorporate classical negation into nonmonotonic reasoning. In this work, we introduce a methodology resembling AFT that can utilize priorly computed upper bounds to more precisely capture semantics. We demonstrate our framework's applicability to hybrid MKNF (minimal knowledge and negation as failure) knowledge bases by extending the state-of-the-art approximator.

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.

Approximation fixpoint theory (AFT) provides an algebraic framework for the study of fixpoints of operators on bilattices and has found its applications in characterizing semantics for various classes of logic programs and nonmonotonic languages. In this paper, we show one more application of this kind: the alternating fixpoint operator by Knorr et al. for the study of the well-founded semantics for hybrid MKNF knowledge bases is in fact an approximator of AFT in disguise, which, thanks to the power of abstraction of AFT, characterizes not only the well-founded semantics but also two-valued as well as three-valued semantics for hybrid MKNF knowledge bases. Furthermore, we show an improved approximator for these knowledge bases, of which the least stable fixpoint is information richer than the one formulated from Knorr et al.'s construction. This leads to an improved computation for the well-founded semantics. This work is built on an extension of AFT that supports consistent as well as inconsistent pairs in the induced product bilattice, to deal with inconsistencies that arise in the context of hybrid MKNF knowledge bases. This part of the work can be considered generalizing the original AFT from symmetric approximators to arbitrary approximators.