Towards a unified theory of logic programming semantics: Level mapping characterizations of selector generated models

Currently, the variety of expressive extensions and different semantics created for logic programs with negation is diverse and heterogeneous, and there is a lack of comprehensive comparative studies which map out the multitude of perspectives in a uniform way. Most recently, however, new methodologies have been proposed which allow one to derive uniform characterizations of different declarative semantics for logic programs with negation. In this paper, we study the relationship between two of these approaches, namely the level mapping characterizations due to [Hitzler and Wendt 2005], and the selector generated models due to [Schwarz 2004]. We will show that the latter can be captured by means of the former, thereby supporting the claim that level mappings provide a very flexible framework which is applicable to very diversely defined semantics.Comment: 17 page

Approximating Operators and Semantics for Abstract Dialectical Frameworks

We provide a systematic in-depth study of the semantics of abstract dialectical frameworks (ADFs), a recent generalisation of Dung\''s abstract argumentation frameworks. This is done by associating with an ADF its characteristic one-step consequence operator and defining various semantics for ADFs as different fixpoints of this operator. We first show that several existing semantical notions are faithfully captured by our definition, then proceed to define new ADF semantics and show that they are proper generalisations of existing argumentation semantics from the literature. Most remarkably, this operator-based approach allows us to compare ADFs to related nonmonotonic formalisms like Dung argumentation frameworks and propositional logic programs. We use polynomial, faithful and modular translations to relate the formalisms, and our results show that both abstract argumentation frameworks and abstract dialectical frameworks are at most as expressive as propositional normal logic programs

Ultimate approximation and its application in nonmonotonic knowledge representation systems

AbstractIn this paper we study fixpoints of operators on lattices and bilattices in a systematic and principled way. The key concept is that of an approximating operator, a monotone operator on the product bilattice, which gives approximate information on the original operator in an intuitive and well-defined way. With any given approximating operator our theory associates several different types of fixpoints, including the Kripke–Kleene fixpoint, stable fixpoints, and the well-founded fixpoint, and relates them to fixpoints of operators being approximated. Compared to our earlier work on approximation theory, the contribution of this paper is that we provide an alternative, more intuitive, and better motivated construction of the well-founded and stable fixpoints. In addition, we study the space of approximating operators by means of a precision ordering and show that each lattice operator O has a unique most precise—we call it ultimate—approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We then discuss applications of these results in logic programming