Revising Undefinedness in the Well-Founded Semantics of Logic Programs

The Well-Founded Semantics (WFS) for normal logic programs associates with each program one single model expressing truth, falsity and undefinedness of atoms. Under the WFS, atoms are said to be undefined if: • Either are part of a two-valued choice (true in some worlds, false in others) but never undeniably true or false; • Or depend on an already undefined literal; • Or are not classically supported. Undefinedness due to lack of classical support could be overcome by the introduction of another form of support, which would allow the WFS to correctly deal with programs requiring non-classical forms of reasoning, and thus gain expressiveness. One of these forms of unclassical support whose application has already been studied in the Revised Stable Models semantics (rSMs) is support by reductio ad absurdum (RAA). This principle states that an hypothesis should be true if by assuming it’s false this assumption leads to a contradition. In this thesis we propose to study the application of the RAA principle to the WFS, thus defining the Revised Well-Founded Semantics (RWFS). Besides this definition we’ll also study the definition of a fixed-point operator Gr, a counterpart of Gelfond-Lifschitz operator G, with support for RAA reasoning, and use this operator to perform the calculation of rSMs and the revised well-founded model of normal logic programs. We will also study a new property of rSMs and the definition of the revised partial stable models. This thesis concludes with the discussion of several open issues and possible next research paths

Ultimate approximations in nonmonotonic knowledge representation systems

We study fixpoints of operators on lattices. To this end we introduce the notion of an approximation of an operator. We order approximations by means of a precision ordering. We show that each lattice operator O has a unique most precise or ultimate approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We apply our theory to logic programming and introduce the ultimate Kripke-Kleene, well-founded and stable semantics. We show that the ultimate Kripke-Kleene and well-founded semantics are more precise then their standard counterparts We argue that ultimate semantics for logic programming have attractive epistemological properties and that, while in general they are computationally more complex than the standard semantics, for many classes of theories, their complexity is no worse.Comment: This paper was published in Principles of Knowledge Representation and Reasoning, Proceedings of the Eighth International Conference (KR2002

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

Space Efficiency of Propositional Knowledge Representation Formalisms

We investigate the space efficiency of a Propositional Knowledge Representation (PKR) formalism. Intuitively, the space efficiency of a formalism F in representing a certain piece of knowledge A, is the size of the shortest formula of F that represents A. In this paper we assume that knowledge is either a set of propositional interpretations (models) or a set of propositional formulae (theorems). We provide a formal way of talking about the relative ability of PKR formalisms to compactly represent a set of models or a set of theorems. We introduce two new compactness measures, the corresponding classes, and show that the relative space efficiency of a PKR formalism in representing models/theorems is directly related to such classes. In particular, we consider formalisms for nonmonotonic reasoning, such as circumscription and default logic, as well as belief revision operators and the stable model semantics for logic programs with negation. One interesting result is that formalisms with the same time complexity do not necessarily belong to the same space efficiency class