The realization of the Semantic Web vision, in which computational logic has a prominent role, has stimulated a lot of research on combining rules and ontologies, which are formulated in different formalisms, into a framework that is more useful for describing semantic content. In particular, combining logic programming with the Web Ontology Language (OWL), which is a standard based on description logics, emerged as an important issue for linking the Rules and Ontology Layers of the Semantic Web. Non-monotonic description logic programs (or dl-programs) were introduced for such a combination, in which a pair (L,P) of a description logic knowledge base L and a set of rules P with negation as failure is given a model-based semantics that generalizes the answer set semantics of logic programs. In this paper, we reconsider dl-programs and present a well-founded semantics for them as an analog for the other main semantics of logic programs. It generalizes the canonical definition of the well-founded semantics based on unfounded sets, and, as we show, lifts many of the well-known properties from ordinary logic programs to dl-programs. Among these properties: our semantics amounts to a partial model approximating the answer set semantics, which yields for positive and stratified dl-programs a total model coinciding with the answer set semantics; it has polynomial data complexity provided the access to the description logic

is relatively straightforward. Another advantage of the ultimate approximation is that in cases where TP is monotone the ultimate well-founded model will be 2-valued and will coincide with the least fixpoint of TP . This is not the case for the standard well-founded semantics. For example in the standard well-founded model of the program: # p. p. p is undefined while the associated TP operator is monotone and p is true in the ultimate well-founded model. One disadvantage of using the ultimate semantics is that it has a higher computational cost even for programs without aggregates. The complexity goes one level higher in the polynomial hierarchy to # 2 for the well-founded model and to 2 for a stable model which is also complete for this class [2]. Fortunately, by adding aggregates the complexity does not increase further. To give an example of a logic program with aggregates we consider the problem of computing the length of the shortest path between two nodes in a direc

Well-known principles of induction include monotone induction and different sorts of nonmonotone induction such as inflationary induction, induction over well-founded sets and iterated induction. In this work, we define a logic formalizing induction over well-founded sets and monotone and iterated induction. Just as the principle of positive induction has been formalized in FO(LFP), and the principle of inflationary induction has been formalized in FO(IFP), this paper formalizes the principle of iterated induction in a new logic for Non-Monotone Inductive Definitions (ID-logic). The semantics of the logic is strongly influenced by the well-founded semantics of logic programming. This paper discusses the formalisation of different forms of (non-)monotone induction by the well-founded semantics and illustrates the use of the logic for formalizing mathematical and common-sense knowledge. To model different types of induction found in mathematics, we define several subclasses of definitions, and show that they are correctly formalized by the well-founded semantics. We also present translations into classical first or second order logic. We develop modularity and totality results and demonstrate their use to analyze and simplify complex definitions. We illustrate the use of the logic for temporal reasoning. The logic formally extends Logic Programming, Abductive Logic Programming and Datalog, and thus formalizes the view on these formalisms as logics of (generalized) inductive definitions. Categories and Subject Descriptors:... [...]:... 1.

Abstract. Approximate reasoning is used in a variety of reasoning tasks in Logicbased Artificial Intelligence. In this abstract we compare a number of such reasoning schemes and show how they relate and differ from the approach of Pawlak’s Rough Sets. 1

Currently, the variety of expressive extensions and di#erent 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 di#erent 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 [17], and the selector generated models due to [24]. 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.

Abstract. The goal of this note is to provide a background and references for the invited lecture presented at Computer Science Logic 2006. We briefly discuss motivations that led to the emergence of nonmonotonic logics and introduce two major nonmonotonic formalisms, default and autoepistemic logics. We then point out to algebraic principles behind the two logics and present an abstract algebraic theory that unifies them and provides an effective framework to study properties of nonmonotonic reasoning. We conclude with comments on other major research directions in nonmonotonic logics. 1 Why nonmonotonic logics In the late 1970s, research on languages for knowledge representation, and considerations of basic patterns of commonsense reasoning brought attention to rules of inference that admit exceptions and are used only under the assumption of normality of the world in which one functions or to put it differently, when things are as expected. For instance, a knowledge base concerning a university should support an inference that, given no information that might indicate otherwise, if Dr. Jones is a professor at