Abstract. The management of uncertain information in logic programs becomes to be important whenever the real world information to be represented is of imperfect nature and the classical crisp true, false approximation is not adequate. A general framework, called Parametric Deductive Databases with Uncertainty (PDDU) framework [10], was proposed as a unifying umbrella for many existing approaches towards the manipulation of uncertainty in logic programs. We extend PDDU with (non-monotonic) negation, a well-known and important feature of logic programs. We show that, dealing with uncertain and incomplete knowledge, atoms should be assigned only approximations of uncertainty values, unless some assumption is used to complete the knowledge. We rely on the closed world assumption to infer as much default "false" knowledge as possible. Our approach leads also to a novel characterizations, both epistemic and operational, of the well-founded semantics in PDDU, and preserves the continuity of the immediate consequence operator, a major feature of the classical PDDU framework.

Abstract. Many frameworks of logic programming have been proposed to manage uncertain information in deductive databases and expert systems. Roughly, on the basis of how uncertainty is associated to facts and the rules in a program, they can be classified into implication-based (IB) and annotation-based (AB). However, one fundamental issue that remains unaddressed in the IB approach is the representation and the manipulation of the non-monotonic mode of negation, an important feature for real applications. Our focus in this paper is to introduce nonmonotonic negation in the parametric IB framework, a unifying umbrella for IB frameworks. The semantical approach that we will adopt is based on the well-founded semantics, one of the most widely studied and used semantics of (classical) logic programs with negation.

Due to the usual incompleteness of information representation, any approach to assign a semantics to logic programs has to rely on a default assumption on the missing information. The stable model semantics, that has become the dominating approach to give semantics to logic programs, relies on the Closed World Assumption (CWA), which asserts that by default the truth of an atom is false. There is a second well-known assumption, called Open World Assumption (OWA), which asserts that the truth of the atoms is supposed to be unknown by default. However, the CWA, the OWA and the combination of them are extremal, though important, assumptions over a large variety of possible assumptions on the truth of the atoms, whenever the truth is taken from an arbitrary truth space.The topic of this paper is to allow any assignment (i.e. interpretation), over a truth space, to be a default assumption. Our main result is that our extension is conservative in the sense that under the "everywhere false" default assumption (CWA) the usual stable model semantics is captured. Due to the generality and the purely algebraic nature of our approach, it abstracts from the particular formalism of choice and the results may be applied in other contexts as well.

We address the problem of defining semantics for logic programs in presence of incomplete and contradictory information coming from different sources. The information consists of facts that a central server collects and tries to combine using (a) a set of logical rules, that is, a logic program, and (b) a hypothesis representing the server's own estimates. In such a setting incomplete information from a source or contradictory information from different sources necessitate the use of manyvalued logics in which programs can be evaluated and hypotheses can be tested. To carry out such activities we propose a formal framework based on bilattices such as Belnap's four-valued logics. In this framework we work with the class of programs defined by Fitting and we propose hypothesisbased semantics for such programs. We also establish an intuitively appealing connection between our hypothesis testing mechanism, on the one hand, and the well-founded semantics and KripkeKleene semantics of Datalog programs with negation, on the other hand.

Abstract. Many frameworks have been proposed to manage uncertain information in logic programming. Essentially, they differ in the underlying notion of uncertainty and how these uncertainty values, associated to rules and facts, are managed. The goal of this paper is to allow the reasoning with non-uniform default assumptions, i.e. with any arbitrary assignment of default values to the atoms. Informally, rather than to rely on the same default certainty value for all atoms, we allow arbitrary assignments to complete information. To this end, we define both epistemologically and computationally the semantics according to any given assumption. For reasons of generality, we present our work in the framework presented in [17] as a unifying umbrella for many of the proposed approaches to the management of uncertainty in logic programming. Our extension is conservative in the following sense: (i) if we restrict our attention to the usual uniform Open World Assumption, then the semantics reduces to the Kripke-Kleene semantics, and (ii) if we restrict our attention to the uniform Closed World Assumption, then our semantics reduces to the well-founded semantics.