Abstract not found

We propose a framework which extends Antitonic Logic Programs [13] to an arbitrary complete bilattice of truth-values, where belief and doubt are explicitly represented. Inspired by Ginsberg and Fitting 's bilattice approaches, this framework allows a precise de nition of important operators found in logic programming, such as explicit and default negation. In particular, it leads to a natural semantical integration of explicit and default negation through the Coherence Principle [38], according to which explicit negation entails default negation. We then de ne Coherent Answer Sets, and the Paraconsistent Well-founded Model semantics, generalising many paraconsistent semantics for logic programs. In particular, Paraconsistent Well-Founded Semantics with eXplicit negation (WFSXp ) [3, 11]. The framework is an extension of Antitonic Logic Programs for most cases, and is general enough to capture Probabilistic Deductive Databases, Possibilistic Logic Programming, Hybrid Probabilistic Logic Programs, and Fuzzy Logic Programming. Thus, we have a powerful mathematical formalism for dealing simultaneously with default, paraconsistency, and uncertainty reasoning. Results are provided about how our semantical framework deals with inconsistent information and with its propagation by the rules of the program.

The stable model semantics has become a dominating approach for the management of negation in logic programming. It relies mainly on the closed world assumption to complete the available knowledge and its formulation has its founding root in the so-called Gelfond-Lifschitz transform. The primary goal of this work is to present an alternative and epistemic based characterization of the stable model semantics, to the Gelfond-Lifschitz transform. In particular, we show that the stable model semantics can be defined entirely as an extension of the Kripke-Kleene semantics and, thus, (i) does rely on the classical management of negation; and (ii) does not require any program transformation. Indeed, we show that the closed world assumption can be seen as an additional source for ‘falsehood ’ to be added cumulatively to the Kripke-Kleene semantics. Our approach is purely algebraic and can abstract from the particular formalism of choice as it is based on monotone operators (under the knowledge order) over bilattices only.

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.

Recently, in (Loyer & Straccia 2005) the Any-World Assumption (AWA) has been introduced for normal logic programs as a generalization of the well-known notions of Closed World Assumption (CWA) and the Open World Assumption (OWA). The AWA allows any assignment (i.e. interpretation), over a truth space (bi-lattice), to be a default assumption and, thus, the CWA and OWA are just special cases. While a declarative and a fixed-point characterization for normal logic programs under the AWA has been given in (Loyer & Straccia 2005), the topic of this pa-per is to provide a simple, yet general top-down query answering procedure for this setting.

Abstract. We define new, both model-theoretical and fixpoint-based, characterizations of the well-founded semantics for logic programs in the general setting of bilattices. This work lights the role of the CWA, used in the well-founded semantics as a carrier of falsehood, and shows that the definition of that semantics does not require any separation of positive and negative information nor any program transformation.

Abstract—In this paper, Ginsberg’s/Fitting’s theory of bilattices, and in particular the associated constructs of bilattice-based squares and triangles, is introduced as an attractive framework for the representation of uncertain and potentially conflicting information, paralleling Goguen’s-fuzzy set theory. We recall some of the advantages of bilattice-based frameworks for handling fuzzy sets and systems, provide the related structures with adequately defined graded versions of the basic logical connectives, and study their properties and relationships. Index Terms—Bilattices, bilattice-based squares and triangles, implicators, MV-algebras, negators, t-norms and t-conorms. I.

ABSTRACT. We introduce a family of preferential logics that are useful for handling information with different levels of uncertainty. The corresponding consequence relations are nonmonotonic, paraconsistent, adaptive, and rational. It is also shown that the formalisms in this family can be embedded in corresponding four-valued logics with at most three uncertainty levels, and that reasoning with these logics can be simulated by algorithms for processing circumscriptive theories, such as DLS and SCAN.

Abstract. The Any-World Assumption (AWA) has been introduced for normal logic programs as a generalization of the well-known notions of Closed World Assumption (CWA) and the Open World Assumption (OWA). The AWA allows any assignment (i.e., interpretation), over a truth space (bilattice), to be a default assumption and, thus, the CWA and OWA are just special cases. To answer queries, we provide a novel and simple top-down procedure.

When integrating data coming from multiple different sources we are faced with the possibility of inconsistency in databases. A paraconsistent approach for knowledge base integration allows keeping inconsistent information and reasoning in its presence. In this paper, we use a paraconsistent logic (LFI1) as the underlying logic for the specification of P-Datalog, a deductive query language for databases containing inconsistent information. We present a declarative semantics which captures the desired meaning of a recursive query executed over a database containing inconsistent facts and whose rules allow infering information from inconsistent premises. We also present a bottom-up evaluation method for P-Datalog programs based on an alternating fixpoint operator.