Epistemic Foundation of Stable Model Semantics

Stable model semantics has become a very popular approach for the management of negation in logic programming. This approach relies mainly on the closed world assumption to complete the available knowledge and its formulation has its basis in the so-called Gelfond-Lifschitz transformation. The primary goal of this work is to present an alternative and epistemic-based characterization of stable model semantics, to the Gelfond-Lifschitz transformation. In particular, we show that stable model semantics can be defined entirely as an extension of the Kripke-Kleene semantics. Indeed, we show that the closed world assumption can be seen as an additional source of `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.Comment: 41 pages. To appear in Theory and Practice of Logic Programming (TPLP

The logic of distributive bilattices

Bilattices, introduced by Ginsberg as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the perspective of abstract algebraic logic . We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú . We also prove some purely algebraic results concerning bilattices, for instance that the variety of distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattice

Efficient paraconsistent reasoning with rules and ontologies for the semantic web

Ontologies formalized by means of Description Logics (DLs) and rules in the form of Logic Programs (LPs) are two prominent formalisms in the field of Knowledge Representation and Reasoning. While DLs adhere to the OpenWorld Assumption and are suited for taxonomic reasoning, LPs implement reasoning under the Closed World Assumption, so that default knowledge can be expressed. However, for many applications it is useful to have a means that allows reasoning over an open domain and expressing rules with exceptions at the same time. Hybrid MKNF knowledge bases make such a means available by formalizing DLs and LPs in a common logic, the Logic of Minimal Knowledge and Negation as Failure (MKNF). Since rules and ontologies are used in open environments such as the Semantic Web, inconsistencies cannot always be avoided. This poses a problem due to the Principle of Explosion, which holds in classical logics. Paraconsistent Logics offer a solution to this issue by assigning meaningful models even to contradictory sets of formulas. Consequently, paraconsistent semantics for DLs and LPs have been investigated intensively. Our goal is to apply the paraconsistent approach to the combination of DLs and LPs in hybrid MKNF knowledge bases. In this thesis, a new six-valued semantics for hybrid MKNF knowledge bases is introduced, extending the three-valued approach by Knorr et al., which is based on the wellfounded semantics for logic programs. Additionally, a procedural way of computing paraconsistent well-founded models for hybrid MKNF knowledge bases by means of an alternating fixpoint construction is presented and it is proven that the algorithm is sound and complete w.r.t. the model-theoretic characterization of the semantics. Moreover, it is shown that the new semantics is faithful w.r.t. well-studied paraconsistent semantics for DLs and LPs, respectively, and maintains the efficiency of the approach it extends

From fuzzy to annotated semantic web languages

The aim of this chapter is to present a detailed, selfcontained and comprehensive account of the state of the art in representing and reasoning with fuzzy knowledge in Semantic Web Languages such as triple languages RDF/RDFS, conceptual languages of the OWL 2 family and rule languages. We further show how one may generalise them to so-called annotation domains, that cover also e.g. temporal and provenance extensions

Formal Frameworks for Circular Phenomena. Possibilities of Modeling Pathological Expressions in Formal and Natural Languages

This dissertation has four parts. The first one is a general introduction into the topic of the work separated in a chapter that explains the used notation, a chapter that discusses typical examples, and a chapter that gives an overview of the three main parts. An important aspect of this first part is the attempt of a conceptual clarification of circularity, in particular in relation to the non-well-foundedness of a phenomenon. This clarification represents the philosophical core of the primarily formal dissertation. In the second part, Kripke's fixed point approach concerning partially defined truth predicates is examined: the algebraic foundations are introduced and problems of the construction are discussed. The main results of this second part are three characterization theorems of subclasses of interlaced bilattices and their applications. In the third part, revision theories are introduced. Their adequacy for the representation of circularity is discussed. Additionally, the complexity of these theories, the relation of revision theories to a wider thematic context, and their empirical properties are examined. In the last part of this dissertation, circularity is introduced on the level of set theory. The crucial idea is the concept of a coalgebraic modeling. In particular, the modeling of truth and the representation of the difference between private and common knowledge is emphasized. A comparison of the different accounts is provided in the last chapter.This dissertation has four parts. The first one is a general introduction into the topic of the work separated in a chapter that explains the used notation, a chapter that discusses typical examples, and a chapter that gives an overview of the three main parts. An important aspect of this first part is the attempt of a conceptual clarification of circularity, in particular in relation to the non-well-foundedness of a phenomenon. This clarification represents the philosophical core of the primarily formal dissertation. In the second part, Kripke's fixed point approach concerning partially defined truth predicates is examined: the algebraic foundations are introduced and problems of the construction are discussed. The main results of this second part are three characterization theorems of subclasses of interlaced bilattices and their applications. In the third part, revision theories are introduced. Their adequacy for the representation of circularity is discussed. Additionally, the complexity of these theories, the relation of revision theories to a wider thematic context, and their empirical properties are examined. In the last part of this dissertation, circularity is introduced on the level of set theory. The crucial idea is the concept of a coalgebraic modeling. In particular, the modeling of truth and the representation of the difference between private and common knowledge is emphasized. A comparison of the different accounts is provided in the last chapter

Reducing fuzzy answer set programming to model finding in fuzzy logics

In recent years, answer set programming (ASP) has been extended to deal with multivalued predicates. The resulting formalisms allow for the modeling of continuous problems as elegantly as ASP allows for the modeling of discrete problems, by combining the stable model semantics underlying ASP with fuzzy logics. However, contrary to the case of classical ASP where many efficient solvers have been constructed, to date there is no efficient fuzzy ASP solver. A well-known technique for classical ASP consists of translating an ASP program P to a propositional theory whose models exactly correspond to the answer sets of P. In this paper, we show how this idea can be extended to fuzzy ASP, paving the way to implement efficient fuzzy ASP solvers that can take advantage of existing fuzzy logic reasoners