Description logic programs under probabilistic uncertainty and fuzzy vagueness

Thomas Lukasiewicz, Umberto Straccia
2009 International Journal of Approximate Reasoning  
This paper is directed towards an infrastructure for handling both uncertainty and vagueness in the Rules, Logic, and Proof layers of the Semantic Web. More concretely, we present probabilistic fuzzy description logic programs, which combine fuzzy description logics, fuzzy logic programs (with stratified default-negation), and probabilistic uncertainty in a uniform framework for the Semantic Web. We define important concepts dealing with both probabilistic uncertainty and fuzzy vagueness, such
more » ... s the expected truth value of a crisp sentence and the probability of a vague sentence. Furthermore, we describe a shopping agent example, which gives evidence of the usefulness of probabilistic fuzzy description logic programs in realistic Web applications. We also provide algorithms for query processing in probabilistic fuzzy description logic programs, and we delineate a special case where query processing can be done in polynomial time in the data complexity. Ó 2009 Elsevier Inc. All rights reserved. Introduction The Semantic Web [1, 8] aims at an extension of the current World Wide Web by standards and technologies that help machines to understand the information on the Web so that they can support richer discovery, data integration, navigation, and automation of tasks. The main ideas behind it are to add a machine-readable meaning to Web pages, to use ontologies for a precise definition of shared terms in Web resources, to use KR technology for automated reasoning from Web resources, and to apply cooperative agent technology for processing the information of the Web. The Semantic Web consists of several hierarchical layers, where the Ontology layer, in form of the OWL Web Ontology Language [31], is currently the highest layer of sufficient maturity. OWL consists of three increasingly expressive sublanguages, namely, OWL Lite, OWL DL, and OWL Full. OWL Lite and OWL DL are essentially very expressive description logics with an RDF syntax. As shown in [13] , ontology entailment in OWL Lite (resp., OWL DL) reduces to knowledge base (un)satisfiability in the description logic SHIF ðDÞ (resp., SHOIN ðDÞ). On top of the Ontology layer, sophisticated representation and reasoning capabilities for the Rules, Logic, and Proof layers of the Semantic Web are developed next. In particular, a key requirement of the layered architecture of the Semantic Web is to integrate the Rules and the Ontology layer. Here, it is crucial to allow for building rules on top of ontologies, that is, for rule-based systems that use vocabulary from ontology knowledge bases. Another type of combination is to build ontologies on top of rules, where ontological 0888-613X/$ -see front matter Ó j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j a r definitions are supplemented by rules or imported from rules. Both types of integration have been realized in recent hybrid integrations of rules and ontologies under the loose coupling, called (loosely coupled) description logic programs (or simply dlprograms), which are of the form KB ¼ ðL; PÞ, where L is a description logic knowledge base, and P is a finite set of rules involving queries to L (see especially [6] and the references therein). Other research efforts are directed towards handling uncertainty and vagueness in the Semantic Web, which are motivated by important Web and Semantic Web applications. In particular, formalisms for handling uncertainty are used in data integration, ontology mapping, and information retrieval, while dealing with vagueness is motivated by multimedia information processing/retrieval and natural language interfaces to the Web. There are several extensions of description logics and Web ontology languages by probabilistic uncertainty and by fuzzy vagueness. Similarly, there are also extensions of hybrid integrations of rules and ontologies by probabilistic uncertainty and by fuzzy vagueness. Clearly, since uncertainty and vagueness are semantically quite different, it is important to have a unifying formalism for the Semantic Web, which allows for dealing with both uncertainty and vagueness. But although there has been some important work in the fuzzy logic community in this direction [9] , to date there are no Semantic Web formalisms that allow for handling both uncertainty and vagueness. In this paper, we try to fill this gap by presenting a novel approach to dl-programs, where probabilistic rules are defined on top of fuzzy rules, which are in turn defined on top of fuzzy description logics. This allows for handling both probabilistic uncertainty and fuzzy vagueness. The main contributions can be briefly summarized as follows:
doi:10.1016/j.ijar.2009.03.004 fatcat:yycaw6pcjvcz7j6xtqrv54fbqm