Undecidability of Fuzzy Description Logics

Fuzzy description logics (DLs) have been investigated for over two decades, due to their capacity to formalize and reason with imprecise concepts. Very recently, it has been shown that for several fuzzy DLs, reasoning becomes undecidable. Although the proofs of these results differ in the details of each specific logic considered, they are all based on the same basic idea. In this report, we formalize this idea and provide sufficient conditions for proving undecidability of a fuzzy DL. We demonstrate the effectiveness of our approach by strengthening all previously-known undecidability results and providing new ones. In particular, we show that undecidability may arise even if only crisp axioms are considered

Fuzzy Description Logics with General Concept Inclusions

Description logics (DLs) are used to represent knowledge of an application domain and provide standard reasoning services to infer consequences of this knowledge. However, classical DLs are not suited to represent vagueness in the description of the knowledge. We consider a combination of DLs and Fuzzy Logics to address this task. In particular, we consider the t-norm-based semantics for fuzzy DLs introduced by Hájek in 2005. Since then, many tableau algorithms have been developed for reasoning in fuzzy DLs. Another popular approach is to reduce fuzzy ontologies to classical ones and use existing highly optimized classical reasoners to deal with them. However, a systematic study of the computational complexity of the different reasoning problems is so far missing from the literature on fuzzy DLs. Recently, some of the developed tableau algorithms have been shown to be incorrect in the presence of general concept inclusion axioms (GCIs). In some fuzzy DLs, reasoning with GCIs has even turned out to be undecidable. This work provides a rigorous analysis of the boundary between decidable and undecidable reasoning problems in t-norm-based fuzzy DLs, in particular for GCIs. Existing undecidability proofs are extended to cover large classes of fuzzy DLs, and decidability is shown for most of the remaining logics considered here. Additionally, the computational complexity of reasoning in fuzzy DLs with semantics based on finite lattices is analyzed. For most decidability results, tight complexity bounds can be derived

Consistency in Fuzzy Description Logics over Residuated De Morgan Lattices

Fuzzy description logics can be used to model vague knowledge in application domains. This paper analyses the consistency and satisfiability problems in the description logic SHI with semantics based on a complete residuated De Morgan lattice. The problems are undecidable in the general case, but can be decided by a tableau algorithm when restricted to finite lattices. For some sublogics of SHI, we provide upper complexity bounds that match the complexity of crisp reasoning

Towards a Tableau Algorithm for Fuzzy ALC with Product T-norm

Very recently, the tableau-based algorithm for deciding consistency of general fuzzy DL ontologies over the product t-norm was shown to be incorrect, due to a very weak blocking condition. In this report we take the first steps towards a correct algorithm by modifying the blocking condition, such that the (finite) structure obtained through the algorithm uniquely describes an infinite system of quadratic constraints. We show that this procedure terminates, and is sound and complete in the sense that the input is consistent iff the corresponding infinite system of constraints is satisfiable

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

Learning in Description Logics with Fuzzy Concrete Domains

Description Logics (DLs) are a family of logic-based Knowledge Representation (KR) formalisms, which are particularly suitable for representing incomplete yet precise structured knowledge. Several fuzzy extensions of DLs have been proposed in the KR field in order to handle imprecise knowledge which is particularly pervading in those domains where entities could be better described in natural language. Among the many approaches to fuzzification in DLs, a simple yet interesting one involves the use of fuzzy concrete domains. In this paper, we present a method for learning within the KR framework of fuzzy DLs. The method induces fuzzy DL inclusion axioms from any crisp DL knowledge base. Notably, the induced axioms may contain fuzzy concepts automatically generated from numerical concrete domains during the learning process. We discuss the results obtained on a popular learning problem in comparison with state-of-the-art DL learning algorithms, and on a test bed in order to evaluate the classification performance

I can get some satisfaction: Fuzzy ontologies for partial agreements in blockchain smart contracts

This paper proposes a novel extension of blockchain systems with fuzzy ontologies. The main advantage is to let the users have flexible restrictions, represented using fuzzy sets, and to develop smart contracts where there is a partial agreement among the involved parts. We propose a general architecture based on four fuzzy ontologies and a process to develop and run the smart contracts, based on a reduction to a well-known fuzzy ontology reasoning task (Best Satisfiability Degree). We also investigate different operators to compute Pareto-optimal solutions and implement our approach in the Ethereum blockchain

Subsumption in Finitely Valued Fuzzy EL

Aus der Einleitung: Description Logics (DLs) are a family of knowledge representation formalisms that are successfully applied in many application domains. They provide the logical foundation for the Direct Semantics of the standard web ontology language OWL2. The light-weight DL EL, underlying the OWL2 EL profile, is of particular interest since all common reasoning problems are polynomial in this logic, and it is used in many prominent biomedical ontologies like SNOMEDCT and the Gene Ontology