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

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

The Complexity of Fuzzy Description Logics over Finite Lattices with Nominals

The complexity of reasoning in fuzzy description logics (DLs) over finite lattices usually does not exceed that of the underlying classical DLs. This has recently been shown for the logics between L-IALC and L-ISCHI using a combination of automata- and tableau-based techniques. In this report, this approach is modified to deal with nominals and constants in L-ISCHOI. Reasoning w.r.t. general TBoxes is ExpTime-complete, and PSpace-completeness is shown under the restriction to acyclic terminologies in two sublogics. The latter implies two previously unknown complexity results for the classical DLs ALCHO and SO

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

Parameterizing the semantics of fuzzy attribute implications by systems of isotone Galois connections

We study the semantics of fuzzy if-then rules called fuzzy attribute implications parameterized by systems of isotone Galois connections. The rules express dependencies between fuzzy attributes in object-attribute incidence data. The proposed parameterizations are general and include as special cases the parameterizations by linguistic hedges used in earlier approaches. We formalize the general parameterizations, propose bivalent and graded notions of semantic entailment of fuzzy attribute implications, show their characterization in terms of least models and complete axiomatization, and provide characterization of bases of fuzzy attribute implications derived from data

Proceedings of the 5th International Workshop "What can FCA do for Artificial Intelligence?", FCA4AI 2016(co-located with ECAI 2016, The Hague, Netherlands, August 30th 2016)

International audienceThese are the proceedings of the fifth edition of the FCA4AI workshop (http://www.fca4ai.hse.ru/). Formal Concept Analysis (FCA) is a mathematically well-founded theory aimed at data analysis and classification that can be used for many purposes, especially for Artificial Intelligence (AI) needs. The objective of the FCA4AI workshop is to investigate two main main issues: how can FCA support various AI activities (knowledge discovery, knowledge representation and reasoning, learning, data mining, NLP, information retrieval), and how can FCA be extended in order to help AI researchers to solve new and complex problems in their domain. Accordingly, topics of interest are related to the following: (i) Extensions of FCA for AI: pattern structures, projections, abstractions. (ii) Knowledge discovery based on FCA: classification, data mining, pattern mining, functional dependencies, biclustering, stability, visualization. (iii) Knowledge processing based on concept lattices: modeling, representation, reasoning. (iv) Application domains: natural language processing, information retrieval, recommendation, mining of web of data and of social networks, etc

Foundations of Fuzzy Logic and Semantic Web Languages

This book is the first to combine coverage of fuzzy logic and Semantic Web languages. It provides in-depth insight into fuzzy Semantic Web languages for non-fuzzy set theory and fuzzy logic experts. It also helps researchers of non-Semantic Web languages get a better understanding of the theoretical fundamentals of Semantic Web languages. The first part of the book covers all the theoretical and logical aspects of classical (two-valued) Semantic Web languages. The second part explains how to generalize these languages to cope with fuzzy set theory and fuzzy logic

Foundations of Fuzzy Logic and Semantic Web Languages

This book is the first to combine coverage of fuzzy logic and Semantic Web languages. It provides in-depth insight into fuzzy Semantic Web languages for non-fuzzy set theory and fuzzy logic experts. It also helps researchers of non-Semantic Web languages get a better understanding of the theoretical fundamentals of Semantic Web languages. The first part of the book covers all the theoretical and logical aspects of classical (two-valued) Semantic Web languages. The second part explains how to generalize these languages to cope with fuzzy set theory and fuzzy logic