Answering Fuzzy Conjunctive Queries over Finitely Valued Fuzzy Ontologies

Fuzzy Description Logics (DLs) provide a means for representing vague knowledge about an application domain. In this paper, we study fuzzy extensions of conjunctive queries (CQs) over the DL $\mathcal{SROIQ}$ based on finite chains of degrees of truth. To answer such queries, we extend a well-known technique that reduces the fuzzy ontology to a classical one, and use classical DL reasoners as a black box. We improve the complexity of previous reduction techniques for finitely valued fuzzy DLs, which allows us to prove tight complexity results for answering certain kinds of fuzzy CQs. We conclude with an experimental evaluation of a prototype implementation, showing the feasibility of our approach.Comment: submitted to the Journal on Data Semantics, v1: 19 pages, v2: 20 pages, improved evaluation sectio

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

Infinitely Valued Gödel Semantics for Expressive Description Logics

Fuzzy Description Logics (FDLs) combine classical Description Logics with the semantics of Fuzzy Logics in order to represent and reason with vague knowledge. Most FDLs using truth values from the interval [0; 1] have been shown to be undecidable in the presence of a negation constructor and general concept inclusions. One exception are those FDLs whose semantics is based on the infinitely valued Gödel t-norm (G). We extend previous decidability results for the FDL G-ALC to deal with complex role inclusions, nominals, inverse roles, and qualified number restrictions. Our novel approach is based on a combination of the known crispification technique for finitely valued FDLs and an automata-based procedure for reasoning in G-ALC

Bisimulation and bisimilarity for fuzzy description logics under the G\"odel semantics

Description logics (DLs) are a suitable formalism for representing knowledge about domains in which objects are described not only by attributes but also by binary relations between objects. Fuzzy extensions of DLs can be used for such domains when data and knowledge about them are vague and imprecise. One of the possible ways to specify classes of objects in such domains is to use concepts in fuzzy DLs. As DLs are variants of modal logics, indiscernibility in DLs is characterized by bisimilarity. The bisimilarity relation of an interpretation is the largest auto-bisimulation of that interpretation. In DLs and their fuzzy extensions, such equivalence relations can be used for concept learning. In this paper, we define and study fuzzy bisimulation and bisimilarity for fuzzy DLs under the G\"odel semantics, as well as crisp bisimulation and strong bisimilarity for such logics extended with involutive negation. The considered logics are fuzzy extensions of the DL $\mathcal{ALC}_{reg}$ (a variant of PDL) with additional features among inverse roles, nominals, (qualified or unqualified) number restrictions, the universal role, local reflexivity of a role and involutive negation. We formulate and prove results on invariance of concepts under fuzzy (resp. crisp) bisimulation, conditional invariance of fuzzy TBoxex/ABoxes under bisimilarity (resp. strong bisimilarity), and the Hennessy-Milner property of fuzzy (resp. crisp) bisimulation for fuzzy DLs without (resp. with) involutive negation under the G\"odel semantics. Apart from these fundamental results, we also provide results on using fuzzy bisimulation to separate the expressive powers of fuzzy DLs, as well as results on using strong bisimilarity to minimize fuzzy interpretations.Comment: This is a revised and corrected version of the publication "Bisimulation and bisimilarity for fuzzy description logics under the G\"odel semantics", Fuzzy Sets and Systems 388: 146-178 (2020

Context-Aware Modeling Using Semantic Web and Z Notation

Surveys in user context modeling have shown that the semantic web is one of the promising approach to represent and structure the contextual information captured from user’s surrounding environment in a context-aware application. A benefit of using semantic web language is that it enables application to reason user contextual information in order to get the knowledge of user’s behavior. However, regarding its notation format, semantic web is suitable for implementation level or to be consumed by application run-time. Context-aware application is a part of distributed computing system. In distributed computing system, the language used for specification should be distinguished from the implementation / run-time purpose. This is known as separation of modeling language. Regarding the context-aware application, for those who are concerned with specification of context modeling, the language that is used for specification should also be distinguished from the implementation one. This thesis aims at proposing the use of formal specification technique to develop a generic context ontology model of user’s behavior at the Computer and Information Sciences Department, Universiti Teknologi PETRONAS. Initially, the context ontology was written in OWL semantic web language. The further process is mapping onto a formal specification language, i.e. onto Z notation. As a result, specification of context ontology and its consistency checking have been developed and verified beyond the semantic web language environment. An inconsistency of context model has been detected during the verification of Z model, which cannot be revealed by current OWL DL reasoner. The context-aware designers might benefit from the formal specification of context ontology, where the designers could fully use formal verification technique to check the correctness of context ontology. Thus, the modeling approach in this thesis has shown that it could complement the context ontology development process, where the checking and refinement are performed beyond the semantic web reasone

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

Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis

Description Logic (abbrv. DL) belongs to the field of knowledge representation and reasoning. DL researchers have developed a large family of logic-based languages, so-called description logics (abbrv. DLs). These logics allow their users to explicitly represent knowledge as ontologies, which are finite sets of (human- and machine-readable) axioms, and provide them with automated inference services to derive implicit knowledge. The landscape of decidability and computational complexity of common reasoning tasks for various description logics has been explored in large parts: there is always a trade-off between expressibility and reasoning costs. It is therefore not surprising that DLs are nowadays applied in a large variety of domains: agriculture, astronomy, biology, defense, education, energy management, geography, geoscience, medicine, oceanography, and oil and gas. Furthermore, the most notable success of DLs is that these constitute the logical underpinning of the Web Ontology Language (abbrv. OWL) in the Semantic Web. Formal Concept Analysis (abbrv. FCA) is a subfield of lattice theory that allows to analyze data-sets that can be represented as formal contexts. Put simply, such a formal context binds a set of objects to a set of attributes by specifying which objects have which attributes. There are two major techniques that can be applied in various ways for purposes of conceptual clustering, data mining, machine learning, knowledge management, knowledge visualization, etc. On the one hand, it is possible to describe the hierarchical structure of such a data-set in form of a formal concept lattice. On the other hand, the theory of implications (dependencies between attributes) valid in a given formal context can be axiomatized in a sound and complete manner by the so-called canonical base, which furthermore contains a minimal number of implications w.r.t. the properties of soundness and completeness. In spite of the different notions used in FCA and in DLs, there has been a very fruitful interaction between these two research areas. My thesis continues this line of research and, more specifically, I will describe how methods from FCA can be used to support the automatic construction and extension of DL ontologies from data

Methodology for integration of fisher's ecological knowledge in fisheries biology and management using knowledge representation (artificial intelligence)

Presentado na International Conference "Putting Fisher's Knowledge to Work", Vancouver, Canadá, 27-30 agosto de 2001[abstract] The fisheries crisis of the last decades and the overexploitation of a great number of stocks (FAO 1995) have been due mainly to the inadequacy of scientific knowledge, uncertainties in assessments and/or failures of the management systems. These problems are critical when the management of coastal ecosystems and artisanal fisheries is involved. These systems possess great complexity due to the high number of human factors that influence their functioning and the fishing activity. Small-scale coastal fisheries have a much greater social significance than offshore industrial fisheries, despite the larger economical importance of the latter (only in macro-economic terms). The artisanal coastal fisheries in Galicia (NW Spain) are in a general state of overexploitation derived from the mismatch between management (derived implicitly from models designed for industrial finfisheries) and the biological and socioeconomic context. Freire and García-Allut (2000) proposed a new management policy (based on the establishment of territorial users’ rights, the involvement of fishers in the assessment and management process in collaboration with the government agencies, and the use of protected areas and minimum landing sizes as key regulations) to solve the above problems. As well as a new management system, research should pay special attention to the design and use of inexpensive and rapid methodologies to get relevant scientific data, and introduce local or traditional ecological knowledge of the fishers to the assessment and management process. In this paper, we analyze the values and characteristics of fishers’ ecological knowledge (FEK). Using the artisanal coastal fisheries of Galicia as a case study, we present the objectives of the integration of FEK in fisheries biology and management and propose a methodology for that goal. The use of Artificial Intelligence (AI) as a tool for the analysis and integration of FEK is discussed, and the role of Knowledge Representation, a branch of AI, is described to show the epistemological and technological adequacy of the chosen languages and tools in a non-computer science foru

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