Fuzzy Description Logics Research Articles

The complexity of fuzzy [formula omitted] under the Łukasiewicz T-norm

Fuzzy Description Logics (DLs) are a family of knowledge representation formalisms designed to represent and reason about vague and imprecise knowledge that is inherent to many application domains. Previous work has shown that the complexity of reasoning in a fuzzy DL using finitely many truth degrees is usually not higher than that of the underlying classical DL. We show that this does not hold for fuzzy extensions of the light-weight DL EL, which is used in many biomedical ontologies, under the finitely valued Łukasiewicz semantics. More precisely, the complexity of reasoning increases from P to ExpTime, even if only one additional truth value is introduced. When adding complex role inclusions and inverse roles, the logic even becomes undecidable. Even more surprisingly, when considering the infinitely valued Łukasiewicz semantics, reasoning in fuzzy EL is undecidable.

Algorithms for reasoning in very expressive description logics under infinitely valued Gödel semantics

Fuzzy description logics (FDLs) are knowledge representation formalisms capable of dealing with imprecise knowledge by allowing intermediate membership degrees in the interpretation of concepts and roles. One option for dealing with these intermediate degrees is to use the so-called Gödel semantics, under which conjunction is interpreted by the minimum of the degrees of the conjuncts. Despite its apparent simplicity, developing reasoning techniques for expressive FDLs under this semantics is a hard task.In this paper, we introduce two new algorithms for reasoning in very expressive FDLs under Gödel semantics. They combine the ideas of a previous automata-based algorithm for Gödel FDLs with the known crispification and tableau approaches for FDL reasoning. The results are the two first practical algorithms capable of reasoning in infinitely valued FDLs supporting general concept inclusions.

Decidability and Complexity of Fuzzy Description Logics

Fuzzy description logics (FDLs) have been introduced to represent concepts for which membership cannot be determined in a precise way, i.e., where instead of providing a strict border between being a member and not being a member, it is more appropriate to model a gradual change from membership to non-membership. First approaches for reasoning in FDLs where based either on a reduction to reasoning in classical description logics (DLs) or on adaptations of reasoning approaches for DLs to the fuzzy case. However, it turned out that these approaches in general do not work if expressive terminological axioms, called general concept inclusions (GCIs), are available in the FDL. The goal of this project was a comprehensive study of the border between decidability and undecidability for FDLs with GCIs, as well as determining the exact complexity of the decidable logics. As a result, we have provided an almost complete classification of the decidability and complexity of FDLs with GCIs.

The Role of Crisp Elements in Fuzzy Ontologies: The Case of Fuzzy OWL 2 EL

In the last years, fuzzy ontologies and fuzzy description logics (DLs) have been applied successfully to several real-world domains. Some recent results on fuzzy DLs show an increase in the theoretical complexity, even for not expressive languages. However, it still may be possible to obtain reasoning algorithms performing well in practice. This paper studies some cases where reasoning with a fuzzy ontology can be reduced to reasoning with a tractable classical language (an OWL 2 profile). Then, building on the assumption that not all concepts and roles in fuzzy ontologies are indeed fuzzy, we show that reasoning algorithms can be optimized by taking into account the existence of possible crisp concepts and roles. We also prove the usefulness on crisp representation reasoning algorithms over an empirical evaluation over fuzzy OWL 2 EL ontologies, and discuss some preliminary techniques to learn which elements should be considered to be crisp or noncrisp.

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.

Capability-oriented architectural analysis method based on fuzzy description logic

A number of problems may arise from architectural requirements modeling, including alignment of it with business strategy, model integration and handling the uncertain and vague information. The paper introduces a method for modeling architectural requirements in a way of ontology-based and capability-oriented requirements elicitation. The requirements can be modeled within a three-layer framework. The Capability Meta-concept Framework is provided at the top level. The domain experts can capture the domain knowledge within the framework, forming the domain ontology at the second level. The domain concepts can be used for extending the UML to produce a domain-specific modeling language. A fuzzy UML is introduced to model the vague and uncertain features of the capability requirements. An algorithm is provided to transform the fuzzy UML models into the fuzzy Description Logics ontology for model verification. A case study is given to demonstrate the applicability of the method.

Mining description logic concepts from relations based on concept lattices

In description logics (DLs), a concept C could be a set of some tuples or a subset of the domain of an attribute. This paper proposed a method for extracting descriptive concepts from a relation, which is based on formal concept analysis (FCA). FCA can be used to support the descriptive concept construction in domains of attributes. Our method can be described as follows: given a relation R and some conceptual partitions on domains of some attributes, by classify-based scaling, one can obtain a new relation R, and correspondingly obtain its concept lattice in which all concept extents are regarded as concepts in a set of tuples. By using these concepts in tuples, one can further construct concepts in domains of attributes. Conversely, one can further construct concepts in tuples. The paper also defined fuzzy concepts, and discussed the connections between fuzzy concepts and those in fuzzy description logics.

On similarity in fuzzy description logics

This paper is a contribution to the study of similarity relations between objects represented as attribute-value pairs in Fuzzy Description Logics. For this purpose we use concrete domains in the fuzzy description logic IALCEF(D) associated either with a left-continuous or with a finite t-norm. We propose to expand this fuzzy description logic by adding a Similarity Box (SBox) including axioms expressing properties of fuzzy equalities. We also define a global similarity between objects from similarities between the values of each object attribute (local similarities) and we prove that the global similarity defined using a t-norm inherits the usual properties of the local similarities (reflexivity, symmetry or transitivity). We also prove a result relative to global similarities expressing that, in the context of the logic MTL∀, similar objects have similar properties, being these properties expressed by predicate formulas evaluated in these objects.

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 knowl- edge. Several fuzzy extensions of DLs have been proposed in the KR field in order to handle im- precise 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 au- tomatically 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.

Reasoning in Fuzzy Description Logics using Automata

Automata-based methods have been successfully employed to prove tight complexity bounds for reasoning in many classical logics, and in particular in Description Logics (DLs). Very recently, the ideas behind these automata-based approaches were adapted for reasoning also in fuzzy extensions of DLs, with semantics based either on finitely many truth degrees or the Gödel t-norm over the interval [0,1]. Clearly, due to the different semantics in these logics, the construction of the automata for fuzzy DLs is more involved than for the classical case. In this paper we provide an overview of the existing automata-based methods for reasoning in fuzzy DLs, with a special emphasis on explaining the ideas and the requirements behind them. The methods vary from deciding emptiness of automata on infinite trees to inclusions between automata on finite words. Overall, we provide a comprehensive perspective on the automata-based methods currently in use, and the many complexity results obtained through them.

A Fuzzy Extension to the OWL 2 RL Ontology Language

Fuzzy extensions to description logics (DLs) have gained considerable attention the last decade. So far most works on fuzzy DLs have focused on either very expressive languages, like fuzzy OWL and OWL 2, or on highly inexpressive ones, like fuzzy OWL 2 QL and fuzzy OWL 2 EL. To the best of our knowledge, a fuzzy extension to the language OWL 2 RL has not been thoroughly studied so far. This language is very relevant since it combines both adequate expressive power as well as efficient reasoning algorithms which can be realized using rule-based (Datalog) technologies. In contrast to previous fuzzy extensions, a fuzzy extension of OWL 2 RL is not a straightforward task for the following reason. The main motivation of OWL 2 RL is that its axioms can be equivalently represented as Datalog rules. Hence, to achieve our goal we need to investigate which OWL 2 RL axioms when interpreted under the fuzzy setting can be transformed to equivalent fuzzy Datalog rules. We show that this is not, in general, possible for all axioms but we show that this ‘issue’ can to a large extent be alleviated. Moreover, we have performed an experimental evaluation with many well-known ontologies which showed that such axioms are not used so often in practice.

Optimising fuzzy description logic reasoners with general concept inclusion absorption

General Concept Inclusion (GCI) absorption algorithms have shown to play an important role in classical Description Logic (DL) reasoners. They allow to transform GCIs into simpler forms to which we may apply specialised inference rules, returning important performance gains. In this work, we develop the first absorption algorithm for fuzzy DLs, implement it in the fuzzyDL reasoner and evaluate it extensively over both classical and fuzzy ontologies. The results show that our algorithm improves the performance of the reasoner significantly.

Research on Fuzzy Extension of Domain Ontology for Emergency Management of Civil Aviation

For the issue that civil aviation emergency domain ontology has not involved in the field of uncertain knowledge, the paper presents a method on fuzzy extension of domain ontology with fuzzy description logic to extend concepts and relations by means of fuzzy concrete domains and fuzzy modifiers. Moreover, it gives the representation of fuzzy ontology and analysis of fulfillment effect on a basis of fuzzy description logic, which not only improves the expression ability of existing domain ontology for emergency management of civil aviation, but offers a methodological support to improve accuracy and scientistic of emergency-rescue plan further.

On the Decidability Status of Fuzzy A ℒ C $\mathcal {A}\mathcal {L}\mathcal {C}$ with General Concept Inclusions

The combination of Fuzzy Logics and Description Logics (DLs) has been investigated for at least two decades because such fuzzy DLs can be used to formalize imprecise concepts. In particular, tableau algorithms for crisp Description Logics have been extended to reason also with their fuzzy counterparts. It has turned out, however, that in the presence of general concept inclusion axioms (GCIs) this extension is less straightforward than thought. In fact, a number of tableau algorithms claimed to deal correctly with fuzzy DLs with GCIs have recently been shown to be incorrect. In this paper, we concentrate on fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\), the fuzzy extension of the well-known DL \(\mathcal {A}\mathcal {L}\mathcal {C}\). We present a terminating, sound, and complete tableau algorithm for fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) with arbitrary continuous t-norms. Unfortunately, in the presence of GCIs, this algorithm does not yield a decision procedure for consistency of fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) ontologies since it uses as a sub-procedure a solvability test for a finitely represented, but possibly infinite, system of inequations over the real interval [0,1], which are built using the t-norm. In general, it is not clear whether this solvability problem is decidable for such infinite systems of inequations. This may depend on the specific t-norm used. In fact, we also show in this paper that consistency of fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) ontologies with GCIs is undecidable for the product t-norm. This implies, of course, that for the infinite systems of inequations produced by the tableau algorithm for fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) with product t-norm, solvability is in general undecidable. We also give a brief overview of recently obtained (un)decidability results for fuzzy \(\mathcal {A}\mathcal {L}\mathcal {C}\) w.r.t. other t-norms.

The limits of decidability in fuzzy description logics with general concept inclusions

Fuzzy description logics (DLs) can be used to represent and reason with vague knowledge. This family of logical formalisms is very diverse, each member being characterized by a specific choice of constructors, axioms, and triangular norms, which are used to specify the semantics. Unfortunately, it has recently been shown that the consistency problem in many fuzzy DLs with general concept inclusion axioms is undecidable. In this paper, we present a proof framework that allows us to extend these results to cover large classes of fuzzy DLs. On the other hand, we also provide matching decidability results for most of the remaining logics. As a result, we obtain a near-universal classification of fuzzy DLs according to the decidability of their consistency problem.

A fuzzy description logic F-SHIQ(G)

It is widely recognized that management of data types is an indispensable component in intelligent knowledge-based applications. Recent research has shown that OWL (Web Ontology Language) has a serious limitation on data types and does not support customized data types and predicates. Some work has been carried out to address this issue in crisp situations. In order to deal with fuzzy information in the Semantic Web, this paper proposes a new kind of fuzzy description logic F-SHIQ(G), which can support not only the representation and reasoning of fuzzy concept knowledge, but also fuzzy data information with customized fuzzy data types and customized fuzzy data type predicates. The syntax, semantics and reasoning algorithm of the F-SHIQ(G) are investigated in the paper, and its decidability, soundness, completeness and complexity of the reasoning algorithm are discussed in the paper also. For the reasoning for F-SHIQ(G), the paper adopts a traditional reasoning architecture for fuzzy data type reasoning and as a result, new fuzzy data type reasoners can be added without changing the fuzzy concept reasoner.

Representation and reasoning of fuzzy ER models with description logic DLR

Description logics have been studied in depth in data modeling. This paper proposes a description logic approach for representing and reasoning on fuzzy ER models. Firstly, an analysis about how to choose suitable description logic as a modeling language for fuzzy ER models is given, which motivates the use of the fuzzy description logic called FDLR the fuzzy extension of DLR for the purpose of this paper. Then, a formal definition of fuzzy ER models is proposed. Furthermore, based on the FDLR and the formalization of fuzzy ER models, we propose a complete formal approach for representing fuzzy ER models with FDLR, i.e., transforming fuzzy ER models into FDLR knowledge bases at both terminological schema and assertional instance levels. On this basis, following the proposed approach, we implemented a prototype tool FER2FDLR. Finally, based on the transformed FDLR knowledge bases, we investigate how to reason on fuzzy ER models with FDLR, and the reasoning tasks of fuzzy ER models can be reasoned through the reasoning mechanism of FDLR.

On finitely-valued Fuzzy Description Logics

This paper deals with finitely-valued fuzzy description languages from a logical point of view. From recent results in Mathematical Fuzzy Logic and following [44], we develop a Fuzzy Description Logic based on the fuzzy logic of a finite BL-chain. The constructors of the languages presented in this paper correspond to the connectives of that logic (containing an involutive negation, Monteiro–Baaz delta and hedges). The paper addresses the hierarchy of fuzzy attributive languages; knowledge bases and their reductions; reasoning tasks; and complexity. Our results regarding decidability together with a summary of the known results related to computational complexity are of particular interest. In Appendix B we also provide axiomatizations for expansions of the logic of a finite BL-chain considered in the paper.

Consistency reasoning in lattice-based fuzzy Description Logics

Fuzzy Description Logics have been widely studied as a formalism for representing and reasoning with vague knowledge. One of the most basic reasoning tasks in (fuzzy) Description Logics is to decide whether an ontology representing a knowledge domain is consistent. Surprisingly, not much is known about the complexity of this problem for semantics based on complete De Morgan lattices. To cover this gap, in this paper we study the consistency problem for the fuzzy Description Logic L-SHI and its sublogics in detail.The contribution of the paper is twofold. On the one hand, we provide a tableaux-based algorithm for deciding consistency when the underlying lattice is finite. The algorithm generalizes the one developed for classical SHI. On the other hand, we identify decidable and undecidable classes of fuzzy Description Logics over infinite lattices. For all the decidable classes, we also provide tight complexity bounds.

基于隶属度比较的描述逻辑SHOIQ模糊扩展

The current fuzzy extensions for description logics do not support representing and reasoning with complex fuzzy knowledge about comparisons between degrees of membership, which is very useful in practical applications. We investigate a novel approach of extending the description logic SHOIQ to SHOIQFC by annotating concepts and roles based on comparison expressions. SHOIQFC provides the ability to represent complex fuzzy knowledge, in addition to the expressive power of the fuzzy description logic FSHOIQ. The main contributions are a decidability proof for the consistency problem of SHOIQFC knowledge bases, as well as its tableau-based reasoning algorithm.