Fuzzy Description Logics Research Articles

A fuzzy description logic based IoT framework: Formal verification and end user programming.

The Internet of Things (IoT) has become one of the most popular technologies in recent years. Advances in computing capabilities, hardware accessibility, and wireless connectivity make possible communication between people, processes, and devices for all kinds of applications and industries. However, the deployment of this technology is confined almost entirely to tech companies, leaving end users with only access to specific functionalities. This paper presents a framework that allows users with no technical knowledge to build their own IoT applications according to their needs. To this end, a framework consisting of two building blocks is presented. A friendly interface block lets users tell the system what to do using simple operating rules such as "if the temperature is cold, turn on the heater." On the other hand, a fuzzy logic reasoner block built by experts translates the ambiguity of human language to specific actions to the actuators, such as "call the police." The proposed system can also detect and inform the user if the inserted rules have inconsistencies in real time. Moreover, a formal model is introduced, based on fuzzy description logic, for the consistency of IoT systems. Finally, this paper presents various experiments using a fuzzy logic reasoner to show the viability of the proposed framework using a smart-home IoT security system as an example.

Minimizing fuzzy interpretations in fuzzy description logics by using crisp bisimulations

The problem of minimizing finite fuzzy interpretations in fuzzy description logics (FDLs) is worth studying. For example, the structure of a fuzzy/weighted social network can be treated as a fuzzy interpretation in FDLs, where actors are individuals and actions are roles. Minimizing the structure of a fuzzy/weighted social network makes it more compact, thus making network analysis tasks more efficient. In this work, we study the problem of minimizing a finite fuzzy interpretation in a FDL by using the largest crisp auto-bisimulation. The considered FDLs use the Baaz projection operator and their semantics is specified using an abstract algebra of fuzzy truth values, which can be any linear and complete residuated lattice. We provide an efficient algorithm with a complexity of O((mlog⁡l+n)log⁡n) for minimizing a given finite fuzzy interpretation I, where n is the size of the domain of I, m is number of nonzero instances of atomic roles of I and l is the number of different fuzzy values used for instances of atomic roles of I. We prove that the fuzzy interpretation returned by the algorithm is minimal among the ones that preserve fuzzy TBoxes and ABoxes under certain conditions.

Fuzzy ontology-based approach for liver fibrosis diagnosis

The domain of the digestive system is prone to severe chronic disease in the form of liver cirrhosis, which is currently a leading cause of mortality. This article presents a new intelligent system for predicting the severity of liver fibrosis in patients with chronic viral hepatitis C. The proposed system is based on the inference capabilities of fuzzy ontology and operates on semantic rule-based techniques. A fuzzy decision tree technique was employed to generate the ontology rule base using a dataset of real fibrosis cases from the Mansoura University Hospital, Egypt. These rules were then encoded into a set of fuzzy semantic rules using the fuzzy description logic format. To evaluate the system’s effectiveness, the proposed ontology was then tested on 47 chronic HCV cases, with an attempt made to see if this correctly diagnosed the patients’ conditions. The performance of the proposed system was compared with that of the now-standard Mamdani fuzzy inference system; while the latter achieved an accuracy of 95.7/%, the proposed fuzzy ontology-based system demonstrated higher performance, with 97.8% accuracy. Furthermore, the proposed system also supports semantic interoperability between clinical decision support systems and electronic health record ecosystems. The positive impacts of this system on the correct prediction of liver fibrosis severity thus suggest that it has the potential to assist medical professionals in diagnosing and treating this dangerous disease.

Logical Characterizations of Crisp Bisimulations in Fuzzy Description Logics

Logical Characterizations of Crisp Bisimulations in Fuzzy Description Logics

Fuzzy Networks for Modeling Shared Semantic Knowledge

Abstract Shared conceptualization, in the sense we take it here, is as recent a notion as the Semantic Web, but its relevance for a large variety of fields requires efficient methods of extraction and representation for both quantitative and qualitative data. This notion is particularly relevant for the investigation into, and construction of, semantic structures such as knowledge bases and taxonomies, but given the required large, often inaccurate, corpora available for search we can get only approximations. We see fuzzy description logic as an adequate medium for the representation of human semantic knowledge and propose a means to couple it with fuzzy semantic networks via the propositional Łukasiewicz fuzzy logic such that these suffice for decidability for queries over a semantic-knowledge base such as “to what degree of sharedness does it entail the instantiation C(a) for some concept C” or “what are the roles R that connect the individuals a and b to degree of sharedness ε.”

Computing the fuzzy partition corresponding to the greatest fuzzy auto-bisimulation of a fuzzy graph-based structure under the Gödel semantics

The similarity measure based on fuzzy bisimulation has the Hennessy-Milner property as a strong logical foundation. It is useful for classification and clustering. In this work, we design an efficient algorithm with the complexity O((mlog⁡l+n)log⁡n) for computing the fuzzy partition corresponding to the greatest fuzzy auto-bisimulation of a finite fuzzy labeled graph G under the Gödel semantics, where n, m and l are the number of vertices, the number of non-zero edges and the number of different fuzzy degrees of edges of G, respectively. Our notion of fuzzy partition is novel, defined for finite sets with respect to the Gödel t-norm, with the aim to facilitate the computation of the greatest fuzzy auto-bisimulation. By using that algorithm, we also provide an algorithm with the complexity O(m⋅log⁡l⋅log⁡n+n2) for computing the greatest fuzzy bisimulation between two finite fuzzy labeled graphs under the Gödel semantics. This latter algorithm is better (has a lower complexity order) than the previously known algorithms for the considered problem. Our algorithms can be restated for other fuzzy graph-based structures such as fuzzy automata, fuzzy labeled transition systems, fuzzy Kripke models, fuzzy social networks and fuzzy interpretations in fuzzy description logics.

A conditional, a fuzzy and a probabilistic interpretation of self-organizing maps

Abstract In this paper we establish a link between fuzzy and preferential semantics for description logics and self-organizing maps (SOMs), which have been proposed as possible candidates to explain the psychological mechanisms underlying category generalization. In particular, we show that the input/output behavior of a SOM after training can be described by a fuzzy description logic interpretation as well as by a preferential interpretation, based on a concept-wise multipreference semantics, which takes into account preferences with respect to different concepts and has been recently proposed for ranked and for weighted defeasible description logics. Properties of the network can be proven by model checking on the fuzzy or on the preferential interpretation. Starting from the fuzzy interpretation, we also provide a probabilistic account for this neural network model.

Computing Fuzzy Bisimulations for Fuzzy Structures Under the Gödel Semantics

Bisimulation is a well-known notion in modal logic and the theory of labeled transition systems. It is used for characterizing indiscernibility between states and has important applications in minimizing structures, separating expressive powers of modal and related logics, as well as concept learning in description logics (DLs). Fuzzy bisimulation is a counterpart of bisimulation for dealing with fuzzy structures. In this article, we present an efficient algorithm with a complexity $O((m+n)n)$ for computing the greatest fuzzy bisimulation between two finite fuzzy interpretations in the fuzzy DL $\mathit {f}\!\mathcal {ALC}$ under the Godel semantics, where $n$ is the number of individuals and $m$ is the number of nonzero instances of roles in the given fuzzy interpretations. We also adapt our algorithm for computing fuzzy bisimulations and simulations between fuzzy finite automata, as well as for dealing with other fuzzy DLs. The resulting algorithms are much more efficient than the previously known ones, as they reduce the complexity from $O(n^5)$ to $O((m+n)n)$ .

Algorithms for Instance Retrieval and Realization in Fuzzy Ontologies

Fuzzy description logics, the formalism behind fuzzy ontologies, are an important mathematical method with applications in many artificial intelligence scenarios. This paper proposes the first specific algorithms to solve two reasoning tasks with respect to a fuzzy ontology: the instance retrieval and the realization problem. Our algorithms are based on a reduction of the number of optimization problems to solve by merging some of them. Our experimental evaluation shows that the novel algorithm to solve the instance retrieval outperforms the previous algorithm, and that in practice it is common to be able to solve a single optimization problem.

Specifying a New Requirement Model for Secure Adaptive Systems

AbstractSecurity is a growing concern in developing software systems. It is important to face unknown threats in order to make the system continue operating properly. Threats are vague and attack methods change frequently. Coping with such changes is a major feature of an adaptive software. Therefore, designing an adaptive secure software is an appropriate solution to address software security challenges. Through estimation of maximum amount of system assets security, one can determine whether the system is protecting the assets or not; if not, reconfiguration can be employed. This paper proposes a new requirement model for secure adaptive systems using fuzzy, goal modeling and Description Logic concepts. The model contains three phases of modeling security aspects of the system, identifying formalizations and relations between the requirements and monitoring and adapting, when needed. To illustrate the relations between the requirements, goal modeling is used in the first phase and fuzzy Description Logic in the second phase. For the third phase, four algorithms are proposed to monitor and determine whether reconfiguration is needed or not. Theorems are given to prove concept satisfaction of the requirements. Furthermore, examples and case studies are discussed to evaluate and show applicability of the proposed model.

Minimizing interpretations in fuzzy description logics under the Gödel semantics by using fuzzy bisimulations

Minimizing interpretations in fuzzy description logics under the Gödel semantics by using fuzzy bisimulations

Bisimulation and bisimilarity for fuzzy description logics under the Gödel 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 Gödel 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 ALCreg (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 TBoxes/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 Gödel 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.

Bisimilarity in Fuzzy Description Logics Under the Zadeh Semantics

Fuzzy description logics (DLs) are extensions of DLs for dealing with imprecise and vague concepts. They found the logical basis for fuzzy ontologies, which are useful for practical applications. Bisimilarity is a natural notion of equivalence between individuals in DLs. In this paper, for the first time, we introduce the notion of bisimilarity in fuzzy DLs under the Zadeh semantics. It is defined using our notion of $p$ -cut simulation between fuzzy interpretations. The considered logics are fuzzy DLs that extend the fuzzy version of the DL $\mathcal {ALC}_{\bf reg}$ (a variant of propositional dynamic logic) with features among inverse roles, the universal role, qualified number restrictions, nominals, and local reflexivity of a role. We provide results on preservation of information by the mentioned simulations, conditional invariance of ABoxes and TBoxes by bisimilarity between witnessed interpretations, as well as the Hennessy–Milner property for fuzzy DLs under the Zadeh semantics.

Fuzzy spatio‐temporal ontologies and formal construction based on fuzzy Petri nets

AbstractFuzzy spatio‐temporal knowledge is required in a wide range of application fields, including GIS, spatio‐temporal database, and artificial intelligence. Ontology, as a formal method of knowledge representation, plays a key role in the Semantic Web. Therefore, how to extend ontologies to represent fuzzy spatio‐temporal knowledge needs to be solved. In this paper, we propose an approach for representing fuzzy spatio‐temporal knowledge with ontologies and investigate a formal construction of fuzzy spatio‐temporal ontologies based on fuzzy Petri nets. First, we propose a formal definition of fuzzy spatio‐temporal ontologies. In addition, based on fuzzy description logic, a description logic named the fuzzy spatio‐temporal description logic, which provides a logical basis for the fuzzy spatio‐temporal ontologies, is presented. Furthermore, a fuzzy spatio‐temporal web ontology language, which is fuzzy and spatio‐temporal extension of standard ontology language OWL, is proposed to make the fuzzy spatio‐temporal ontologies easy to use and support efficient reasoning. On the basis of fuzzy spatio‐temporal ontologies, we investigate a formal approach for constructing fuzzy spatio‐temporal ontologies from fuzzy spatio‐temporal Petri nets, ie, transforming fuzzy spatio‐temporal Petri nets (including Petri net model and Petri net instance) into fuzzy spatio‐temporal ontologies at both structure and instance levels. Finally, we prove the correctness of the transformation and provide a detailed transformation example.

Integration of Weighted Terminological Concepts and Vague Knowledge in Ontologies for Decision Making

A well-known family of logics for managing structured knowledge is Description logics (DLs). They form the basis for a wide variety of ontology languages. Experience with the use of DLs in applications has, however, shown that their capabilities are insufficient for some domains. In particular, the decision-making process requires the assessment of two, possibly contradictory, influences on decision factors. First, there are items belonging to certain classes or fulfillling certain roles within complex logical constructs, but these memberships are to some extent vague. Secondly, individual preferences may change depending on the person who controls the decision-making process. Therefore, the challenge in building a decision making framework is to appropriately account for these variable influences by depicting and incorporating both aspects. This paper shows how these influences can be best modeled using a combination of fuzzy description logic and weighted description logic. Fuzzy logic is used to represent vagueness and ambiguity in ontologies, weighted description logic expresses individual preferences. In addition, the paper shows how to engineer an appropriate architecture for the suggested model.

Automation of Recognizing Old Slavonic Manuscripts with help of Description Logics

[full article, abstract in Russian; abstract in Lithuanian and English]
 In the paper an approach to recognition of old Slavonic symbols is proposed, which includes the stages of ternary segmentation, detecting edges of areas, synthesis of two skeleton variants, forming a fuzzy graph, creating a logical description in a fuzzy description logic, and recognition with help of a subsystem of formal automatic reasoning. Results of preliminary experiments are provided, which confirm the perspective of the proposed approach.

Towards a Dempster–Shafer Fuzzy Description Logic—Handling Imprecision in the Semantic Web

Vague information has been emerged as a main issue in Semantic Web community. Vagueness is traditionally represented by fuzzy set theory. Besides vagueness, Semantic Web queries often have to deal with information incompleteness, aka uncertainty. This kind of information can be represented through Dempster–Shafer theory, that also enables distributed information fusion. Vagueness along with information incompleteness are often referred to under the common term imprecise information . Imprecise information should be represented and manipulated under a common framework. We propose such a framework by defining a fuzzy Description Logic extended with Dempster–Shafer theory. Furthermore, we regard our method as a DL extension and we implemented it by a metaontology that captures Dempster–Shafer Fuzzy statements.

Verification of fuzzy UML models with fuzzy Description Logic

The fuzzy UML model has been introduced to enable the conceptual modeling of imprecise data in many applications. How to prevent, detect, and correct errors as early as possible in the modeling process by verifying the correctness of fuzzy UML models is desirable. But it is difficult to manually verify the correctness of the models. Thanks to the effective reasoning service of fuzzy Description Logics (DLs), they are considered as natural candidates for supporting the verification of fuzzy UML models. In this paper we propose a complete and practical approach for verifying fuzzy UML models with fuzzy DLs. In detail, we first propose an approach for transforming a fuzzy UML model into a fuzzy DL knowledge base. Then we further reduce the verification tasks of the fuzzy UML model to reasoning problems of the transformed fuzzy DL knowledge base. Also, the correctness of the approach is proved, and a running example is provided to explain the approach in detail. Final we implemented a prototype that can transform fuzzy UML models into fuzzy DL knowledge bases. Case studies show that the proposed approach is feasible and the prototype translator actually works. By means of our approach and tool, the transformed fuzzy DL knowledge base and reasoning tasks can be input into the existing fuzzy DL reasoners, and the reasoning results reported by the reasoners can be passed back to the designer to improve the quality of the fuzzy UML model.

Reasoning within Fuzzy OWL 2 EL revisited

Description Logics (DLs) are logics with interesting representational and computational features and are at the core of the Web Ontology Language OWL 2 and its profiles among which there is OWL 2 EL. The main feature of OWL 2 EL is that instance/subsumption checking can be decided in polynomial time. On the other hand, fuzzy DLs have been proposed as an extension to classical DLs with the aim of dealing with fuzzy concepts and we focus here on Fuzzy OWL 2 EL under standard and Gödel semantics. We provide some reasoning algorithms showing that instance/subsumption checking decision problems remain polynomial time for Fuzzy OWL 2 EL. We also identify some issues in previous related work (essentially incompleteness problems).

On the relationship between fuzzy description logics and many-valued modal logics

In this paper we study the relationships between a family of ALC-like fuzzy description logics (FDLs) defined over left-continuous t-norms and some many-valued multi-modal logics (MMLs). We analyze these relationships in both directions, that is, how to merge FDLs into MMLs and vice-versa. The analysis starts from the relationships between the languages to reach systematically the deeper level of reasoning tasks. At this level we are able to truly investigate both formalisms from each other point of view. Finally, the results concerning translations between reasoning tasks are applied in order to get decidability and complexity bounds.