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

We study the problem of minimizing interpretations in fuzzy description logics (DLs) under the Gödel semantics by using fuzzy bisimulations. The considered logics are fuzzy extensions of the DL 𝒜ℒ𝒞 reg (a variant of propositional dynamic logic) with additional features among inverse roles, nominals and the universal role. Given a fuzzy interpretation ℐ and for E being the greatest fuzzy auto-bisimulation of ℐ w.r.t. the considered DL, we define the quotient ℐ/ E of ℐ w.r.t. E and prove that it is minimum w.r.t. certain criteria. Namely, ℐ/ E is a minimum fuzzy interpretation that validates the same set of fuzzy terminological axioms in the considered DL as ℐ. Furthermore, if the considered DL allows the universal role, then ℐ/ E is a minimum fuzzy interpretation bisimilar to ℐ, as well as a minimum fuzzy interpretation that validates the same set of fuzzy concept assertions in the considered DL as ℐ.

Logical characterizations of fuzzy bisimulations in fuzzy modal logics over residuated lattices

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 ALC <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">reg</sub> (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.

The paper describes the relation between fuzzy and non-fuzzy description logics. It gives an overview about current research in these areas and describes the difference between tasks for description logics and fuzzy logics. The paper also deals with the transformation properties of description logics to fuzzy logics and backwards. While the process of transformation from a description logic to a fuzzy logic is a trivial inclusion, the other way of reducing information from fuzzy logic to description logic is a difficult task, that will be topic of future work.

Bipolarity is an important feature of spatial information, involved in the expression of preferences and constraints about spatial positioning or in pairs of opposite spatial relations such as left and right. Another important feature is imprecision which has to be taken into account to model vagueness, inherent to many spatial relations (as for instance vague expressions such as close to, to the right of), and to gain in robustness in the representations. In previous works, we have shown that fuzzy sets and fuzzy mathematical morphology are appropriate frameworks, on the one hand, to represent bipolarity and imprecision of spatial relations and, on the other hand, to combine qualitative and quantitative reasoning in description logics extended with fuzzy concrete domains. The purpose of this paper is to integrate the bipolarity feature in the latter logical framework based on bipolar and fuzzy mathematical morphology and description logics with fuzzy concrete domains. Two important issues are addressed in this paper: the modeling of the bipolarity of spatial relations at the terminological level and the integration of bipolar notions in fuzzy description logics. At last, we illustrate the potential of the proposed formalism for spatial reasoning on a simple example in brain imaging.

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 DLs can be used for such domains when data and knowledge about them are vague. 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 (fuzzy) DLs, it can be used for concept learning. In this paper, for the first time, we define fuzzy bisimulation and (crisp) bisimilarity for fuzzy DLs under the Godel semantics. The considered logics are fuzzy extensions of the DL \(\mathcal {ALC}_{reg}\) with additional features among inverse roles, nominals, qualified number restrictions, the universal role and local reflexivity of a role. We give results on invariance of concepts as well as conditional invariance of TBoxes and ABoxes for bisimilarity in fuzzy DLs under the Godel semantics. We also provide a theorem on the Hennessy-Milner property for fuzzy bisimulations in fuzzy DLs under the Godel semantics.

Bisimulation and bisimilarity for fuzzy description logics under the Gödel semantics

Description logics are known to be a family of logic-based knowledge representation formalisms, and fuzzy description logics are expressive description logics for representing and handling fuzzy (vague or imprecise) knowledge bases. A sequential fuzzy description logic, which is introduced in this paper, is an extended fuzzy description logic where a sequence modal operator is introduced. In this paper, a translation from the proposed sequential fuzzy description logic to a standard fuzzy description logic is defined. Further, a theorem for embedding the sequential fuzzy description logic into the standard fuzzy description logic is proved using this translation. A theorem for relative decidability of the sequential fuzzy description logic with respect to the standard fuzzy description logic is established using the embedding theorem. The proposed logic and translation are intended for effective handling of fuzzy knowledge bases with sequential information (i.e., information expressed as sequences). Moreover, using the translation, existing methods and algorithms for the standard fuzzy description logic can be reused to effectively handle fuzzy knowledge bases with sequential information described by the sequential fuzzy description logic.

Description Logics (DLs) are knowledge representation languages built on the basis of classical logic. DLs allow the creation of knowledge bases and provide ways to reason on the contents of these bases. Fuzzy Description Logics (FDLs) are natural extensions of DLs for dealing with vague concepts, commonly present in real applications. Following the ideas of Hajek in [17] and Garcia-Cerdana et al. in [15] we develop a family of FDLs whose underlying logic is the fuzzy logic of a finite linearly ordered residuated lattice, that is, an n-graded fuzzy logic defined by a divisible finite t-norm over a finite chain. Moreover, the role of the constructor of implication in the languages for FDLs is discussed, and a hierarchy of AL-languages adapted to the behavior of the connectives in the fuzzy logics underlying these description languages is proposed. Finally, we deal with reasoning tasks within the framework of finitely valued DLs.

Reasoning within intuitionistic fuzzy rough description logics

Reasoning within expressive fuzzy rough description logics

Typical description logics are limited to dealing with crisp concepts and crisp roles. However, Web applications based on description logics should allow the treatment of the inherent imprecision. Therefore, it is necessary to add fuzzy features to description logics. A family of extended fuzzy description logics is proposed to enable representation and reasoning for complex fuzzy information. The extended fuzzy description logics introduce the cut sets of fuzzy concepts and fuzzy roles as atomic concepts and atomic roles, and inherit the concept and role constructors from description logics. The definitions of syntax, semantics, reasoning tasks, and reasoning properties are given for the extended fuzzy description logic. The extended fuzzy description logics adopt a special fuzzify-method with more expressive power than the previous fuzzy 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.

Fuzzy Description Logics and t-norm based fuzzy logics

Description logics (DL) constitute a family of formal knowledge representation languages used for ontology grounding, each of which comes with well-understood computational properties. A crucial design principle in description logics is to establish a favorable trade-off between expressivity and scalability and, when needed, maximize expressivity. Since the best balance between DL expressivity and reasoning complexity depends on the intended application, several description logics have been developed. The expressivity of DL-based ontologies is determined by the mathematical constructors available in the underlying description logic. Description logics used in multimedia representation and reasoning include not only general-purpose description logics, but also spatial, temporal, and fuzzy description logics.