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### Reasoning within Fuzzy Description Logics

Description Logics (DLs) are suitable, well-known, logics for managing structured knowledge. They allow reasoning about individuals and well defined concepts, i.e., set of individuals with common properties. The experience in using DLs in applications has shown that in many cases we would like to extend their capabilities. In particular, their use in the context of Multimedia Information Retrieval (MIR) leads to the convincement that such DLs should allow the treatment of the inherent imprecision in multimedia object content representation and retrieval. In this paper we will present a fuzzy extension of ALC, combining Zadeh's fuzzy logic with a classical DL. In particular, concepts becomes fuzzy and, thus, reasoning about imprecise concepts is supported. We will define its syntax, its semantics, describe its properties and present a constraint propagation calculus for reasoning in it.

### Reasoning within Fuzzy Description Logics

Description Logics (DLs) are suitable, well-known, logics for managing structured knowledge. They allow reasoning about individuals and well defined concepts, i.e., set of individuals with common properties. The experience in using DLs in applications has shown that in many cases we would like to extend their capabilities. In particular, their use in the context of Multimedia Information Retrieval (MIR) leads to the convincement that such DLs should allow the treatment of the inherent imprecision in multimedia object content representation and retrieval. In this paper we will present a fuzzy extension of ALC, combining Zadeh's fuzzy logic with a classical DL. In particular, concepts becomes fuzzy and, thus, reasoning about imprecise concepts is supported. We will define its syntax, its semantics, describe its properties and present a constraint propagation calculus for reasoning in it.

### On the Non-Monotonic Description Logic $\mathcal{ALC}$+T$_{\mathsf{min}}$

In the last 20 years many proposals have been made to incorporate non-monotonic reasoning into description logics, ranging from approaches based on default logic and circumscription to those based on preferential semantics. In particular, the non-monotonic description logic $\mathcal{ALC}$+T$_{\mathsf{min}}$ uses a combination of the preferential semantics with minimization of a certain kind of concepts, which represent atypical instances of a class of elements. One of its drawbacks is that it suffers from the problem known as the \emph{property blocking inheritance}, which can be seen as a weakness from an inferential point of view. In this paper we propose an extension of $\mathcal{ALC}$+T$_{\mathsf{min}}$, namely $\mathcal{ALC}$+T$^+_{\mathsf{min}}$, with the purpose to solve the mentioned problem. In addition, we show the close connection that exists between $\mathcal{ALC}$+T$^+_{\mathsf{min}}$ and concept-circumscribed knowledge bases. Finally, we study the complexity of deciding the classical reasoning tasks in $\mathcal{ALC}$+T$^+_{\mathsf{min}}$.

### Fusions of Description Logics and Abstract Description Systems

Fusions are a simple way of combining logics. For normal modal logics, fusions have been investigated in detail. In particular, it is known that, under certain conditions, decidability transfers from the component logics to their fusion. Though description logics are closely related to modal logics, they are not necessarily normal. In addition, ABox reasoning in description logics is not covered by the results from modal logics. In this paper, we extend the decidability transfer results from normal modal logics to a large class of description logics. To cover different description logics in a uniform way, we introduce abstract description systems, which can be seen as a common generalization of description and modal logics, and show the transfer results in this general setting.

### A Uniform Framework for Concept Definitions in Description Logics

More precisely, an interpr etation I ( I; I) consists of a domain of interpr etation I, and an interpr etation function I mapping ev ery atomic concept A to a subset of I and ev ery atomic role R to a subset of I I . The in terpretation function I is extended to complex concepts of ALC Q (note that in ALC Q roles are alw a ys atomic) as follo ws: I I? I; (: C) I I C I ( C 1 u C 2) I C I 1 \ C I 2 ( C 1 t C 2) I C I 1 [ C I 2 ( 9 R: C) I f s 2 I j9 s 0: ( s; s 0) 2 R I and s 0 2 C I g ( 8 R: C) I f s 2 I j8 s 0: ( s; s 0) 2 R I implies s 0 2 C I g ( nR: C) I f s 2 I j # f s 0 j ( s; s 0) 2 R I and s 0 2 C I g n g ( nR: C) I f s 2 I j # f s 0 j ( s; s 0) 2 R I and s 0 2 C I g n g 89 De Gia como & Lenzerini where # S denotes the cardinalit y of the set S . A c onc ept C is satis able i there exists an in terpretation I suc h that C I 6;, otherwise C is unsatis able . A c onc ept C 1 is subsume d by a c onc ept C 2, written as C 1 v C 1, i for ev ery in terpretation I, C I 1 C I 2 . Our kno wledge expressed in terms of concepts and roles is assem bled in to a sp ecial kno wledge base, traditionally called TBox, whic h consists of a nite (p ossibly empt y) set of assertions. In order to b e as general as p ossible, w e assume that ev ery assertion has the form of an inclusion assertion (or simply inclusion): C 1 v C 2 without an y restriction on the form of the concepts C 1 and C 2 . A pair of inclusions of the form f C 1 v C 2;C 2 v C 1 g is often written as C 1 C 2 and is called e quivalenc e assertion .