1 Introduction
Nowadays, many applications and domains use some form of knowledge representation language in order to improve their capabilities. Encoding human knowledge and providing means to reason with it can benefit applications a lot, by enabling them provide intelligent answers to complex user defined tasks. Examples of modern applications that have recently adopted knowledge representation languages are the World Wide Web [BernersLee, Hendler, LassilaBernersLee et al.2001, Baader, Horrocks, SattlerBaader et al.2002b], where knowledge is used to improve the abilities of agents and the interoperability between disparate systems, multimedia processing applications [Alejandro, Belle, SmithAlejandro et al.2003, Benitez, Smith, ChangBenitez et al.2000], which use knowledge in order to bridge the “gap” between human perception of the objects that exist within multimedia documents, and computer “perception” of pixel values, configuration applications [McGuinessMcGuiness2003], etc. Unfortunately, there are occasions where traditional knowledge representation languages fail to accurately represent the concepts that appear in a domain of interest. For example, this is particularly the case when domain knowledge is inherently imprecise or vague. Concepts like that of a “near” destination [BernersLee, Hendler, LassilaBernersLee et al.2001], a “highQuality” audio system [McGuinessMcGuiness2003], “many” children, a “faulty” reactor [Horrocks SattlerHorrocks Sattler1999], “soon” and many more, require special modelling features. In the past many applications of various research areas, like decision making, image processing, robotics and medical diagnosis have adopted special mathematical frameworks that are intended for modelling such types of concepts [ZimmermannZimmermann1987, Larsen YagerLarsen Yager1993, Krishnapuram KellerKrishnapuram Keller1992]. One such a mathematical framework is fuzzy set theory [ZadehZadeh1965]. Though fuzzy extensions of various logical formalisms, like propositional, predicate or modal logics have been investigated in the past [HajekHajek1998], such a framework is is not yet well developed for Description Logics and much research work needs to be done. More precisely, there is the need for reasoning in very expressive fuzzy Description Logics.
In order to achieve knowledge reusability and high interoperability, modern applications often use the concept of an “ontology” [BernersLee, Hendler, LassilaBernersLee et al.2001] to represent the knowledge that exists within their domain. Ontologies are created by encoding the full knowledge we possess for a specific entity of our world using a knowledge representation language. A logical formalism that has gained considerable attention the last decade is Description Logics [Baader, McGuinness, Nardi, PatelSchneiderBaader et al.2002a]. Description Logics (DLs) are a family of classbased (conceptbased) knowledge representation formalisms, equipped with well defined modeltheoretic semantics [TarskiTarski1956]. They are characterized by the use of various constructors to build complex concept descriptions from simpler ones, an emphasis on the decidability of key reasoning problems, and by the provision of sound, complete and empirically tractable reasoning services. Both the welldefined semantics and the powerful reasoning tools that exist for Description Logics makes them ideal for encoding knowledge in many applications like the Semantic Web [Baader, Horrocks, SattlerBaader et al.2002b, PanPan2004], multimedia applications [Meghini, Sebastiani, StracciaMeghini et al.2001], medical applications [Rector HorrocksRector Horrocks1997], databases [Calvanese, De Giacomo, Lenzerini, Nardi, RosatiCalvanese et al.1998] and many more. Interestingly, the current standard for Semantic Web ontology languages, OWL [Bechhofer, van Harmelen, Hendler, Horrocks, McGuinness, PatelSchneider, eds.Bechhofer et al.2004], is based on Description Logics to represent knowledge and support a wide range of reasoning services. More precisely, without regarding annotation properties of OWL, the OWL Lite species of OWL is equivalent to the DL, while OWL DL is equivalent to [Horrocks, PatelSchneider, van HarmelenHorrocks et al.2003]. Although DLs provide considerable expressive power, they feature limitations regarding their ability to represent vague (fuzzy) knowledge. As obvious, in order to make applications that use DLs able to cope with such information we have to extend them with a theory capable of representing such kind of information. One such important theory is fuzzy set theory. Fuzzy Description Logics are very interesting logical formalisms as they can be used in numerous domains like multimedia and information retrieval [FaginFagin1998, Meghini, Sebastiani, StracciaMeghini et al.2001] to provide ranking degrees, geospatial [Chen, Fellah, BishrChen et al.2005] to cope with vague concepts like “near”, “far” and many more.
In order to make the need to handle vagueness knowledge more evident and the application of fuzzy set theory more intuitive, let us consider an example. Suppose that we are creating a knowledgebased image processing application. In such application the task is to (semi)automatically detect and recognize image objects. Suppose also that the content of the images represents humans or animals. For such a domain one can use standard features of Description Logics to encode knowledge. For example, a knowledge base describing human bodies could contain the following entities
where is a subsumption relation and is obviously a transitive relation. This knowledge can be captured with the aid of the DL [SattlerSattler1996]. Moreover, one might want to capture the knowledge that the role is the inverse of the role , writing , thus being able to state that something that is a body and has a tail is also an animal as,
For this new feature one would require the DL [Horrocks SattlerHorrocks Sattler1999]. The new axiom gives us the ability to recognize that the concept is subsumed by . Finally, the DL can be further extended with role hierarchies and number restrictions. Hence, one is able to capture the fact that the role is a subrole of the role , by writing , while we can also provide a more accurate definition of the concept by giving the axiom,
stating that the body is a direct part of a human and it also has exactly two arms.
Up to now we have only used standard Description Logic features. Now suppose that we run an image analysis algorithm. Such algorithms usually segment the image into regions and try to annotate them with appropriate semantic labels using low level image features. This process involves a number of vague concepts since an image region might be red, blue, circular, small or smooth textured to some degree or two image regions might not be totally but only to some degree adjacent (since not all of their pixels are adjacent), one contained within the other, etc. Hence we can only decide about the membership of a region to a specific concept only to a certain degree [Athanasiadis, Mylonas, Avrithis, KolliasAthanasiadis et al.2007]. For example, in our case we could have that the object the object to a degree of , that to a degree of , that is an to a degree of and that is a to a degree of 0.85. From that fuzzy knowledge one could deduce that belongs to the concept to a degree of 0.75. This together with a definition of the form , where represents equivalence, means that there is a good chance that is a . Observe, that in this definition, in order for someone to be a human, we do not force a to explicitly have a part that is an . This is a reasonable choice in the present application, because depending on the level of the segmentation, there might be several segmented regions between and . As it is obvious is such applications handling the inherent vagueness certainly benefits the specific application.
In this paper we extend the well known fuzzy (f) DL [StracciaStraccia2001] to the fuzzy DL (f), which extends the f DL with the inverse role constructor, transitive role axioms, role hierarchies and the number restrictions constructor. Moreover, we prove the decidability of the f DL by providing a tableaux algorithm for deciding the standard DL inference problems. In order to provide such an algorithm we proceed in two steps. First, we focus on the f language studying the properties of fuzzy transitive roles in value and existential restrictions, as well as the applicability of the techniques used in the classical language to ensure the termination of the algorithm [Horrocks SattlerHorrocks Sattler1999]. As we will see there is great difficulty on handling such axioms on the context of fuzzy DLs, but after finishing our investigation we will see that similar notions as in classical language can be applied. Secondly, we extend these results by adding role hierarchies and number restrictions. We provide all the necessary extensions to the reasoning algorithm of f, thus providing a reasoning algorithm for the f language. Discarding datatypes, is slightly more expressive than (OWL Lite) and slightly less expressive than (OWL DL). In order to achieve our goal we again extend the techniques used for the classical language and which ensure correctness of the algorithm [Horrocks SattlerHorrocks Sattler1999, Horrocks, Sattler, TobiesHorrocks et al.2000]. Finally, we prove the decidability of the extended algorithm. There are many benefits on following such an approach. On the one hand we provide a gradual presentation to the very complex algorithm of f, while on the other hand we provide a reasoning algorithm for a less expressive, but more efficient fuzzy DL language, f. The classical language is known to be Pspacecomplete, in contrast to the Exptimecompleteness of [TobiesTobies2001], hence our algorithm for f can be used for future research and for providing efficient and optimized implementations.
Please note that fuzzy DLs [StracciaStraccia2001] are complementary to other approaches that extend DLs, like probabilistic DLs [Koller, Levy, PfefferKoller et al.1997, Giugno LukasiewiczGiugno Lukasiewicz2002, Ding PengDing Peng2004], or possibilistic DLs [HollunderHollunder1994]. More precisely, these theories are meant to be used for capturing different types of imperfect
information and knowledge. Fuzziness is purposed for capturing vague (fuzzy) knowledge, i.e. facts that are certain but which have degrees of truth assigned to them, like for example the degree of truth of someone being tall. On the other hand, possibilistic and probabilistic logics are purposed for capturing cases where knowledge is uncertain due to lack of information or knowledge about a specific situation or a future event, like for example a sensor reading or a weather forecast. These facts are assigned degrees of possibility, belief or probability, rather than truth degrees. Dubois01 provide a comprehensive analysis on these theories along with their different properties.
The rest of the paper is organized as follows. Section 2 briefly introduces the DL and provides some preliminaries about the notion of a fuzzy set and how set theoretic and logical operations have been extended to the fuzzy set framework. Section 3 introduces the syntax and semantics of the fuzzy DL, which we call f.^{1}^{1}1In a previous approach to fuzzy DLs the notation is used [StracciaStraccia2004], but this notation is not so flexible to represent fuzzy DLs which use different norm operations, as we will see later on. In some other approaches [Tresp MolitorTresp Molitor1998, Hölldobler, Khang, StörrHölldobler et al.2002] the naming is used but this can easily be confused with ( extended with functional restrictions, Horrocks99), when pronounced. language. Section 4 provides an investigation on the semantics of fuzzy DLs when fuzzy transitive relations participate in value and existential restrictions. In section 5 we give a detailed presentation of the reasoning algorithm for deciding the consistency of a fuzzy ABox and we provide the proofs for the termination, soundness and completeness of the procedure. Then, in section 6 we extend the previous results by adding role hierarchies and number restrictions. More precisely, the results of section 4 are enriched by considering transitive roles and roles hierarchies in value and existential restrictions. Using this results we extend the algorithm of section 5 to handle with these new feature and finally we prove its soundness, completeness and termination. At last, in section 7 we review some previous work on fuzzy Description Logics while section 8 concludes the paper.
2 Preliminaries
In the current section we will briefly introduce classical DLs and fuzzy set theory, recalling some mathematical properties of fuzzy set theoretic operators.
2.1 Description Logics and the Dl
Description Logics (DLs) [Baader, McGuinness, Nardi, PatelSchneiderBaader et al.2002a] are a family of logicbased knowledge representation formalisms designed to represent and reason about the knowledge of an application domain in a structured and wellunderstood way. They are based on a common family of languages, called description languages, which provide a set of constructors to build concept (class) and role (property) descriptions. Such descriptions can be used in axioms and assertions of DL knowledge bases and can be reasoned about with respect to (w.r.t.) DL knowledge bases by DL systems.
In this section, we will briefly introduce the DL, which will be extended to the f DL later. A description language consists of an alphabet of distinct concept names (), role names () and individual (object) names (); together with a set of constructors to construct concept and role descriptions.
Now we define the notions of roles and concepts.
Definition 2.1
Let be a role name and a role. role descriptions (or simply roles) are defined by the abstract syntax: . The inverse relation of roles is symmetric, and to avoid considering roles such as , we define a function which returns the inverse of a role, more precisely,
The set of concept descriptions (or simply concepts) is the smallest set such that:

every concept name is a concept,

if and are concepts and is a role, then , , , and are also concepts, called general negation (or simply negation), disjunction, conjunction, value restrictions and existential restriction, respectively, and

if a simple^{2}^{2}2A role is called simple if it is neither transitive nor has any transitive subroles. This is crucial in order to get a decidable logic [Horrocks, Sattler, TobiesHorrocks et al.1999]. role and , then and are also concepts, called atmost and atleast number restrictions.
By removing point 3 of the above definition we obtain the set of concepts.
Description Logics have a modeltheoretic semantics, which is defined in terms of interpretations. An interpretation (written as ) consists of a domain (written as ) and an interpretation function (written as ), where the domain is a nonempty set of objects and the interpretation function maps each individual name to an element , each concept name to a subset , and each role name to a binary relation . The interpretation function can be extended to give semantics to concept and role descriptions. These are given in Table 1.
Constructor  Syntax  Semantics 

top  
bottom  
concept name  
general negation  
conjunction  
disjunction  
exists restriction  
value restriction  
atmost restriction  
atleast restriction 
A knowledge base (KB) consists of a TBox, an RBox and an ABox. A TBox is a finite set of concept inclusion axioms of the form , or concept equivalence axioms of the form , where are concepts. An interpretation satisfies if and it satisfies if . Note that concept inclusion axioms of the above form are called general concept inclusions [Horrocks SattlerHorrocks Sattler1999, BaaderBaader1990]. A RBox is a finite set of transitive role axioms (), and role inclusion axioms (). An interpretation satisfies if, for all , , and it satisfies if . A set of role inclusion axioms defines a role hierarchy. For a role hierarchy we introduce as the transitivereflexive closure of . At last, observe that if , then the semantics of role inclusion axioms imply that , while the semantics of inverse roles imply that . A RBox is obtained by a RBox if we disallow role inclusion axioms. A ABox is a finite set of individual axioms (or assertions) of the form , called concept assertions, or , called role assertions, or of the form . An interpretation satisfies if , it satisfies if , and it satisfies if . A ABox is obtained by a ABox by disallowing inequality axioms . An interpretation satisfies a knowledge base if it satisfies all the axioms in . is satisfiable (unsatisfiable) iff there exists (does not exist) such an interpretation that satisfies . Let be concepts, is satisfiable (unsatisfiable) w.r.t. iff there exists (does not exist) an interpretation of s.t. ; subsumes w.r.t. iff for every interpretation of we have . Given a concept axiom, a role axiom, or an assertion , entails , written as , iff for all models of we have satisfies .
2.2 Fuzzy Sets
Fuzzy set theory and fuzzy logic are widely used today for capturing the inherent vagueness (the lack of distinct boundaries of sets) that exists in real life applications [Klir YuanKlir Yuan1995]. The notion of a fuzzy set was first introduced by Zadeh65. While in classical set theory an element either belongs to a set or not, in fuzzy set theory elements belong only to a certain degree. More formally, let X be a collection of elements (called universe of discourse) i.e . A crisp subset A of X is any collection of elements of X that can be defined with the aid of its characteristic function (x) that assigns any to a value 1 or 0 if this element belongs to X or not, respectively. On the other hand, a fuzzy subset A of X, is defined by a membership function (x), or simply A(x), for each . This membership function assigns any to a value between 0 and 1 that represents the degree in which this element belongs to . Additionally, a fuzzy binary relation over two crisp sets and is a function . For example, one can say that belongs to the set of people to a degree of , writing , or that the object is part of the object to a degree of 0.6, writing . Several properties of fuzzy binary relations have been investigated in the literature [Klir YuanKlir Yuan1995]. For example, a binary fuzzy relation is called  transitive if , while the inverse of a relation is defined as [Klir YuanKlir Yuan1995].
Using the above idea, the most important operations defined on crisp sets and relations, like the boolean operations (complement, union, and intersection etc.), are extended in order to cover fuzzy sets and fuzzy relations. Accordingly, a sound and complete mathematical framework that plays an important role in the management of imprecise and vague information has been defined and used in a wide set of scientific areas including expert systems and decision making [ZimmermannZimmermann1987]
[KandelKandel1982], image analysis and computer vision
[Krishnapuram KellerKrishnapuram Keller1992], medicine [Oguntade BeaumontOguntade Beaumont1982], control [SugenoSugeno1985], etc.2.3 Fuzzy Set Theoretic Operations
In this section, we will explain how to extend boolean operations and logical implications in the context of fuzzy sets and fuzzy logics. These operations are now performed by mathematical functions over the unit interval.
The operation of complement is performed by a unary operation, , called fuzzy complement. In order to provide meaningful fuzzy complements, such functions should satisfy certain properties. More precisely, they should satisfy the boundary conditions, and , and be monotonic decreasing, for , . In the current paper we will use the Lukasiewicz negation, , which additionally is continuous and involutive, for each , holds. In the cases of fuzzy intersection and fuzzy union the mathematical functions used are binary over the unit interval. These functions are usually called norm operations referred to as tnorms (), in the case of fuzzy intersection, and tconorms (or snorms) (), in the case of fuzzy union [Klement, Mesiar, PapKlement et al.2004]. Again these operations should satisfy certain mathematical properties. More precisely, a tnorm (tconorm) satisfies the boundary condition, , is monotonic increasing, for then , commutative, , and associative, . Though there is a wealth of such operations in the literature [Klir YuanKlir Yuan1995] we restrict our attention to specific ones. More precisely, we are using the Gödel tnorm, and the Gödel tconorm, . Additionally to the aforementioned properties, these operations are also idempotent, i.e. and , hold. Finally, a fuzzy implication is performed by a binary operation, of the form . In the current paper we use the KleeneDienes fuzzy implication which is provided by the equation, . The reason for restricting our attention to these operations would be made clear in section 5.1. We now recall a property of the norm operation that we are going to use in the investigation of the properties of transitive relations under the framework of fuzzy set theory.
Lemma 2.2
[HajekHajek1998] For any , where takes values from the index set , the operation satisfies the following property:

.
3 The Dl
In this section, we introduce a fuzzy extension of the DL presented in Section 2.1. Following Stoilos05, since we are using the KleeneDienes (KD) fuzzy implication in our language, we call it . This presentation follows the standard syntax and semantics of fuzzy DLs, that has been introduced in the literature [StracciaStraccia2001, Hölldobler, Khang, StörrHölldobler et al.2002, Sánchez TettamanziSánchez Tettamanzi2004]. More precisely, was first presented by Straccia05. For completeness reasons we will also present the language here. Please also note that our presentation differs from that of Straccia05 in the semantics of concept and role inclusion axioms.
As usual, we consider an alphabet of distinct concept names (), role names () and individual names (). The abstract syntax of concepts and roles (and respectively of concepts and roles) is the same as their counterparts; however, their semantics is based on fuzzy interpretations (see below). Similarly, keeps the same syntax of concept and role axioms as their counterparts in . Interestingly, extends individual axioms (assertions) into fuzzy individual axioms, or fuzzy assertions (following, Straccia01), where membership degrees can be asserted.
Firstly, by using membership functions that range over the interval , classical interpretations can be extended to the concept of fuzzy interpretations [StracciaStraccia2001]. Here we abuse the symbols and define a fuzzy interpretation as a pair ,^{3}^{3}3In the rest of the paper, we use to represent fuzzy interpretations instead of crisp interpretations. where the domain is a nonempty set of objects and is a fuzzy interpretation function, which maps

an individual name to an element ,

a concept name to a membership function ,

a role name to a membership function .
For example, if then gives the degree that the object belongs to the fuzzy concept , e.g. . By using the fuzzy set theoretic operations defined in section 2.3, the fuzzy interpretation function can be extended to give semantics to concepts and roles. For example, since we use the function for fuzzy union the membership degree of an object to the fuzzy concept is equal to . Moreover since, according to Table 1, a value restriction is an implication of the form, , we can interpret as [HajekHajek1998], and as the KleeneDienes fuzzy implication and finally have the equation, . The complete set of semantics is depicted in Table 2. We have to note that there are many proposals for semantics of number restrictions in fuzzy DLs [Sánchez TettamanziSánchez Tettamanzi2004, StracciaStraccia2005b]. We choose to follow the semantics proposed by Straccia05 since they are based in the FirstOrder interpretation of number restrictions [Baader, McGuinness, Nardi, PatelSchneiderBaader et al.2002a]. Moreover, as it is shown by Stoilos05c and as we will see in section 6, under these semantics all inference services of stay decidable and reasoning can be reduced to a simple counting problem, yielding an efficient algorithm. Note that, although most of the above semantics have been presented elsewhere [Sánchez TettamanziSánchez Tettamanzi2004, StracciaStraccia2005b], we include them here simply for the sake of completeness.
Constructor  Syntax  Semantics 
top  
bottom  
general negation  
conjunction  
disjunction  
exists restriction  
value restriction  
atmost  
atleast  
inverse role  

An knowledge base consists of a TBox, an RBox and an ABox. Let be a concept name and an concept. An TBox is a finite set of fuzzy concept axioms of the form , called fuzzy inclusion introductions, and of the form , called fuzzy equivalence introductions. A fuzzy interpretation satisfies if . A fuzzy interpretation satisfies if . A fuzzy interpretation satisfies an TBox iff it satisfies all fuzzy concept axioms in ; in this case, we say that is a model of .
There are two remarks here. Firstly, we give a crisp subsumption of fuzzy concepts here, which is the usual way subsumption is defined in the context of fuzzy sets [Klir YuanKlir Yuan1995]. In contrast, Straccia05 defines a fuzzy subsumption of fuzzy concepts. As it was noted by Bobillo06 in fDLs fuzzy subsumption is sometimes counterintuitive. Secondly, as we can see, we are only allowing for simple TBoxes. A TBox is called simple if it neither includes cyclic nor general concept inclusions, i.e. axioms are of the form or , where is a concept name that is never defined by itself either directly or indirectly. A procedure to deal with cyclic and general TBoxes, in the context of fuzzy DLs, has been recently developed by Stoilos06ate, while also in parallel a slightly different technique was presented by Li06a. This process involves additional expansion rules and a preprocessing step called normalization, which are not affected by the expressivity of the underlying fuzzy DL. Hence, in order to keep our presentation simple we will not consider general TBoxes in the following, but we will focus on the decidability and reasoning of and , which involve many technical details. At the end of section 6 we will comment more on the issue of handling GCIs in the language.
An RBox is a finite set of fuzzy transitive role axioms of the form and fuzzy role inclusion axioms of the form , where are roles. A fuzzy interpretation satisfies if , , while it satisfies if , . Note that the semantics result from the definition of  transitive relations in fuzzy set theory. A fuzzy interpretation satisfies an RBox iff it satisfies all fuzzy transitive role axioms in ; in this case, we say that is a model of . Similarly with the classical language, the semantics of inverse roles and role inclusion axioms of imply that from and it holds that and .
An ABox is a finite set of fuzzy assertions [StracciaStraccia2001] of the form or , where stands for and , and or of the form . Intuitively, a fuzzy assertion of the form means that the membership degree of the individual a to the concept is at least equal to . We call assertions defined by positive assertions, while those defined by negative assertions. Formally, given a fuzzy interpretation ,
satisfies  if  , 
satisfies  if  , 
satisfies  if  , 
satisfies  if  , 
satisfies  if  . 
The satisfiability of fuzzy assertions with is defined analogously. Observe that, we can also simulate assertions of the form by considering two assertions of the form and [Hölldobler, Khang, StörrHölldobler et al.2002, StracciaStraccia2001]. A fuzzy interpretation satisfies an ABox iff it satisfies all fuzzy assertions in ; in this case, we say that is a model of .
Furthermore, as it was noted by Straccia01,Straccia05, due to the mathematical properties of the norm operations defined in section 2.3, the following concept equivalences are satisfied: , , , , and . Furthermore, since the Lukasiewicz complement is involutive it holds that, . Moreover, the De Morgan laws: , are satisfied. As a consequence of the satisfiability of the De Morgan laws and the use of the KleeneDienes fuzzy implication the following concept equivalences also hold.
,  ,  
, 
At last note that the classical laws of contradiction and excluded middle , do not hold.
Example 3.1
Let us revisit the fuzzy knowledge base () that we informally introduced in section 1. Formally, the knowledge base can be defined as follows: , where
=  ,  
,  
=  ,,  
, ,  

= 
Now, in order for some fuzzy interpretation to be a model of it should hold that and , . Furthermore, if , , and , then is also a model of . As a model of the RBox , should also satisfy that , .
Now let us consider the concept that we mentioned in Section 1. Due to the semantics of existential restrictions presented in Table 2, we have that
=  


= 
Hence would belong in the intersection of the two concepts with the minimum membership degree which is greater or equal than 0.75, as we claimed in Section 1.
Following Straccia01, we introduce the concept of conjugated pairs of fuzzy assertions to represent pairs of assertions that form a contradiction. The possible conjugated pairs are defined in Table 3, where represents a assertion. For example, if , then the assertions and conjugate.
Furthermore, due to the presence of inverse roles and role inclusion axioms the definition should be slightly extended from that of Straccia01 and hence, one should also take under consideration possible inverse roles or a role hierarchy when checking for conjugation two role assertions. For example, if , then the assertion , conjugates with ; similarly for the rest of the inequalities.
Now, we will define the reasoning problems of the DL.
A fuzzy interpretation satisfies an knowledge base if it satisfies all axioms in ; in this case, is called a model of . An knowledge base is satisfiable (unsatisfiable) iff there exists (does not exist) a fuzzy interpretation which satisfies all axioms in . An concept is satisfiable (unsatisfiable) w.r.t. an RBox and a TBox iff there exists (does not exist) some model of and for which there is some such that , and . In this case, is called nsatisfiable w.r.t. and [NavaraNavara2000]. Let and be two concepts. We say that is subsumed by w.r.t. and if for every model of and it holds that, . Furthermore, an ABox is consistent w.r.t. and if there exists a model of and that that is also a model of . Moreover, given a fuzzy concept axiom or a fuzzy assertion , an knowledge base entails , written , iff all models of also satisfy .
Furthermore, by studying Table 3, we can conclude that an ABox can contain a number of positive or negative assertions without forming a contradiction. Therefore, it is useful to compute lower and upper bounds of truthvalues. Given an knowledge base and an assertion , the greatest lower bound of w.r.t. is , where . Similarly, the least upper bound of w.r.t. is , where . A decision procedure to solve the best truthvalue bound was provided by Straccia01. In that procedure the membership degrees that appear in a ABox, together with their complemented values and the degrees 0, 0.5 and 1, were collected in a set of membership degrees and subsequently the entailment of a fuzzy assertions and , for all was tested, thus determining glb and lub. Obviously this procedure is independent of the expressivity of the DL language, and thus also applicable in our context.
Remark 3.2
From Table 2 we see that the semantics of the value and existential restrictions in fuzzy DLs are defined with the aid of an infimum and a supremum operation. This means that we can construct an infinite interpretation , i.e. an interpretation where contains infinite number of objects, for which is nsatisfiable ( for some ) but for all , . This is possible since although the maximum of the membership degrees involved for each individual object is strictly greater than the limit of the infinite sequence could converge to . This fact was first noted for fuzzy DLs by Hajeck05, introducing the notion of witnessed model for fuzzy DLs. A model is called witnessed if for there is some such that either or , i.e. there is some that witnesses the membership degree of to . Fortunately, there are fuzzy logics that have an infinite model if and only if they have a witnessed model. More precisely, Hajek proves this property for the Lukasiewicz fuzzy logic^{4}^{4}4Lukasiewicz fuzzy logic uses the tnorm , the tconorm , the Lukasiewicz complement and the fuzzy implication . He then concludes that the same proofs can be modified to apply to the fuzzy logic defined by the fuzzy operators we are using in the current paper. That is because these operators are definable in the Lukasiewicz logic [Mostert ShieldsMostert Shields1957]. For the rest of the paper, without loss of generality, we are going to consider only witnessed models.
In this paper, we will provide an algorithm to decide the fuzzy ABox consistency problem w.r.t. an RBox in very expressive fuzzy DLs. Many other reasoning problems can be reduced to this problem. Firstly, concept satisfiability for a fuzzy concept can be reduced to consistency checking of the fuzzy ABox {}. Secondly, in this paper, we only consider unfoldable TBoxes, where KB satisfiability can be reduced to ABox consistency w.r.t. an RBox. A TBox is unfoldable if it contains no cycles and contains only unique introductions, i.e., concept axioms with only concept names appearing on the left hand side and, for each concept name , there is at most one axiom in of which appears on the left side. A knowledge base with an unfoldable TBox can be transformed into an equivalent one with an empty TBox by a transformation called unfolding, or expansion [NebelNebel1990]: Concept inclusion introductions are replaced by concept equivalence introductions , where is a new concept name, which stands for the qualities that distinguish the elements of from the other elements of . Subsequently, if is a complex concept expression, which is defined in terms of concept names, defined in the TBox, we replace their definitions in . It can be proved that the initial TBox with the expanded one are equivalent.
Moreover, the problem of entailment can be reduced to the problem of fuzzy knowledge base satisfiability [StracciaStraccia2001]. More precisely, for , iff is unsatisfiable. With , we denote the “negation” of inequalities; e.g., if then , while if then . Finally, the subsumption problem of two fuzzy concepts and w.r.t. a TBox can also be reduced to the fuzzy knowledge base satisfiability problem. More formally, Straccia01, proved that iff , for both , and , is unsatisfiable. The above reduction can be extended in order for a fuzzy knowledge base to also include an RBox. Please note that, in crisp DLs, in order to check if a concept is subsumed by a concept we check for the unsatisfiability of the concept, . This reduction to unsatisfiability is not applicable to fDLs since the fuzzy operations that we use do not satisfy the laws of contradiction and excluded middle.
We conclude the section with an example.
Example 3.3
Consider again our sample knowledge base (). By applying the transformation of unfolding, defined earlier, one would obtain the following expanded fuzzy TBox:
=  ,  
while the respective fuzzy assertions would be transformed to
=  ,  
Now, let us formally specify the query we introduced in section 1. If is our modified knowledge base, after unfolding, the query would have the form . According to our previous discussion in order to check for the entailment of such a query one should check for the consistency of the fuzzy ABox , w.r.t. the RBox , since after the expansion we can remove . Our task in the following sections is to provide a procedure that decides the consistency of a fuzzy ABox w.r.t. an RBox.
4 Transitivity in Fuzzy Description Logics
In classical DLs, a role is transitive iff for all , and imply . Sattler96 shows that, for , if is transitive, is an successor of , are the successors of , and , then all successors of should be instances of , e.g., because: (i) (as is transitive), (ii) (as ) and (iii) (due to the semantics of ). In other words, this means that the following concept subsumption holds, .
The above property suggests that value restrictions on transitive relations () are propagated along the path of individuals. This propagation is crucial for reasoning algorithms in order to retain the treemodel property [Baader, McGuinness, Nardi, PatelSchneiderBaader et al.2002a], which is a property that leads to decidable decision procedures [VardiVardi1997]. Our goal in the rest of the section is to investigate this property in the context of fuzzy Description Logics that allow for transitive role axioms. We have to determine if similar propagation occurs and if it is, to find out the membership degree that the propagation carries to subsequent objects. This is the first time that such an investigation is presented in the literature.
In fuzzy DLs, objects are instances of all possible fuzzy concepts in some degree, ranging over the interval . As we have shown in Section 3, a fuzzy role is transitive iff, for all
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