Undefined 1 (2012) 1–5 1 IOS Press Efficient Query Answering over Fuzzy EL-OWL Based on Crisp Datalog Rewritable Fuzzy EL++ Afef Bahri a;¤, Rafik Bouaziz a and Faïez Gargouri a a MIRACL Laboratory, University of Sfax, Tunisia E-mail: [email protected], [email protected], [email protected] Abstract. OWL EL is an extension of the tractable EL++ description logic. Despite their inference capabilities over TBoxes, DL reasoners have a high ABox reasoning complexity which may constitute a serious limitation in the Semantic Web where we rely mainly on query answering (i.e. instance checking). The subsomption algorithm used in fuzzy EL++ reduce instance checking into concept satisfiability. To allow efficient instance checking and query answering over fuzzy EL-OWL, we propose in this paper two approaches for integrating fuzzy EL++ and crisp Datalog programs: homogenous and hybrid approaches. In the homogenous approach, we define crisp EL++ rules based on a crisp rewriting of fuzzy EL++. To preserve decidability, crisp EL++ rules are then written with Datalog safe ones. In the hybrid approach, fuzzy EL++ axioms and assertions are defined with EDB facts and Datalog rules are used to denote fuzzy EL++ instance checking deduction rules. That is, DL axioms and assertions are translated into Datalog EDB facts and Datalog rules are used to derive conclusions about them. Keywords: Fuzzy EL++, Instance checking, Homogenous and hybrid approaches, Rule based fuzzy OWL reasoning, Datalog entailment rules 1. Introduction logic programming has attracted the interest of many researches [10,12,11,8,13,9]. The rules languages which The OWL is a knowledge representation standard are subject of theses integration are the ones based proposed by the W3C Consortium. Its semantics is on Horn clausal logics. The integration of the two based on description logics and particularly on the ex- paradigms should play an important role in the Se- pressive SROIQ(DL). This expressivity lead to high mantic Web. In fact, despite their inference capabili- reasoning complexity and make reasoning algorithm ties over complex TBoxes, DL reasoners have a high impractical in real life applications. To solve this is- ABox reasoning complexity which may constitute a sue, three lightweight sublangages of OWL have been serious limitation in the Semantic Web where we rely introduced. We talk about OWL RL, OWL QL and mainly on query answering (i.e. instance checking). OWL EL profiles. OWL RL is the rule based fragment The subsomption algorithm used in fuzzy EL++ re- of OWL, OWL QL is a query language and OWL EL duce instance checking into concept satisfiability. That is used for conceptual modelling. OWL EL is an ex- is, to retrieve the instances of a given concept, we need ++ tended version of the EL description logic. OWL to run the subsumption algorithm for each individual EL supports, among others, local reflexivity and con- in the ABox. In Datalog systems, query answers are cept products. computed in one pass (i.e. bottom-up, top-down). Two The integration of description logics with rules and principal integration approaches are used in the lit- erature: the hybrid and the homogenous approaches. *Corresponding author. E-mail: [email protected] In the hybrid approach, the two paradigms are used 0000-0000/12/$00.00 °c 2012 – IOS Press and the authors. All rights reserved 2 to represent a knowledge bases. That is, a knowledge we begin by transforming fuzzy concepts and roles are base KB is defined as KB = hLD; PLi where LD is mapped into crisp ones based on [4]. We define after defined with description logic and PL is defined with that, crisp EL++ rules based on DL axioms and as- Logic Programming. In hybrid approach, the reason- sertions. To preserve decidability, the obtained EL++ ing over DL ontologies is performed only by the DL rules are then mapped into safe Datalog ones. In the reasoner. The rules are used to define constraints on second approach, DL constructors are defined with the defined ontology and rule engine are just used for EDB facts and Datalog rules are used to denote fuzzy rule execution. EL++ instance checking deduction rules. DL axioms While, in hybrid approaches, the rules and ontolo- and assertions are translated into Datalog EDB facts gies are treated separately, in homogenous ones rules and Datalog rules are used to derive conclusions about and ontology are combined in a new single logic lan- them. guage. In practice, the description logic is mapped into a rule based formalisms known as description logic programs. Such an integration approach may lead to ++ undecidability of reasoning problems due the opposite 2. Fuzzy EL and crisp Datalog programs assumption (Closed world and Open world assump- tion) of the two paradigms. Decidability may be ob- EL++ is a tractable (polynomial-time decidable) tained by restricting rules to DL-safe ones [12]. Ho- description logic proposed in [2]. The semantic of mogenous approaches perform reasoning only by rules fuzzy description logic is based on an interpretation engine. I = (¢I ; ?I ). ¢I is a nonempty set called the domain The problem of combination of rules and ontologies whereas ?I is a function that associates to every con- has equally been treated for fuzzy ontologies. The ma- cept C a membership function CI :¢I ! [0; 1]; and to jority of the works realized on this subject propose ho- every role R a membership function RI :¢I £ ¢I ! mogenous integration approaches in which fuzzy DL [0; 1]; and as for the crisp case, to every individual an programs are defined as a result of the integration of element of ¢I [14]. CI (resp. RI ) is thus interpreted fuzzy DLs and fuzzy rule languages [15]. The reason- as the membership degree function of fuzzy concept ing should be performed by fuzzy inference engines. C (respectively role R). CI (d) gives the degree of d The fact that there is no common implementation of (d 2 ¢I ) being an element of the fuzzy concept C fuzzy rule engines, theses works have only focused on under interpretation I. A concrete domain is consid- the theoretical aspects of the integration. ered as a fuzzy set. A fuzzy domain is defined with a In this paper we focus on the problem of combina- pair (¢D,©D) where ¢D is an interpretation domain tion of the fuzzy EL++ with crisp rule language for and ©D is a set of fuzzy predicates d of arity n and tractable query answering over large scale ABoxes. We D n an interpretation d : ¢D ! [0; 1]. For example, propose in this paper two integration approach of the Large is a fuzzy predicate which measures the degree two paradigms: in the first one we adopt a tight inte- of largeness of the width of a given country c and may gration approach in the sense that a unified language be defined with a trapezoidal membership function as and semantics are used to represent the two paradigms shown in figure 1. We may define the fuzzy concept (homogenous like). The crisp Datalog rule language GreatCountry is a fuzzy concept and is defined as acts as a unified language and Datalog based system follows: correspond to the unique framework in which various reasoning tasks are realized. In the second, a loose GreatCountry = Country u 9width:Large coupled integration approach (hybrid like), the two paradigms act separately but in a complementary way. That is, fuzzy DL reasoners are used to reason over the TBox of the ontology and crisp Datalog inference The fuzzy predicates used in this paper are defined as engines reason over the ABoxes for instance checking follows : and query answering. As we want to work with crisp Datalog program, we – Trapezoidal : trz(u,v,w,o) need to represent fuzzy ontology axioms and asser- – Triangular : tri(u,v,w) tions with crisp Datalog predicates. Two approaches – Left shoulder : ls(u,v) are equally used in this paper. In the first approach, – Right shoulder : rs(u,v) 3 3. Homogenous approach for integrating fuzzy ¹Large(width) EL++ and Datalog programs 6 We propose in this section an homogenous approach ++ 1 ¯ L for integration fuzzy EL description logic and crisp ¯ L Datalog programs. Fuzzy concept and roles axioms ¯ L and assertions are then represented used crisp Data- ¯ L log rules. The algorithm that we adopt to transform an ¯ L - ++ u v w o width EL fuzzy knowledge base into a Datalog program is defined as follows: Fig. 1. Trapezoidal representation of the linguistic term Large 1. Normalization of fuzzy EL++. ++ The fuzzy EL++ description logic is based on four dis- 2. Mapping of normalized fuzzy EL into crisp EL++. joint finite sets : the set of fuzzy concept names NC , 3. Definition of EL++ rules. the set of fuzzy role names NR, the set of individual 4. Mapping EL++ rules into Datalog rules. names NI and the set of fuzzy predicates ©D. We give in table 1 the syntax and the semantics of fuzzy EL++. A fuzzy CBox is in normal form if: 1. all GCIs have one of the following forms: Syntax DL Semantic C1 v® D, C1 v® 9r:C1, ??I = 0 > >I = 1 C1 u C2 v® D, 9r:C1 v® D I I I C1 u C2 (C1 u C2) (a) = min(C1 (a),C2 (a)) I where, C1, C2 2 BCC and D 2 BCC [ f?g. f®1=a1; :::; ®n=ang f®1=a1; :::; ®n=ang (a) = supija=aI ®i i I I I 2. all RIs are of the form r v s or r1 ± r2 v s (we 9 r.C (9r:C) (a) = supb2¢I fmin(r (a; b);C (b))g I I D use a crisp definition of RIs).