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1. rdfs range Hence it is natural for the RDF Schema institution S3 RDFS to be developed on top of 3 RDF with some more semantical constraints added to capture the semantics of RDF Schema language constructs The RDF Schema theory RDFS Vzprs Ikprs is composed of the RDF Schema vocabulary Vanes including V together with rdfs domain rdfs range rdfs Resource rdfs Literal rdfs Datatype rdfs Class rdfs subClassOf rdfs subPropertyOf rdfs member rdfs Container rdfs ContainerMembershipProperty and the triples Tpprs including Tppp together with rdf type rdfs domain rdfs Resource rdfs domain rdfs domain rdf Property rdfs range rdfs domain rdf Property rdfs subPropertyOf rdfs domain rdf Property rdfs subClassOf rdfs domain rdfs Class rdf subject rdfs domain rdf Statement rdf predicate rdfs domain rdf Statement Here we present only a part of triples added to T p The complete set of triples defining Tpprs can be found in 9 As for RDF we suppose that there is given a set Rpprs of RDF Schema resources and a 7 RDF framework institutions function Sppro Vrors gt Reps Which associates a resource with each RDF symbol and that satisfies Srnesly Sror Rrors and Sans can be extended to a RDFS model I pprs For each theory such that there is a theory morphism f RDFS RR T we consider the semantical constraint RR T pps as consisting of those RR T2. models 7 such that 1 R includes R and the restriction of S coincides with S RDFS RDFS 2 ext cdfs Resource R 3 Vx y u ve R x y E ext rdfs domain u v ext x gt uE ext y 4 Vx y u vE R X x y e ext cdfs range A u v e ext x gt ve ext y 5 Vx ye R x y ext crdfs subClassOf gt ext x c ext y 6 Vxe ext rdfs Class x rdfs Resource e ext rdfs subClassof 7 Vx ye R x y ext rdfs subPropert yOf gt ext x c ext y Vxe ext rdfs ContainerMembershipProperty x rdfs member e ext rdfs subPropertyOf Proposition 2 There is an institution morphism 3 RDFS gt 3 RDF Proof The institution morphism is given by the inclusion RDF RDFS Theorem 4 S RDFS is exact It is known that unsorted first order logic is only semiexact Hence the embedding of RDF Schema in first order logic can be inappropriate for many software applications In turn the sorted first order is exact and therefore is more appropriate to embed RDF Schema 4 CONCLUSION RDF and RDF Schema define the basic vocabularies for describing Semantic Web resources They are the foundation on which more expressive ontology languages are constructed Institutions and institution morphisms are used to formally represent and reason about the migration of different logical systems In this paper we use institutions to represent languages RDF and RDF Schema and use institution morphisms to re
3. 1 2 such that R R then there is no an RR model J with Mody XD 1 j 1 2 Theorem 2 BRDF is semiexact Proof Let W RR gt RR y RR RR be the pullback of the following diagram in Set Q RR gt RR RR If I is a RR model j 1 2 such that Mod 1 Mod 1 then R R We can define an RR model 7 withMod y U 1 j 1 2 We setR R P P VP gt S rr 8 rr if rre RR and ext p ext p if pe P j 1 2 Corollary BRDF is semiexact 3 2 RDF INSTITUTION The RDF institution RDF is defined using the construction we defined in Section 2 starting from a theory RDF and a semantical constraint _ kpp RDF theory is RDF Vppr Ikpr where the RDF vocabulary Vpr includes the following items rdf type rdf Property rdf value rdf Statement rdf subject rdf predicate rdf object rdf List rdf first rdf rest rdf nil rdf seg rdf Bag rdf Alt rdf _1 rdf _2 and Tp includes the following triples rdf type rdf type rdf Property rdf first rdf type rdf Property rdf subject rdf type rdf Property rdf rest rdf type rdf Property rdf predicate rdf type rdf Property rdf nil rdf type rdf List A rdf rdf object rdf type rdf Property 1 rdf type rdf Property rdf _2 rdf type rdf Property rdf value rdf type rdf Property We suppose that there is given a set Rp of RDF resources and a function Sppp Vans g
4. Z F Th where the objects are theory morphisms f X gt F and the arrows f f are consisting of theory morphisms 2 F gt 2 F such that f f 2 the model functor Mod S lt F maps each signature f 2 F gt F into the subcategory 2 F 3 the sentence functor Sen S X F maps a signature f 2 F 2 F into the set of L sentences 4 the satisfaction relation is defined by M iffM 9 Note that the semantical constraint is required only for the theories F for that there exists a theory morphism f gt F If F gt EF is a theory morphism then there is an institution morphism B a 3 x F gt S A 3 INSTITUTION OF RDF FRAMEWORK In this section we define the institutions for the languages RDF and RDF Schema The construction of these institutions is divided into three parts Firstly we construct a bare bone institution for RDF logic capturing only the very essential concepts in RDF namely the resource references and the triples format This logic then serves as the basis on which the institutions of the actual RDF and RDF Schema are constructed 3 1 BARE RDF INSTITUTION The Bare RDF institution BRDF is a bare bone institution with resource references sets as the only signatures and triples as sentences We use it as a basis over which we develop the other institutions involved in Semantic Web A signature RR in BRDF is
5. for each amp modelM Mod o M is M viewed as a model and it is called the reduct of M along The functor Sen abstracts the way the sentences are constructed from signatures vocabularies The satisfaction condition may be read as M satisfies the translation of iffM viewed as a model satisfies i e the meaning of Mis not changed by the translation A specification presentation is a way to represent the properties of a system independent of model implementation Formally a specification is a pair F where is a signature and F is a set of L sentences A X F model is a X model M such that M for each e F A theory is a specification 2 F with F closed i e satisfying the property Voa Z sentence VM a F model M o gt EF A signature morphism 2 is a theory morphism F F iff for each sentence Q we haveg e F gt e F We denote by Th the category of theories Given an institution the theoroidal institution 3 of is the institution Th Mod Sen where Mod is the extension of Mod to theories Sen is sign Senwith sign Th gt Sig the functor amp which forgets the sentences of a theory and sign An important property of the logical systems is that the finite limits are preserved by the model functor This property is called in literature exactness We recall that the functor Mod is defi
6. Berners Lee in Fig 1 Digital Signature Figure 1 Based on mature technologies such_as XML Unicode and URI Uniform Resource Identifier The Resource Description Framework RDF id is the foundation of later languages in the SW RDF is a model of metadata defining a mechanism for describing resources that makes no assumptions about a particular application domain It allows structured and semi structured data to be mixed and shared across applications It provides a simple way to make statements about Web resources An RDF document is a collection of triples statements of the form subject predicate object where subject is the resource we are interested in predicate specifies the property or characteristic of the subject and object states the value of the property RDF also defines vocabularies for constructing containers such bag sequence list etc RDF Schema B provides additional vocabularies for describing RDF documents It defines semantical entities such as resource class property literal and various properties about these entities such as subClassOf domain range etc In RDF Schema resource is the universe of description It can be further categorized as class property datatype or literal With these semantical constructs RDF Schema can be regarded as the basic ontology language 2 2 INSTITUTIONS Institutions supply a uniform way for structuring the theories in various logical systems Many logical systems have been p
7. F fo and RDF Schema RDFS EI are the cornerstone languages of the Semantic Web They define the set of basic vocabularies and syntax for later ontology languages Directly sitting on RDF and RDFS ontology languages such as DAML OIL IEI OWL fi and the Horn style rules extension SWRL Edi have been developed to improve the expressivity of the Semantic Web These languages extend the expressivity of RDF and RDFS by defining more language constructs to further classify and relate Web resources The Web Ontology language OWL is the mainstream ontology language It has three increasingly expressive sub languages OWL Lite DL and Full By imposing certain restrictions on the use of RDF RDFS and OWL Full constructs OWL Lite and DL are decidable Recently researchers have observed some issues related to the layering of OWL languages on top of RDFS 51 It has been argued by some researchers that the layering of OWL Lite and DL on top of RDFS is not very proper as they redefine the semantics of some of the RDFS vocabularies 3 14 Such improper layering may cause interoperability problems between ontologies written in these languages Hence it is of fundamental importance to thoroughly study the properties of RDF and RDFS to provide more insight into investigating the layering issue The notion of institutions was proposed to formalize the concepts of logical systems Institutions provide a means of reasoning about software specifications regardless of
8. THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Series A H OF THE ROMANIAN ACADEMY Volume 7 Number 1 2006 pp 000 000 RDF FRAMEWORK INSTITUTIONS Dorel LUCANU Yuan Fang LI Jin Song DONG Faculty of Computer Science Alexandru Ioan Cuza University Romania School of Computing National University of Singapore Singapore Corresponding author Dorel Lucanu Jdlucanu info uaic ro The Semantic Web is a vision of enabling software agents to autonomously understand process and integrate Web resources This ability is based on semantically marking up Web resources using ontologies Ontology languages are based on Resource Description Framework RDF and RDF Schema RDFS which form the foundation of the Semantic Web In this paper we define the RDF framework in terms of institution theory and investigate the exactness property Key words Semantic Web RDF RDF Schema institution 1 INTRODUCTION The Semantic Web has attracted much attention as a new generation of the current World Wide Web It attempts to realize the full potential of the Web by semantically marking up Web resources so that they can be readily processed by software agents on the Web Ontologies provide such markups for the Semantic Web As depicted in Fig 1 the development of Semantic Web languages takes a layered approach in which languages higher in_the stack are developed based on those below them The Resource Description Framework RD
9. a set of resource references A signature morphism RR RR is an arrow in Set The RR sentences are triples of the form sn pn on where sn pn on RR Usually sn is for subject name pnis for property predicate name and on is for object name RR models I are tuples I R P S ext where R is a set of resources P is a subset of R the set of properties S RR R is a function that maps each resource reference to some resource and ext P R X R is an extension function mapping each property to a set of pairs of resources that it relates An RR homomorphism h I I between two RR models is a function A R R such thath P Ch P S h S and ext hxh ext If RR gt RR is a signature morphism and 5 RDF framework institutions an RR model then the reduct of J along is the RR model I defined as follows R Rp P P S Siiger ext ext The satisfaction is defined as follows I Ege sn pn on iff S sn S on ext S pn that sn pn on is satisfied if and only if the pair consisting of the resources associated with the subject name sn and the object name on is in the extension of pn Since the signatures in BRDF are sets it follows that both the category of signatures and category of theories are cocomplete Theorem 1 BRDF is not exact Proof Let RR RR be a discrete diagram and y RR gt RR W RR RR its limit in Set If I is a RR model j
10. erence html March 2001 Received October 17 2005
11. late them The exactness property of various RDF institutions is also discussed One interesting conclusion we reached is that the institution for RDFS is exact As unsorted first order logic is known to be semiexact the exactness of RDF Schema implies that there might be issues of applying some first order logic reasoning tools to RDF Schema ontologies As stated previously the layering of the ontology language OWL on top of RDF Schema has been criticized to be improper possible follow up work would be to further investigate the layering issue REFERENCES 1 BARR M WELLS Ch Category Theory for Computing Science Les Publications CRM Montreal third edition 1999 2 BERNERS LEE T HENDLER J LASSILA O The Semantic Web Scientific American May 2001 3 BRICKLEY D GUHA R V Description Framework RDF Schema Specification 1 0 http www w3 org TR rdf schema February 2004 4 DIACONESCU R Grothendieck institutions Applied Categorical Structures 10 pp 383 402 2002 5 de BRUIN J LARA R POLLERERS A FENSEL D OWL DL vs OWL flight conceptual modeling and reasoning the semantic Web Proc of the 14th International Conference on World Wide Web WWW2005 Chiba Japan May 10 14 2005 pp 623 632 Dorel LUCANU Yuan FANG LI Jin SONG DONG 8 o 11 12 13 14 15 GOGUEN J Three Perspectives on Information Integration Proc of Semantic Interoperability and Integration Schloss Dagst
12. ned over Sig and the limits in this category are in fact colimits in Sig If the model functor preserves only the pullbacks then the institution is semiexact Let S Sig Mod Sen and 3 Sig Mod Sen be two institutions An institution morphism B a 3 3 consists of 1 afunctor Sig Sig 2 a natural transformation B Mod gt Mod i e a natural family of functors b Mod gt Mod 2 and 3 a natural transformation Sen Sen ie a natural family of functions a Sen B X gt Sen X such that the following satisfaction condition holds M a 9 amp B M Fa Q Dorel LUCANU Yuan FANG LI Jin SONG DONG 4 for any model M in 3 and X sentence inS3 Usually the institution morphisms are used to express the embedding relationship An example of institution morphism is B a 3 3 which express the embedding of 3 in3 We show how a theory 2 and a semantical constraint can define an institution S X F A semantical constraint is a map _ which associates a subcategory Z F Mod F with each theory X F such that Mod M je F for all 2 F gt F andM e F The constraints defined in 7 are a particular case of semantical constraints defined here when the subcategory can be syntactically represented The institution S F is defined as follows 1 the category of signatures is the comma category
13. roved to be institutions Recent research showed that institutions are useful in designing tools supporting verification over multiple logics The basic reference for institutions is 7 A well structured approach of the various institution morphisms and many other recent constructions can be found in B A recent application of institutions in formalizing the information integration is given in 6 The institutions use intensively category theory we recommend for a detailed presentation of categories and their applications in computer science An institution is a quadruple 3 Sig Mod Sen where Sig is 3 RDF framework institutions a category whose objects are called signatures Mod Sig Cat is a functor which associates with each signature a category whose objects are called models Senis a functor Sen Sig Set which associates with each signature a set whose elements are called X sentences and is a function which associates with each signature Xa binary relation S Mod X x Sen X called satisfaction relation such that for each signature morphism X the satisfaction condition Mod o M P M s Sen 9 Q holds for each model M e Mod x and each sentence 9 Sen X The functor Mod is defined over the opposite category Sig because a translation between vocabularies amp X defines a forgetful functor Mod Mod Mod X such that
14. t Rgpr which associates a resource with each RDF symbol It is easy to see that Rpr and Sppr can be extended to a RDF model I ppr For each theory such that there is theory morphism we consider the semantical constraint RR T app as consisting of those RR T models 7 such that 1 R includes Rep and the restriction of S coincides with Srp gt Dorel LUCANU Yuan FANG LI Jin SONG DONG 6 2 if pe P then p rdf Property e ext rdf type Since f is theory morphism the restriction of J to Vay is an RDF model Example The reification is given by the RDF signature reification RDF gt r s p o T where T further to Tppr includes the triples r rdf type rdf Statement r rdf subject s r cdf predicate p r rdf object o We denote by S RDF the institution defined by the theory RDF together with the semantical constraint _ pp using the method presented in Section 2 Proposition 1 There is an institution morphism RDF gt BRDF Theorem 3 S RDF is exact Proof Any two RDF signatures f RDF gt RR 7 and f RDF RR T have a pushout in Th BRDF and we apply Corollary of Theorem 2 3 2 RDF SCHEMA INSTITUTION RDF Schema defines additional language constructs for the RDF language It expands the expressivity of RDF by introducing the concept of universe of resources rdfs Resource the classification mechanism rdfs Class and a set of properties that relate them rdfs subClassOf rdfs domain
15. the logical system Hence it serves as a natural candidate to study the relationship among the various SW languages as they are based on different logical systems semantics Institution co morphisms capture the migration from one logical system to another The rest of the paper is organized as follows In Section 2 we briefly present the background information on the Semantic Web the RDF and RDFS languages institution and institution morphisms In Recommended by Mihai DRAGANESCU member of the Romanian Academy Dorel LUCANU Yuan FANG LI Jin SONG DONG 2 Section 3 we present three institutions The first one is for the bare RDF which is a bare bone RDF we constructed that only includes the triple format and resource references Building on the institution of bare RDF we then develop the institution for the actual RDF including its vocabularies and defining their semantics Thirdly we construct the institution for RDFS by defining semantics for its language constructs The exactness properties of these institutions are discussed Finally Section 4 concludes the paper and discusses future works 2 PRELIMINARIES 2 1 RDF FRAMEWORK The Semantic Web is a vision as the new generation of the current Web in which information is semantically marked up so that intelligent software agents can autonomously understand process and aggregate data This ability is realized through the development of a stack of languages as depicted by
16. uhl Germany 2005 GOGUEN J BURSTALL R Institutions Abstract Model Theory for Specification and Programming Journal of the Association for Computing Machinery 39 1 pp 95 146 1992 GOGUEN J RO U G Institution Morphism Formal Aspects of Computing 13 pages 274 307 2002 HAYES P RDF Semantics http www w3 org TR rdf mt February 2004 HORROCKS I PATEL SCHNEIDER P F BOLEY H TABET S GROSOF B DEAN M SWRL A semantic web rule language combining OWL and RuleML http www daml org 2004 11 fol rules all html November 2004 HORROCKS I PATEL SCHNEIDER P F van HARMELEN F From SHIQ and RDF to OWL The making of a web ontology language Journal of Web Semantics 1 1 7 26 2003 KLYNE G CARROLL J editors Resource Description Framework RDF Concepts and Abstract Syntax W3C Recommendation http www w3 org TR rdf concepts 2004 LUCANU D LI Y F DONG J S Web Ontology Verification and Analysis in the Z Framework Technical Report TR 05 01 University Alexandru Ioan Cuza of Iasi Romania January 2005 fattp thor info uaic ro tr tr05 01 ps PATEL SCHNEIDER P F HAYES P HORROCKS I editors OWL Web Ontology Semantics and Abstract Syntax ttp www w3 org TR 2004 REC owl semantics 20040210 2004 van HARMELEN F PATEL SCHNEIDER P F HORROCKS I editors Reference description of the DAML OIL ontology markup language ttp www daml org 2001 03 ref
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