Modelling and Implementation Methodology in the IoT/LD Approach to the Semantic Web

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1 Modelling and Implementation Methodology in the IoT/LD Approach to the Semantic Web Terje Aaberge Western Norway Research Institute, P. O Box 169, 6851 Sogndal, Norway Abstract. The paper presents a modelling and implementation methodology compatible with the Internet of Things and Linked Data paradigm. Linked data are naturally depicted by a directed graph whose nodes represent Things and edges or arrows relations. The conceptual models of intensional interpretations of first order formal languages are directed graphs. It is therefore natural to base the description language of the domain represented by a linked data structure on an intensional interpretation. In this paper I outline how to define an intentional interpretation of an object language and how to construct an object metalanguage. The construction is canonical; moreover, the metalanguage has the characteristics of an ontology language. I use this to show how to represent linked data on a machine readable format such that they more easily and effectively can be exploited by smart human oriented apps. Keywords: semantic web, internet of things, linked data, iot, ld, lod, semantic, modelling 1. Introduction The Internet of Things (IoT), in the context of the Semantic Web, is the idea of referring to things of the world on the Web by Universal Resource Identifiers (URI). A URI comes as dereferenceable URI or URN. Technically they both function as IDs, though one denote the Web-location (URL) or Webidentifier of a resource and the other serves as its name. There are two kinds of resources; those that have a digital representation located on the Web and things of the world that can only be referred to. For example, a digital representation of a book may be stored on the Web and located by a URL and it may have a unique name (URN), its isbn-number. On the other hand, things of the world, i.e. physical objects, persons, towns, services etc. can only be referred to by URIs. Linked Data (LD) is a method of publishing loosely connected data. It builds upon standard Web technologies, such as HTTP and URIs, but rather than using them to serve web pages for human readers, allow data from different sources to be connected and queried, thus enabling automated access to information [1,]. A set of URIs 1 representing objects, relations between them and information resources about them is an example of a set of linked data. It is a concrete realisation of a conceptual model of the corresponding domain of objects and information resources pictured by a directed graph (Figure 1): 1 To simplify the presentation throughout I will not insist on the difference between the dereferenceable URI identifying an individual and the URL locating an index card describing the individual, though this difference is important both conceptually and technically.

2 where the relations between documents and individuals are of one kind, About. The other relations are internal to the object sub domain. The standard methodology of LD does however, not exploit the graph as the conceptual model on which to base the semantics of a formal description language of the domain. This is what will be done in the following. I will outline a modelling methodology by applying intensional formal semantic for which the conceptual models are precisely directed graphs []. The result will then be a modelling methodology taking into account the specificities of the IoT/LD and which moreover, distinguishes modelling from the encoding [3]. An intensional interpretation is represented by maps from the domain to the vocabulary of the language: an isomorphism that maps the individuals (or relations) to names and observables that maps individuals (or relations) to predicates. The observables are identified by mutual exclusion of properties. Two properties that cannot simultaneously be possessed by an individual are represented by predicates belonging to the range of the same observable; an individual cannot at the same time be red and green, colour is therefore an observable. Other observables are weight, position in space, temperature etc. The interpretations of physical theories are intensional and the notion of observable is taken from physics. A philosophical justification of intensional semantic is moreover, found in Wittgenstein s Tractatus [4] where the picture theory points to an intensional interpretation of logic. Intensional interpretations conceive the structure of language as being imposed by the structure of the domain, contrary to extensional (model theoretic) interpretations for which the structure of language imposes the structure of the domain. In fact, the conceptual model of the domain in the latter case is a set consisting of the individuals, sets of individuals, sets of pairs of individuals and the interpretation is represented by a map (isomorphism) that to a name associates an individual and to a predicate its extension, i.e. to a one-place predicate a set of individuals and to a two-place predicate a set of ordered pairs of individuals etc. The extensional interpretation can be derived from an intensional interpretation. By taking the inverse images of the predicates with respect to the observables we get the extensions of the predicates. The opposite construction is however, not possible. We cannot reconstruct an intensional interpretation from an extensional one. The reason is that intensional definitions contain more information than extensional enumerations. This is an important bonus when applying intensional interpretations. The semantic structure of an intensionally interpreted object language is expressed by a set of commutative diagrams. Their instances provide a conceptual (directed graph) model for the domain of the object metalanguage which thus has a canonical construction. The truth conditions, being related to the commutativity of the diagrams, can be formulated in this metalanguage without the generation of a hierarchy of languages. The metalanguage is by construction an ontology language. The linguistic representation of all the instances of the diagrams gives a complete description of the individuals of the object domain and the semantic relations of the intensional representation. It can be implemented as four separate components, ontology index knowledge base domain model The index is constituted by a set of index cards, one for each individual and located by a URL redirected to from the URI of the individual. The index card lists all atomic sentences attaching properties and attributes to the individual. The ontology is a set of implicit definitions (axioms) and terminological (intensional and extensional) definitions that picture structural properties of the domain and put restrictions on the possible interpretation of the vocabulary. The knowledgebase is constituted by the collection of information resources properly described and the domain model is a linguistic representation of the conceptual model. In the following I will give an outline of the intensional interpretation of an object language, present a construction of the canonical metalanguage, describe the above four components and explain their role in information retrieval.. Object language Let L ( N N V,P P ) D 1 1 stand for the object language for a domain D modelled as a directed graph (Figure 1). N 1 denotes the set of names of individuals, N the set of names of relations, V the set of variables, P 1 the set of unary predicates, and P the set of binary predicates. The language is endowed with an ontology.

3 The names of the individuals are given by a map ν : ν:d N1 N ;d ν( d) (1) drd s t ν ( drd s t) that to an individual d or relation drd s t in the domain d n n,n by D associates the name n by ν ( ) = or ( s t) ( drd) ( n,n) ν =, where n s and n t are the names s t s t of the individuals that constitutes source d s and target dt of the relation. I will assume, for simplicity, that ν is an isomorphism; then, there is a unique name for every individual (or relation) and there is exactly the same number of names as of relations and individuals. If there is more than one relation between two individuals one will have to index relation names with respect to the kind of relation. Each observable δ relates an individual d D or relation drd s t Dto a predicate δ δ 1:D P;d 1 1( d) :D P ;d s rd t ( d s rd t) δ δ () Moreover, for each δ there exists a unique map π defined by the condition of commutativity [5] of the diagram: πi Ni Pi ν δi D ( ) () s t i.e. π ν (). = δ., d, d rd D (3) where i=1 for properties and i= for relations. The diagram simulates measurements determining atomic facts and thereby assigns a property to an individual or relation to a pair of individuals. They are formulated as atomic sentences by the juxtaposition of names and predicates, e.g. the sentence pn where n is the name of the individual d and p the predicate referring to a property of d expresses a fact about d d d d n π n = p. The ( ) ( ) ( ) ( ) iff πν ( ) =δ, for ν ( ) = and ( ) instances ( d) d πν =δ of the commutativity condi- tions are therefore truth condition for the atomic object language sentences. An instance equates a proposition about the system d with a statement of the result of a measurement on d with respect to the observable δ. The instances of the diagrams thus express the content of object language sentences and of the semantic relations determining the interpretation of the object language. 3. Ontology An ontology for an object language is a set of axioms intensional definitions extensional definitions An axiom is an implicit definition that relates the primary terms of the vocabulary of the object language. The axioms picture structural properties of the domain and express restrictions on the possible meaning of predicates. The intensional and extensional definitions are terminological. They define secondary predicates, from the primary terms 3, that serve to facilitate the discourse, e.g. instead of having to repeat the properties that an individual must possess to be of a certain kind an intensional definition will introduce a predicate to denote the kind. An intensional definition of a predicate (definiendum) is thus the conjunction of atomic sentences (definientia) stating which properties that an individual must possess for the predicate to apply. When the meaning of the definientia is given the definition explains the meaning of definiendum. An extensional definition of a predicate on the other hand, is simply the list of the names of the individuals that constitute its extension. When the individuals referred to are known, the extension of the predicate representing its meaning is given. From an intensional definition of a predicate an extensional one can be derived; the extension of the predicate is the class of individuals that satisfies definientia in the intensional definition. It follows that the interpretation of the vocabulary is determined by the interpretation of the primary vocabulary. Though an ontology does not concern the semantics per se, the construction of an ontology has to be made in an interpreted language and evaluated accor- I use standard mathematical notation. denotes a map and points to the value of the map for a given argument. 3 Or rather, all terminological definitions can be expressed by means of the primary terms.

4 dingly. Its construction should therefore be justified by (formal) semantic considerations. 4. Object metalanguage/ontology language The relation between the object metalanguage and its domain is pictured by Figure : Figure where the direction of the arrows indicates intensional semantic [5]. The metalanguage for the object language is denoted L ( M M,Q) G 1 where the domain G consists of the set of instances of the diagrams (3) and of variables, sentences and formulae as isolated nodes; M=L 1 D, the names 4 of the nodes; M, the names of the relations (arrows d n etc. in (3)); and Q, the predicates of the metalanguage. The names of the individuals, relations between individuals, terms, sentences, and relations between these objects in the metalanguage, are given by the naming map (4), η:g M1 M ; d η ( d) = d n η ( n) = n i i (4) 4 I apply the convention that the string of symbols representing a name, a predicate, a sentence, a formula, a node or a relation serves as its name. This is justified by the fact that a sentence as symbolic entity embodies a syntactic form and expresses a proposition. The proposition belongs to the language and the syntactic form to the metalanguage. Since the sentences of the language are only mentioned and not used in the metalanguage the convention does not lead to problems. ( ( ) ) ( ) 1( ) ( ) = (( n,n s t),p) ( ) ( ) ν d = n η( ν d = n) = ( d,n) ( π n = p) η( π ( n) = p) = ( n,p) ( π n,n s t = p) η( π ( n,n s t) = p) ( δ d = p) η( δ ( d) = p) = ( d,p) ( δ drd s t = p) η( δ ( drd s t) = p) = ( drd,p s t ) where ( d) ν = ndenotes relations (arrows: d n ) etc. Predicates of the metalanguage are Individual, Property, Type, Class, Name, Relation and Sent (for sentence) that express properties of the elements of its domain and NameOf, π i Of and δ i Of that express (semantic) relations. One should notice that since an individual is identified by a URI and a URN one needs two predicates in the metalanguage to distinguish between them, Individual to denote URIs of individuals and Name to denote the URNs. Similarly, Relation and Property has been introduced in order be able to distinguish between URLs denoting kinds of relations and their linguistic representatives. Notice also that with respect to concrete applications δ i Of will be expressed in terms of the name of the observable, for example ColourOf. I have used Property and Class as a reminder of the OWL vocabulary. Other words might be more natural in the present intensional setting. In first order logic notation a complete representation of the object language atomic sentence pn in the metalanguage, including statements about syntactic values and semantic relations, is ( ) ( ) ( ) ( ) ( ) ( ) Individual d Name n NameOf n,d Class p δ1of p,d π 1Of p,n (5) Thus, the truth conditions for atomic object language sentences are of the form ( ) ( ) ( ) ( ) δ1of ( p,d) ( ) Individual d Name n Class p NameOf n,d π1of p,n Tpn (6) for T being the truth predicate; similarly for the object language sentences expressing relations. The left hand side of this implication is the linguistic representations of an instance of the commutativity conditions (3). This shows that the translation of the con-

5 ceptual content of the object language sentence pn to the metalanguage is π 1Of ( p,n), e.g. the sentence n is red to red is the colour of n. For a fixed d the sentences of the form (7) whose atomic components satisfy (8) are the descriptive sentences of the individual represented by the URI d. Reasoning takes place in the metalanguage, on the basis of the ontology formulated in the object language, by means of deduction rules like modus ponens, ( ) ( ) ( ) ( ) Sent h1 Sent h Th1 T h1 h Th (7) It depends solely on the syntactic structure of the object language sentences, not on their meaning. Deductions will not be discussed in this paper. 5. Index The sentences of L G are uniquely expressible in a N3 representation by blocks of RDF triples 5. Thus, (5) translates to ml:d owl:type owl :Individual ol:n owl:type ml :Name ol:p owl:type owl:class ol:n ml:nameof ml:d ol:p ml: δ 1 Of ml:d ol:p ml: π 1 Of ol:n and ml:r owl:type ml:relation ol: ( n,n s t) owl:type ml:name ol: p owl:type owl:property ol: ( n,n s t) ml:nameof ml: nrn s t ol: p ml: δ Of ml: drd s t ol: p ml: π Of ol: ( n,n s t ) ml: d s ml: r ml: d t (8) (9) is a translation of a corresponding sentence involving a relation. The prefixes ol, ml, and owl refer to the object language, metalanguage and OWL namespaces respectively. The triples of the kind 5 I will refer to the corresponding representation language as the ontology language. ml:d owl:type owl :Individual ol:n owl:type ml :Name ol:p owl:type owl:class ol:n ml:nameof ml:d are sentences in the ontology language that express the syntactic role of the terms of the vocabulary. The triples ol:p ml: π 1 Of ol: n ol: p ml: π ol: ( n,n ) Of s t are sentences in the ontology language that describe properties and relations of the individual referred to by the URI d s, and the triples ol:n ml:nameof ml:d ol:p ml: δ 1 Of ml:d ol: ( n,n s t) ml:nameof ml: nrn s t ol: p ml: δ Of ml: drd s t are sentences in the ontology language that describe semantic relations. And finally, triples of the kind ml: d s ml: r ml: d t are sentences in the metalanguage that express relations between individuals, i.e. they link the descriptions of the individuals of the domain. The index card for an individual d is constituted by the set of blocks like (8) which satisfies (6). The individuals of the object domain are thus represented as bundles of properties 6. The index is the set of index cards. 6. Knowledge base Web documents generated from a content management system (CMS) are located by a URL and named by a unique name (identifier). Though this might seem unnatural, the way they are represented shows that they should be considered to be a bundle of properties. A document in a CMS is named by an ID and described the by the values of the field names, e.g. Title, Abstract, Author, and Text. The values are the titles, author names, abstracts, and texts. If the title, author, abstract and text of a document named 6 In many practical cases the objects are quite complex and it will not be possible to pursue this ideal, i.e. describe the objects as a bundle of properties, but make more indirect descriptions.

6 n is, respectively, x, y, z and u, then the object language description of the document is xn yn zn un (10) The semantic structure of the object language is defined by the commutative diagrams (3) where D is the domain of documents, N the identifiers, P the predicates (i.e. in my example, the sets of titles, author names, abstracts, and texts): π N P ν δ D i.e. ( ) It follows that a N3 description of a document is 7 ml: d owl:type owl :Individual dl: n owl:type ml:name dl: x owl:type owl:class dl: n ml:nameof ml: d dl: x ml:titleof ml: d dl: x ml:titleof dl: n dl:y owl:type owl:class dl:y ml:authorof ml: d (1) dl:y ml:authorof dl: n dl:z owl:type owl:class dl:z ml:abstractof ml: d dl:z ml:abstractof dl: n dl:u owl:type owl:class dl:u ml:textof ml: d dl:u ml:textof dl: n that also describe the formal meaning of its elements. The set of such descriptions of the documents is an N3 representation of the knowledge base, i.e. a description (and representation) in an ontology language. 7. Domain model ( ) ( ) πν d =δd, d D The domain model is the linguistic representative of the conceptual graph model (Figure 1). It consists of a set of N3 triples ml: d s ml: r ml: d t that represent the relations between individuals, and the set of triples 7 dl stands for the document language name space. (11) ml: d ml:about ml: d that represent the relations between documents and individuals. In addition we have to declare the syntactic role of the terms, ml: d owl:type owl:individual ml: d owl:type owl:individual ml: r owl:type ml:relation ml:about owl:type ml:relation The domain model contains the structural information about the domain. For reasons that will become clear it is most appropriate for the construction of apps to keep this information separated from the information in the index cards. 8. Methodology The modelling and implementation methodology to establish the description language of a domain is the canonically given stepwise procedure: 1. delimit precisely the object domain. make a conceptual model of the object domain 3. identify a vocabulary for the description of the elements of the object domain 4. establish an ontology for the object language 5. describe the individuals of the domain in the object language 6. construct the object metalanguage 7. express the linguistic representatives of the instances of the diagrams (3) in the metalanguage 8. constitute the knowledge base 9. establish a document description language and document metalanguage for the knowledge base 10. encode a. ontology b. index cards c. knowledge base d. model of the hole domain (Figure 1) The steps 1-3 are preliminary but important and practically inseparable. The choice of conceptual model of the domain determines a unique interpretation. Step 4 must take into account what an ontology is, i.e. the difference between axioms and terminological definitions. To describe the individuals (step 5) is an empirical task. The metalanguage is canonically constructed (step 6). Step 7 results in descriptions of the individuals and an account of the conceptual model of the object domain. 8 and 9 are in prac-

7 tice inseparable. The last step consists in formulating a machine readable representation of the ontology, index, knowledge base and model of the total domain. The methodology strictly distinguishes between modelling and coding of the knowledge base from an abstract point of view. 9. Retrieval mechanism The retrieval mechanism is search through the ontology. This means that the search term is compared with the ontology. If it is a name or a primary term it will pass through and is then compared with the content of the index cards. This determines a set of URIs representing individuals that satisfy the search criteria. The domain model then pick the documents (or sub parts of documents) that are about these individuals, i.e. have the relation About to the individuals. If it is a secondary predicate defined by an intensional definition then the search engine is comparing the definition of the predicate with the descriptions of the individuals on the index cards. It then picks the individuals that satisfy the definitions. The next operation is identical to the former case. The retrieval mechanism may include deductions. The problem that has triggered the research is how to publish on the Web in such a way that the information is easily exploitable by third party applications combining information from different sources. Clearly, such applications involve many issues, from translations between vocabularies and ontology alignment to privacy and copyrights. Modulo these issues, the paradigm of IoT/LD opens up for many possible applications, technically quite feasible at present. In fact to build such an application one may follow the methodology in 8. This means to constitute a domain by choosing URIs representing individuals and documents from several published repositories. One then establishes a domain model and constructs an ontology adapted to the tasks to be performed by the application. If indexes do not exist, an index must be established from scratch. The ontology, domain model and the index are implemented in the application which will retrieve information from the published knowledge bases by means of the mechanisms already described. The smartness of the applications depends on the semantic information available in the index cards and on the structure of the ontology. How it can present the information will depend on the way documents are described. 11. Concluding Remarks In this paper I have presented a modelling and implementation methodology for IoT/LD based on intensional semantic. Given the structure of the Web and the specificities of the IoT/LD this seems to me to be a more appropriate choice of semantic theory on which to base the modelling than extensional (model theoretic) semantic. Coherently pursued it leads to a canonical construction of a metalanguage and associated ontology language in which one may formulate all the true sentences of the individuals, assess the syntactic role of the terms of the vocabulary and express the semantic relations. The methodology uses extensively diagrams that are easy to read and interpret such that the structure of the framework is clearly exposed; and they are easy to apply, thus facilitating modelling and encoding. 10. Applications References [1] Bizer, Ch., Heath, T. and Berners-Lee T.: Linked Data - The Story So Far, lee-ijswis-linked-data.pdf [] Bizer,., Cyganiak, R. and Heath, T.:How to Publish Linked Data on the Web, [3] Aaberge, T., 009. On Intensional Interpretations of Scientific Theories, In: Münz, V., Puhl, K. and Wang, J. (eds.): The 3nd International Wittgenstein Symposium, LWS, Kirchberg [4] Kuhn, W., 010. Modeling vs encoding for the Semantic Web, Journal of Web Semantics 1 (11 15) [5] Wittgenstein, L., Tractatus logico-philosophicus, Routledge and Kegan Paul, London [6] Aaberge, T., 010. Picturing Semantic Relations, In: Heinrich, R., Nemeth, E. and Pichler, W. (eds.): The 33d International Wittgenstein Symposium, LWS, Kirchberg [7] Lawvere, F. W. And Schanuel, S. H., Conceptual Mathematics, Cambridge University Press, Cambridge