Binary Relations in Educational Ontologies Seremeti Lambrini Aggelopoulou Nikolitsa Pierrakeas Christos Kameas Achilles Mathematician, Researcher Mathematician, Researcher, Lecturer, Dept. of Business Associate Professor, School Hellenic Open University Hellenic Open University Administration, of Science and Technology, (HOU) (HOU) Technological Educational Hellenic Open University +30 2610 367 963 +30 2610 367 963 Institute (TEI) of Western (HOU) seremeti@cti.gr naggelop@eap.gr Greece and +30 2610 367 735 Hellenic Open University kameas@eap.gr (HOU) +30 2610 367 730 pierrakeas@eap.gr ABSTRACT conceptualization is based on subjective statements of the kind Educational ontologies are classified into οnthologies of Student (subject, verb, object) triples that experts provide, it describes the Learning Outcomes (SLO), Learning Objects (LO) and Cognitive basic concepts of each CoD and the relations among them Domains (CoD). In contrast to the conceptualization and between concepts. The classification of these statements in a implementation of SLO and LO ontologies, based on standards specific ontology could help to avoid polysemy and ambiguity of available in the literature, the CoD ontologies involve subjectivity relations used to describe CoD. These relations are binary and derived from the analysis of basic concepts of each CoD and their formal representation by means of ontology will restrict the relational expressions that experts use in order to associate these use of inappropriate definitions of relations during the basic concepts. This subjectivity can create inconsistent implementation of CoD ontologies. ontologies. The aim of this paper is to establish a set of binary Several existing ontology population techniques able to extract relations to be used in the official representation of CoD. These arbitrary semantic relations from text corpora focused exclusively relations consist of triples (subject, verb, and object) and can be in binary relations. Ontologies present binary relations (called classified into a Binary Relation (BR) ontology. properties in OWL). In this paper, we conceptualize an ontology Binary Relations Keywords (BR), which officially represents the relations needed to describe Educational Ontologies, Binary Relations CoD concepts, under the HOU framework. The ultimate goal is to provide a minimum set of binary relations that are necessary to 1. INTRODUCTION implement CoD ontologies. In this way, experts should restrict to In the last ten years technology offers opportunities to the proposed binary relations in order to conceptualize CoD. Universities to reconsider how to extend the teaching, to students The remainder of the paper is organized as follows: Section 2 beyond the traditional teaching and not limited by boundaries. explains the need for formally describing relational expressions Hellenic Open University (HOU) aims to bring together leading used in CoD’s description. Section 3 focuses on binary relations technologies and pedagogical approaches to implement e-learning by giving their mathematical definition and their usage in environments, specialized to the needs of adult users with ontology engineering and Section 4 describes related work for different knowledge background, skills and biases. In the binary relations. Section 5 describes the main points of BR realization of this objective, ontologies play a key role. They are ontology engineering, and Section 6 concludes the paper. machine readable representations of the content of educational material, users’ profiles, and taxonomy of learning outcomes, which enables to the creation of individualized learning paths [1]. 2. COGNITIVE DOMAIN (CoD) For this purpose the educational ontologies constructed for HOU ONTOLOGIES [2], [3], [4], [5], [6], can be divided into ontologies for Learning Initially, domain experts define the basic concepts of cognitive Objects (LO), ontologies for Student Learning Outcomes (SLO) domain and create relationships between basic concepts of CoD and ontologies for Cognitive Domains (CoD). Regarding the ontologies. Afterwards, they develop concept maps based on the engineering of SLO and LO ontologies, problems do not exist. concepts and relationships that have defined. [6] The conceptualization of LO ontologies is based on standards available in the literature such as the official description of IEEE These relations are been expressed through individual relations, LOM standard [7]. Tthe conceptualization of SLO ontologies is known as properties. Properties are divided as object and datatype based on the Bloom’s taxonomy [8], a widely accepted taxonomy properties. Datatype properties link an individual to a specific of learning domains that is often used in the design of educational value, namely an XML Schema datatype or an RDF literal. [2] processes. −1 More specifically, the pair-wise inverse properties X and X In contrast, when designing CoD ontologies, because their are used to declare a parent-child relation between two concepts. They can be a) functional, meaning it is a property that can have Copyright © 2015 for the individual papers by the papers' authors. Copying permitted only for private and academic purposes. only one (unique) value y for each instance x, b) transitive, meaning that if a pair (x,y) is an instance of P, and the pair (y,z) is In: A. Bădică, M. Colhon (eds.): Proceedings of the 2015 Balkan also instance of P, then we can infer the pair (x,z) is also an Conference on Informatics: Advances in ICT 45 instance of P or c) symmetric meaning if the pair (x,y) is an criterion, which is described in subsection 5.2, of the extracted instance of P, then the pair (y,x) is also an instance of P. The binary relations. This can facilitate ontology experts and domain instance property connects class with its members, whilst Y experts to avoid mistakes in coding CoD. correlates any individual with a certain modifier. Finally, to define The resulting ontology can also promote interoperability of the particular relation of a concept with a reserved keyword, the Z educational ontologies and support automated reasoning in e- property is used. learning environments. Figure 1 presents as an example, based on a part of the concept map that has been created for the cognitive domain CoD: PLI30 3. BINARY RELATIONS as we can see in [6]. The relational expressions that domain experts use to provide the This conceptual map represents 32 identified basic relevant formal description of a CoD as we saw previously are sentences concepts (see the nodes of Fig. 1) and 5 relations (see the edges of that simply indicate a relation between two basic concepts of the Fig. 1; for example, “Associative Network includes Node”). same cognitive domain, without any further information. These sentences are typically described by binary relations. 3.1 Definition of Binary Relations We will give a formal definition for binary relation. Binary relations are important, since relations of arity greater than 2 can be studied in terms of binary relations. Mathematically speaking, if X and Y are non-empty sets, a binary relation from X to Y is a subset R ⊆ X × Y . We write ( x, y ) ∈ R or xRy to denote that ( x, y ) ∈ X × Y and we say that X is related to Y through R . For example, in the accounting CoD, the natural language expressions “Slot represents Concept”, “Slot represents Object”, “Slot represents Event” can be formulated as the binary relation R = {represents} from the set X = {Slot} to the set Y = {concept , object , event} . For some binary −1 relation R ⊆ X × Y , we can define its inverse R ⊆Y×X , −1 such that yR x ⇔ xRy . An interesting point to consider about binary relations is their composition which is defined as follows: let R ⊆ X × Y and S ⊆ Y × Z binary relations. Their composition is a binary relation S oR ⊆ X ×Z defined by x ( S o R ) z ⇔ ∃y ∈ Y such that xRy and ySz . Figure 1: Part of the concept map expressing concept “Frame We are also interested in certain properties satisfied by these Systems” relations, such as: (a) reflexivity ( xRx for all x in X ), (b) It is necessary to formalize the terms of relations (verbs), which are binary relations, in a rigorous machine readable format, with symmetric ( xRx′ implies x′Rx for all x, x′ in X ), and (c) the aim of understanding the knowledge expressed by domain transitivity ( x′′Rx′ and x′Rx imply x′′Rx for all x, x′, x′′ experts in concept maps. There are several ontology development tools that implement concept model ontology in different in X ). languages. Ontology experts can apply Protégé [9] to convert a The main point is that to uniquely describe a relation R , the conceptual map produced by the experts into a formal model by using the formal OWL language. To evaluate the graphic collection of all ordered pairs ( x, y ) such that x is related to representations of the concepts of the course through the concept maps in OWL.Then, they can use off-the-shelf automated y by R , must be listed. reasoning tools. 3.2 Binary Relations in Ontologies Note that the same natural language relation (verb) can be used by The relations contained in ontologies are usually binary. They experts to connect different concepts in the same field or in have two arguments; the first is called the domain of relation, and different cognitive domains. One way to develop consistency and the second is called range. These relations are mainly related to clear standard definitions of relational expressions used in the classes of the ontology and usually initialized using the educational ontologies concerning CoD, is to develop an ontology knowledge from the domain representing the ontology. For providing definition and classification, according to a certain example, to express that “the x processor executes the y software”, 46 the relation “executes” should be designed and should have a 5. BR ONTOLOGY ENGINEERING class “Processor” as the domain and a class “software” as the The main questions arising when engineering the ontology of range. On occasion, the same relations used to relate classes, are binary relations used in the HOU context are: Which are the also used to express attributes of specific classes. These are also intended uses of the BR ontology? Which are the entities that the binary relations, where domain is a certain class and their require a unique categorization? According to what criterion? range is a datatype, such as string, number, etc. What kinds of binary relations are used in the literature? What In the case of n-ary relations, that is, relations which link an kind of relations can we formally describe? What are the individual to more than one individual or values are represented properties of the described relations? The BR ontology is by creating an intermediary entity that serves as the subject for the engineered according to commonly accepted engineering entire set of all relations [9]. In our approach, we refer only to methodologies, based on specification, conceptualization, binary relations, which are the most common type of relation implementation and evaluation phases, where all the questions mapping a single subject to a value. stated above are answered [10]. 4. RELATED WORK 5.1 Specification of the BR Ontology The most common type of relation is a binary relation that The CoD ontologies in the framework of HOU are designed to connects two concepts. Discussions for binary relations have been provide reference points for the expression of the basic concepts researched in [9], [11], [12], and [13]. The problem of of each cognitive object in a machine readable format. Their representing a binary relation is not new. construction is based on natural language statements gathered by In 2014 [9] Vinu, Sherimon, Krishnan and Tarkoni discuss the the domain experts, which are expressed in sentences of the form issues in modelling n-ary relations. They support that the main (subject, verb, object). These sentences of the kind “A -relation- elements of ontology are concepts, relations and individuals. W3C B” (where A and B are concepts belonging to the same CoD provides several patterns to represent n-ary relations. They ontology and “relation” symbolize connects for associating these examine the issues in n-ary relations, the concept of RDF concepts) can be considered as binary relations between semantic reification and provide an appropriate pattern to represent the n- concepts in a vocabulary that is specified for a certain cognitive ary relations. The examples of n-ary relations are taken from domain. Seafood Ontology they developed. In contrast to our work, they Our task is to develop a minimum set of coherently define binary focus on the issues of n-ary relations. It explains the ontology relations involved in the formal representation of cognitive languages followed by the n-ary relation, the issues in n-ary domains through ontologies and the scope to capture the relations relations,reification and its drawbacks and outline an appropriate currently expressed in the context of the CoD ontologies. This is pattern to represent the n-ary relations. important, since (a) the inability to distinguish relational Welty and Fikes in [11] discuss the standard approach to deal expressions which are close in meaning, results in an erroneous with relationships that change over time, such as OWL that are reasoning process, and (b) the polysemy of relational expressions biased towards binary relations. Their approach involves treating impedes interoperability between educational ontologies entities in the domain of discourse as four dimensional with developed in the HOU. temporal parts that participate in the relation, corresponds to and stablished ontological position in analytical metaphysics called 5.2 Conceptualization of the BR Ontology perdurantism. In the literature, binary relations are distinguished in the following Martin and Benard in [12] propose an ontology design pattern for three kinds. The categorization of binary relations based on their leading knowledge to represent knowledge in a more normalized domain and range. way. This pattern is: “using binary relation types directly derived from concept types, especially role types or types of process with • class, class : for example, statements such as the nominal expressions as names”. It provides an ontology deriving class “Slot” represents (relation) the class “Object” or relation types from concept types; this derivation reduces having the class “Slot” represents (relation) the class “Event”. to introduce new relation types. It explains, formalizes and illustrates the different parts of ABP (advocated best practice) and • ins tan ce, class : for example statements such as relates this practice to other ODPs (Ontology Design Patterns). the instance “current assets” includes (relation) the class In contrast to our work, in [13] Banek, Juric and Skocir introduce “requirements” or the instance “current assets” includes an unsupervised method for learning domain n-ary relations from (relation) the class “inventories” and Wikipedia articles. They claim that providing ontologies with n- ary relations instead of the standard binary relations built on • ins tan ce, ins tan ce : for example, statements subject –verb- object paradigm results in preserving the initial context of time, space, cause, reason that otherwise would be lost. such as the instance “unit of manure” contains (relation) They discuss the use of n-ary relations for discovering richer the instance “80 Kg N” , since they cannot be semantic context, the relation extraction process and the considered as sets of objects. evaluation of this approach. 5.2.1 BR Ontology Our work is consistent with Martin and Bernard in [12]. We By following our ontology engineering methodology [6], we attempted to define the relations created from concept maps as constructed an ontological model for the relations of the cognitive binary relations in order to enable us to better construction of domains in HOU. The BR ontology was constructed with the aid educational ontologies. of Protégé based on the most recent version of the Web Ontology Language (OWL) and W3C standard, OWL 2. 47 The main classes of the BR ontology (see in Figure 2) are: • the class “Relation”, which is divided into three different subclasses: “ClassClassRelation”, “ClassInstanceRelation” and “InstanceInstanceRelation” illustrates the main types of relations. Specific relations such as “Contains”, “Involves”, “Uses”, “Determines”, etc. are subclasses of the class “ClassClassRelation”. • the class “DomainRange”, which is divided into two subclasses: “Class” and “Instance”, and • the class “CognitiveObject” Figure 4: The binary relation “represents” from the BR ontology This structure categorizes the relation “Represents” as a binary relation with domain and range classes. It corresponds to a specific cognitive domain and has properties, such as transitive, functional and symmetric. Synonyms and description of its semantics are also provided. 5.2.3 Description of Instances in BR ontology The natural language statement “knowledge_representation_language_represents_sentence_of_ propositional_logic” is an instance of the class “Represents” of Figure 2: The class hierarchy of the BR ontology the BR ontology. Although this statement is understandable by humans, it has no meaning for a machine. Using the structure of 5.2.2 Description of Properties in BR ontology the BR ontology, the meaning of this statement can also become The various types of interaction among ontology concepts are machine readable. We can see the instance expressed through respective relations, known as properties (see “knowledge_representation_language_represents_sentence_of_ in Figure 3). propositional_logic” in Figure 5. We have defined six (6) object properties and five (5) datatype properties. More specifically, the class “Relation” relating with the class “CognitiveObject” with the object property correspondsTo, the class “InstanceInstanceRelation” relating with the class “Instance” with the object property hasDomainInstance etc. The data property isSymmetric determine if the “Relation” is symmetric or not. Figure 5. The statement “Knowledge representation language Represents Sentence of propositional logic” as an instance of the class “Represents” According to the structure of the BR ontology, the natural Figure 3: The properties hierarchy of the BR ontology language statement “Knowledge representation language The structure of the BR ontology, conceptualizing a specific Represents Sentence of propositional logic” is conceptualized as binary relation is depicted in Figure 4. an instance of the class “Represents”. 5.3 Implementation of the BR Ontology The idea behind the structure of the BR is that the various statements considered as instances of the relation can be considered as a binary relation, and are categorized depending on the domain and range. For example, an instance of the relation “Determines” implemented in Protégé [14] is depicted in Figure 6. 48 Figure 11: Competency question answers if the relation “uses_for_resolution” is Functional The second example is for a ClassInstanceRelation the relation represents. In the next Figures we can see the individual: “knowledge_representation_language_represents_sentence_of_ propositional_logic”. This individual hasDomain: knowledge_representation_ language (Figure 12), correspondsTo: pli31_CoD1 (Figure 13), hasLabel: αναπαριστώ (Figure 14), Figure 6. An instance of the class “Determines” implemented hasRange: sentence_of_propositional_ logic (Figure 15) and in Protégé isSymmetric relation (Figure 16). The BR ontology can be found at http://ontologies.eap.gr/webprotege/#Edit:projectId=4be4a475- b9ff-4b46-ab40-b884c0bf18fa. 5.4 The BR Ontology Figure 12: Competency question which answers what is the The BR has been assessed, using the same competency questions, domain of the relation “represents” as in the specification phase. The questions answered concern finding the inverse of a relation, its instantiations, its domain and range, etc. We present two examples of competency questions submitted to BR. The first example is for an InstanceInstanceRelation the relation usesForInstanceInstance. In the next Figures we can see Figure 13: Competency question answers to what cognitive the individual domain belongs the relation “represents” “backward_chaining_uses_for_resolution_conjuctive_normal_for m”. This individual hasDomain: backward_chaining (Figure 7), correspondsTo: pli31_CoD1 (Figure 8), hasLabel: χρησιµοποιεί (Figure 9), hasRange: conjuctive_normal_form (Figure 10) and isFunctional relation (Figure 11). Figure 14: Competency question answers which is the label of the relation “represents” Figure 7: Competency question which answers what is the domain of the relation “uses_for_resolution” Figure 15: Competency question answers what is the range of the relation “represents” Figure 8: Competency question answers to what cognitive domain belongs the relation “uses_for_resolution” Figure 16: Competency question answers if the relation “represents” is Symmetric 6. CONCLUSION In this paper we aim at systematically representing the binary Figure 9: Competency question answers which is the label of relations involved while coding CoD ontologies in the HOU the relation “uses_for_resolution” context, in order to avoid polysemy (the interpretation of a specific relation must be clear and unambiguous) and homonymy (different nomenclature may refer to the same relation). To this end, we have developed the BR ontology which is used to solve interoperability issues, as well as a reference point from Figure 10: Competency question answers what is the range of where a minimum set of binary relations, that are used in machine the relation “uses_for_resolution” readable relational expressions of cognitive objects are extracted. 49 7. ACKNOWLEDGMENTS of the 14th IEEE International Conference on Advanced This research described in this paper has been co-financed by the Learning Technologies, 2014, pp. 716-718. 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