Comparison of knowledge during the assembly process of learning objects

Abstract

This paper describes the conceptual framework OntoGlue for the assembly of learning objects (LOs). To permit a coherent assembling process from the point of view of requirements and competencies, OntoGlue enhances the definition of LOs by including associated knowledge (i.e. requirements and competencies) in their conceptual data schema. This associated knowledge is defined in terms of classes of educational ontologies (used as taxonomies), possibly related by mappings. There are several advantages associated with the OntoGlue approach. Firstly, it provides an enhanced description of the LOs, which permits their search and reuse by considering requirements and competencies. Secondly, during the assembly process of two LOs, OntoGlue checks that the competencies of the first LO cover the requirements of the second LO, guaranteeing a coherent assembling process from the requirements and competencies’ point of view. Thirdly, OntoGlue automatically calculates the meta-data of the resulting assembled LO. Finally, the definition of the associated knowledge in terms of classes of ontologies, possibly related by mappings, permits an advanced comparison of requirements and competencies during assembly and search processes.

Appendices

Appendix A: Algorithm coveredKnowledge

This Appendix provides an algorithm which calculates the knowledge covered by a class c. In order to define this algorithm some functions must be defined.

Definition 9.1

(directReachablesC) This function calculates all the classes that can be directly reached from class c via the application of any mapping with class c in its domain.

$$\begin{array}{rcl} directReachablesC: \Omega_{Classes} & \to & 2^{\Omega_{Classes}} \\ c & \to & \bigcup\limits_{m \in \{m' \in \Omega_{mappings} | isDefined(m',c)\}} m(c) \end{array}$$

Algorithm coveredKnowledge

This algorithm calculates the knowledge covered by a class. It has a functional appearance in order to be as language-independent as possible. This function bears in mind the transitive closure for the relation of specialization and for the relations induced by mappings. It is defined in: $coveredKnowledge: \Omega_{Classes} \to 2^{\Omega_{Classes}}$.

To calculate coveredKnowledge(c) we have to define the following sets:

$$\begin{array}{rcl} Reachables_0 & = & \{c\} \in 2^{\Omega_{Classes}} \\ Subclasses_0 & = & subClasses(c) \in 2^{\Omega_{Classes}} \\ Currents_0 & = & \left(Reachables_0 \bigcup Subclasses_0\right) \in 2^{\Omega_{Classes}} \end{array}$$

The function coveredKnowledge uses the recursive function calculateC to carry out its functionality. Thus, coveredKnowledge(c) = calculateC(Currents ₀,Reachables ₀, Subclasses ₀).

The function calculateC is defined below. This function keeps calculating the transitive closure of both relationships by considering the new reachable classes from the current subclasses, and the new subclasses of the new reachable classes.

Definition 9.2

(calculateC) The function $calculateC: 2^{\Omega_{Classes}} \times 2^{\Omega_{Classes}} \times 2^{\Omega_{Classes}} \to 2^{\Omega_{Classes}}$ is defined as follows (k ≥ 0):

calculateC(Currents _k , Reachables _k , Subclasses _k ) =

• calculateC(Currents_k + 1, Reachables_k + 1, Subclasses_k + 1), if Reachables_k + 1 ≠ ∅

• Currents _k , if Reachables_k + 1 = ∅

Where:

$$\begin{array}{rcl} Reachables_{k+1} & = & \Bigg(\Bigg(\bigcup\limits_{c \in (Reachables_k \bigcup Subclasses_k)} directReachablesC(c)\Bigg) \setminus Currents_k\Bigg) \in 2^{\Omega_{Classes}} \\ Subclasses_{k+1} & = & \Bigg(\Bigg(\bigcup\limits_{c \in Reachables_{k+1}} subClasses(c)\Bigg)\setminus Currents_k\Bigg) \in 2^{\Omega_{Classes}} \\ Currents_{k+1} & = & \left(Currents_k \bigcup Reachables_{k+1} \bigcup Subclasses_{k+1} \in\right) 2^{\Omega_{Classes}} \end{array}$$

The set Reachables _k keeps count of the transitive closure of the relation image of of a certain class by all the mappings that have this class in their domain. The set Subclasses _k keeps count of the transitive closure of the relation subclass of of a certain class. Likewise, the set Currents _k keeps count of the original class and its transitive closures by both relations. Finally, note that although from a theoretical point of view the length of coverage could be potentially infinite, in practice (i.e. in the context of a specific computing system) this length is necessarily finite.

Appendix B: Algorithm sufficientKnowledge

This appendix provides an algorithm which calculates the sufficient knowledge for a class c. In order to define this algorithm some functions must be defined.

Definition 10.1

(directReachablesS) As in the case of function directReachablesC, this function calculates all the classes that can be directly reached from class c via the application of any mapping with class c in its domain. As we mention above, now it is necessary both to calculate the reachable classes and to take into account the representations of these classes across mappings. For this reason, this function takes into account the classes into which, possibly, a class can be split due to the mappings.

$$\begin{array}{rcl} directReachablesS: 2^{\Omega_{Classes}} & \to & 2^{2^{\Omega_{Classes}}} \\ directReachablesS(\{c_1, c_2,\dots, c_n\}) & = & \bigcup\limits_{m \in \{m' \in \Omega_{mappings} | \forall i, 1 \leq i \leq n, isDefined(m', c_i)\}} \Bigg\{\bigcup\limits_{c_i \in\{c_1, \dots, c_n\}} m(c_i)\Bigg\} \end{array}$$

Algorithm sufficientKnowledge

This algorithm calculates the sufficient knowledge for a class. It has a functional appearance in order to be as language-independent as possible. This function bears in mind the transitive closure for the relation of generalization and for the relations induced by mappings. It is defined in: $sufficientKnowledge: \Omega_{Classes} \to 2^{2^{\Omega_{Classes}}}$. In order to define function sufficientKnowledge(c), the following sets are defined:

$$ \begin{array}{rcl} Reachables_0 & = & \{\{c\}\} \in 2^{2^{\Omega_{Classes}}} \\ Superclasses_0 & = & superClasses(\{c\}) \in 2^{2^{\Omega_{Classes}}} \\ Currents_0 & = & \left(Reachables_0 \bigcup Superclasses_0\right) \in 2^{2^{\Omega_{Classes}}} \end{array} $$

The function sufficientKnowledge uses the recursive function calculateS to carry out its functionality. Thus, sufficientKnowledge(c) = calculateS(Currents ₀, Reachables ₀, Superclasses ₀).

This function keeps calculating the transitive closure of the two relationships by considering the new reachable classes from the current superclasses, and the new superclasses of the current reachable classes.

Definition 10.2

(calculateS) Function $calculateS: 2^{2^{\Omega_{Classes}}} \times 2^{2^{\Omega_{Classes}}} \times 2^{2^{\Omega_{Classes}}} \to 2^{2^{\Omega_{Classes}}}$ is defined as follows (k ≥ 0):

calculateS(Currents _k , Reachables _k , Superclasses _k ) =

• calculateS(Currents_k + 1, Reachables_k + 1, Superclasses_k + 1,), if Reachables_k + 1 ≠ ∅

• Currents _k , if Reachables_k + 1 = ∅

Where:

$$ \begin{array}{rcl} Reachables_{k+1} &=& \Bigg(\Bigg(\bigcup\limits_{X \in (Reachables_k \bigcup Superclasses_k)} directReachablesS(X)\Bigg) \setminus \\ && Currents_k\Bigg) \in 2^{2^{\Omega_{Classes}}} \\ Superclasses_{k+1}& =& \Bigg(\Bigg(\bigcup\limits_{X \in Reachables_{k+1}} superClasses(X)\Bigg) \setminus Currents_k\Bigg) \in 2^{2^{\Omega_{Classes}}} \\ Currents_{k+1} &=& \left(Currents_k \bigcup Reachables_{k+1} \bigcup Superclasses_{k+1}\right) \in 2^{2^{\Omega_{Classes}}} \end{array} $$

The set Reachables _k keeps count of the transitive closure of the relation image of of a certain class by all the mappings that have this class in their domain. The set Superclasses _k keeps count of the transitive closure of the relation superClass of of a certain class. Likewise, the set Currents _k keeps count of the original class and its transitive closures by both relations.