Examples

The Ontograph visualization [Bens2014] in figure mobjects represents an extended modeling of the MObjects example by Neumayr et al. from [NeSc2008], page 9, figure 3. In their example, potency-based deep instantiation is used. The consequence of their use of potency is that ‘Additional abstraction levels for some domain concept cannot be introduced without requiring global model changes’. In our modeling we show that such global changes can be avoided: The ^Product class hierarchy contains at the top the classes that are related through »subClassOf. Starting with the class ^Porsche911, the vehicle domain contains a subhierarchy in which the classes are linked by »subModelOf. The following applies: ◊subModelOf ⊆ ◊subClassOf ⊆ ◊is. The inheritance of the Data Properties (DP) applies to the entire ^Product class hierarchy including the models. In classic Conceptual Modeling (CM), the class ^Car and the associated class ^Car_Model would be modeled in two parallel branches of the class hierarchy. ^Car_Model is then referred to as the PowerType (PT) of the base class ^Car with ^Car_Model = PT (^Car). Using our method of Meta-Level Modeling (MLM), these two classes were merged into the single class ^Car. The prerequisite for this is that the definition and instantiation of the data properties comply with the restrictions described in the section Data Property Types. It is always possible to return from MLM to CM, for example, by separating two classes ^Car and ^Car_Model from the one class ^Car. The two classes must then be connected by OPs such as (^Car_Model, ◊isCarModelOf, ^Car). Now we give concrete examples for the different types of data properties.

The graph in figure lubm2 represents the information in table 2 of [NgBo2015]. To quote Nguyen et al. [NgBo2015]: "A singleton property is defined as a property instance identifying the unique relationship between the subject and the object". The relationship between >ProfessorA and the universities >University1 and >University2 is represented by the two relators »worksFor#1 and »worksFor#2. So, we interpret the two relators as singletons in the above sense. Both are instantiated from the ◊worksFor Object Property (OP), which is a ◊subOpOf the OP ◊TimeRelation. The latter relation is denoted by the »opi OP. Thus, it is possible to model any number of working relationships for >ProfessorA, to each of which the PDPs .from and .to can be assigned. The underlying Object Property Definition (OPD) is (^Professor, ◊worksFor, ^University), which is equivalent to (◊worksFor, »hasDomain, ^Professor) ∧ (◊worksFor, »hasRange, ^University). In the example, ◊worksFor represents a binary relationship type. Note that »worksFor#1 is an OP instance of the OP ◊worksFor, which means that it is also an element of ◊TimeRelation and ◊ObjectProperty. But (>ProfessorA, »TimeRelation, >University1) and (>ProfessorA, »ObjectProperty, >University1) cannot be entailed.

An n-ary relationship type can be modeled by having one domain class, e.g. ^Transport and n range classes, e.g., ^Agent, ^Product, ^Route, ^Instrument, ^Period. This can be interpreted as ^Transport being a reification of n binary relationship types. Other typical examples of n-ary relations are ^Surgery, ^Employment, ^Marriage, ^ClassEnrollment and so on. In [Bens2023b] it is described how arbitrary complex state of affairs can be modeled with so-called Cascaded Role Sets (CRS). Additionally, the OPs in figure lubm2 form an OP hierarchy where for their extensions it holds that ◊hasAdvisor ⊆ ◊TimeRelation, ◊worksFor ⊆ ◊TimeRelation, and ◊TimeRelation ⊆ ◊ObjectProperty. In future work we will elaborate on our insight that the instantiation for MDPs, UDPs and TDPs can also be applied for OPs in analogy to classes. Currently, we are also investigating candidates for two additional attribute types in order to capture the possibility of supporting Value Assignment Propagation (VAP) for universals (..∆MetaTransparentDataProperty), and another one for the representation of meta information on universal (..^MetaUniversalsDataProperty). The example in the figure lubm2 can be extended by a modeling option for partitioning classes that is only possible through the use of Transparent Data Properties (TDPs). This is done by declaring the .ΔGender attribute in the ^.Gender class. In the subclasses ^.female and ^.male the values female/weiblich and male/männlich are assigned for English and German. Particulars then get the corresponding value via Value Assignment Propagation (VAP), e.g. for >StudentB the value male can be entailed as assigned in ^.male. Thus, compared to [FoAl2021], the explicit modeling of powertypes such as PersonType, PersonTypeByGender and EmployeeType can be simplified.