The importance of relations for conceptual modelling motivates an evaluation of Conceptual Graphs (CG) in this respect. This analysis is presented on the formal ontological basis provided by the General Formal Ontology (GFO). On the basis of a simple example domain, modelling problems are identified and analyzed in connection with more sophisticated relational concepts like roles, relators, and player universals. This leads to a proposal for enhancing CGs and their diagrammatic modelling framework in order to capture the example domain more adequately. The newly introduced Conceptual Graphs with Relators allow for expressing roles and relators and help to clarify the ambiguous translation of classical CG relations to relators and roles. From a more general point of view, the overall approach provides an example of applying formal ontological theories in the meta-analysis of modelling language semantics.
The General Formal Ontology (Gfo) is a top-level formal ontology (also known as upper level or core ontology) and part of the ontological framework which is being developed by the Research Group Onto-Med at the University of Leipzig1.
Gfo is chosen as ontological background for this work because of is subtle modelling of
relations and roles [
The following sections will utilize the Cg framework to model a practical example domain: the
situation of trust as formalized by Coleman and Buskens [
Trust is a quaternary relation trust(X, Y, S, AG) between two social agents X and Y, which participate together in the contextual situation S. This situation involves an action A that involves a good G belonging to X and which is currently at the disposal of Y. The relation trust reads: “X trusts Y in the situation S to apply action AG”. Normally, the action lies a certain amount of time in the future which accounts for the risk the trustor must take. The relational roles of X and Y will be labelled trustor and trustee 3.
This relation holds in the situation of lending a book, i.e., lending is a special case of trust (by adding additional constraints on X, Y, S, AG and their interrelation). The two agents are the person lending the book (a Mr. Norrell), called lender, and the borrower (Mr. Strange) who is trusted to return the book (AG) – a book that has the id 314 – after a certain amount of time.
Fig. 1 introduces graphs that try do model the concrete example above with proceeding complexity: starting from trust as a simple dyadic relation between concrete persons (G1), the object of trust and its relation to the participating agents is introduced (G2 and G3). Leaving aside for a moment the modelling of the action and its embedding in time which would require advanced temporal modelling techniques, G3 is lacking the assignment of the roles which describe the positioning of the related persons towards the relation.
(a) graph G1 (b) graph G2
(c) graph G3
Further, the problem of how to describe (meta-)relations between relations arises when trying to establish the specialization of trust to borrow beyond simple subsumption.
In order to make these modelling demands and their possible inexpressiveness with Cgs
explicit, a more detailed view onto relations as entities sui generis is inevitable.
1 http://www.onto-med.de
2 As worked out more detailed in [
“Relations are very peculiar entities; [. . . ] [Many philosophers] have thought that relations are nothing other than the relata and their features or that they are merely appearances. But others have conceived relations as the very stuff from which the world is ultimately constituted.” [12, p. 58f] Regarding this quotation of Jorge Garcia, relations are basic entities that heavily depend on the underlying general, ontological paradigm. From the variety of different approaches to formalize relations, Gfo’s relator model will be introduced in the following as it includes a very subtle approach towards relational roles. 2.1
In brief, Gfo relations “bind [a finite number of] things of the real world together” [4, p. 33].
These are the relata of the relation and their number is the arity of the relation. Moreover, the
relata can play the same or a different role in the context of the relation. Relations exhibit a
categorial character, i.e., they generalize a kind of entities which form the “glue” among other
entities. In other words, a relator is the distinct entity that assigns additional capabilities to
interrelated entities, these are described by the relator’s roles. The crux lies in the modelling of
these (relational) roles which describe the mediation between the arguments and the relation
or relator, respectively. The (meta-)relation between the (categorial) roles of a relation and
the corresponding relata is named plays which is subsumed by the ontological basic relation
dependent-on because roles depend on their player and on complementary roles, viz the totality
of roles involved in the relator, cf. [4, p. 33f] [
As relators can be seen as instantiations of (categorial) relations, the corresponding roles of a relator are instances of a relation’s (categorial) roles. Fig. 2 summarizes these new aspects in an Uml-style diagram which introduces the classical view4 on relations as derivable (the entities marked by “/ ”) from the relator or the relation, resp.; the diagram can be read bivalently as either class or object diagram depending on focussing either relations or relators. For simplicity, the following diagrams will only depict the case of dyadic relations but can be extended to arbitrary arity.
The problematic nature of roles resides in the simple fact that they are highly dynamic entities (e.g., roles can change over time, one entity can stand in two different roles to the same relator and needs to be treated differently regarding both roles), whereas the classical conceptual modelling approach prefers the dissection of a domain into more or less static and discrete entities. Therefore, roles prefer to be separated from material entities (“natural kinds”) and in the following will be assumed to form a hierarchy of their own. Nevertheless, the connection of the roles’s (part-of) hierarchy and the classical material subsumption hierarchy adds additional aspects to the above model.
As one of a role’s most important effect is its restriction of the super-type of its player, Fig. 2 includes an abstract universal named player universal which can be regarded as a compositum of all the types of the objects that can be in the plays relation towards this role, and, hence, serves as a constraint for the type of the relatum. Fig. 2: Extended Diagram of Gfo’s Relation/Relator 2.2
In the light of the preceding considerations, G3 of Fig. 1 still lacks the information of the relational roles of the participants of the trust-relation. Further, as one “not consider[s] the mere collection of the arguments which respect to a single fact [i.e., the entirety of relator and relata as instance of a relation]” [4, p. 33], relations tend to resemble Cg-concepts instead of Cg-relations. 4 Relators/relations are assumed to hold between the material relata and not the roles.
The following requirements would additionally underpin the choice of relational concepts: the demand to model subsumption between relations, e.g., the relation borrow as sub-relation of trust as well as the composition of relations which is not possible with Cgs as only a simple, partial-ordered subsumption hierarchy is admitted [14, p. 481], and the necessity to annex a relation with additional information, like attributive properties.
Another important subject is the difference between relations that include individuals as the relata and the definition of abstract (universal) relations. As a Cg-concept is related by default to the existence of an entity of that concept, this distinction does not carry weight in the following Cg enhancement as – regarding the terminology of Gfo – the entity representing a relation is bound to the instance level, i.e., has to be a relator not a relation.5 However, a formal way to introduce new relational concepts via abstraction would be necessary to grasp the abstract definition of the trust relator in the example. 2.3
As elaborated by Frank Loebe [
An important change to the previous considerations is the refinement of the definition of player universal as the maximal type constraint of a role bearing entity into a class; this step lifts a role-player from the instance level and will be called (role) player universal. This class is accompanied by a natural kind that constrains the types of the role-bearers.
The prototypical trust relation between two player instances takes place in the lowest row of Fig. 3: Mr. Norrell, as the individual entity subsumed under the player universal, plays the individual role (depicted as object) that instantiates the role category Trustor. Further, this role individual is in the roleOf association towards the relator individual that instantiates the relation Trusts. The important feature is the differentiation between instantiation and generalization: Trusts is a relation (via generalization) that is simultaneously an instance of the (meta-)category relator.
Another important distinction lies between the similarly named associations of the instanceand the categorial level: the plays relations between instances has another semantic grounding than the categorial relation of the same name, nevertheless they depend on each other.
A general, abstract definition of a special relation conforming to the example domain has to give a role base, i.e., a relation (Trusts) with its relational roles (Trustor, Trustee) and the natural category which the according player universal specializes (both are Persons). The 5 The different modes of defining the referrent of a concept node would allow to approximate an abstract entity by the general referent ∗, for example in (“something”). differentiation between role and class types is hidden behind the demand of a player universal to subsume natural kinds contrary to relational roles. 3
By recapitulating the previous excursion into an ontological theory of relations, the following requirements towards the expressiveness of the modelling language can be extracted: it should be able to represent roles distinct from the entities of role-bearers, as well as relators between roles and meta-relations between these relators; further, one needs to introduce new relators in an abstract way (like a role base), and, as player universals are rather complex abstract entities, to express at least their effects as type restrictions on the role-bearers. 3.1
As already explained above, the mixture of relation and object hierarchies, i.e., relation
concepts and classical Cg concept, must be avoided. Therefore the approach of Ribi`ere et al. [
Ribi`ere et al. proceeded as follows: first, there remains only one Cg relation which connects an element of the link type hierarchy with a classic concept; second, both the link type hierarchy and the concept ontology are disjointly combined into a concept lattice whereas both sub-hierarchies only share > and ⊥. This leads to the situation depicted in Fig. 4.
As there remains only one Cg relation, the corresponding
Fig. 4: Type Hierarchy nodes will be omitted in the graphical representation. Further, a new style of vertices is introduced to depict link concepts. Therewith, can be shortened to .
Hence, the approach of [
These improvements will allow to model the relations of the domain more fine grained than with classical Cgs. 3.2
Another requirement is the possibility to name the roles of a certain relator. Cg relations
were already introduced as roles: “Conceptual relations specify the role that each percept
[or the concept representing this percept, resp.] plays” [2, p. 70f]. Consequently, the graph
has to be interpreted as “Concept2 plays the role described by hasRole
towards Concept1”[ibid.]. A formal foundation of the approach based on has<Rolename> is
given in [14, Sect. “Classifying Roles”] and [
This application of Cg relations overlaps with the approach of utilizing them as conceptual relations itself. Even the original work of John Sowa did not distinguish these clearly: Cg relations are applied in both ways – as roles (see above example) and relations (cf. classical “cat (being) on mat” [14, p. 477]).
Besides the problem of expressing complex relations via simple role-names, this approach has the disadvantage of intermingling roles with the relator which were both assumed to be separated due to the general ontological considerations above. 3.3
The proposed solution will be a combination of most previously mentioned approaches to model
relations: first, relators will be modelled by link types with the appropriate relator taxonomy;
second, the relations of conceptual graphs model the relational roles between a (classical Cg)
6 John Sowa already introduced links and a link type hierarchy based on Aristotle’s analysis of
relational links but without a rigorous foundation [
Concept Graphs with Relators are finite, tri partite, directed, not necessarily connected multigraphs G = (V, E) with vertices V = C ∪· L ∪· R and edges E ⊆ V × V . The vertices of the graph are segregated into three types: concepts C, relators (links) L, and roles R. An edge walk connects a relator node to either a concept node or a relator node via a single role node7, hence E ⊆ L × R ∪ R × C ∪ L × L; additionally, there are no other edges than those participating in a walk, and walks do not cross in roles, i.e., the degree of role vertices is always two.
The special role named hasRelatum is the maximal element of a lattice-order ≤R on the
roles. Further, both concepts and relators form a lattice-order ≤C / ≤L with maximal
element >C / >L. These two orders are combined into a single lattice with an additional
element > such that > ≤R/L >R/L serves as new maximal element whereas the bottom
elements coincide ⊥ = ⊥C = ⊥L.
7 Without a formal semantic basis of roles, roles between two relators seem dispensable and will be
omitted; nevertheless, these entities could describe a new kind of object which could turn out to be
useful in conceptual modelling.
The heart of the abstraction
are two types of coreference:
first, w refers to the
definiendum but further allows to
include subsumption by giving
a type more special than >L
(viz. later Fig. 8 which derives
borrow from trust); second,
the (free) variables x, y, s, and
a are the relator’s arguments
whose roles are given by role
vertices and whose player
uniFig. 7: Defining the Abstract Trust Relator versal is given by the type
of the corresponding concept
node. Thus, the argument x plays the role hasTrustor towards the definiendum w and must
be an object of type PERSON. Therefore, player universals are hidden and only their effect of
constraining concept type subsumption is represented. In the spirit of [
A relator type abstraction written ”relator R(r)(a1, . . . , an) is G” declares a new relator R ∈ L of arity n which is given by the (n + 1)-adic abstraction [2, Def. 3.6.1] of the form λ r, a1, ..., an : G whereas the concept graph G includes one relator node r (the definiendum) representing R which is related via roles to concept nodes a1 to an whose type expresses the constraints by the according player universal of the role bearer. The type of r can be used to inherit an already defined relator or set to >L.
To conclude, the simple borrow relation which was mentioned as a prototypical example of a trust relation can be formalized on top of the above relator abstraction as in Fig. 8 whereas the epistemic relators and the (temporal) sequence have to be read “intuitively” without an accompanying, appropriate Cg ontology. Thus, this graph highlights the transition from a situation in which the trustee possesses the object to a situation in which the trustor believes that this object has been returned.
Hence, relator type abstraction allows to introduce new, complex relator which derive from already existing ones by giving a role base and additional constraints beyond simple subsumption. The previous sections are an example for the utilization of a formal ontology to the task of making the differences between a formal semantics and the semantics intended by the engineer explicit, as well as to feed back these results into an appropriate enhancement of the modelling language.
The ontologically coined view allows to express a catalogue of requirements that one would want to express when trying to represent the trust domain (or domains including relations and roles in general). As the standard Cg framework does not provide the necessary features to express these demands (particularly relators and roles), an example-tailored enhancement of Cgs is introduced as Conceptual Graphs with Relators which fulfills both the requirements of modelling certain aspects of the domain (instance level description and abstract introduction of relations) as well as the catalogue derived from a closer look onto relations via Gfo.
The choice of a particular underlying ontological approach influenced the enhancement
as it mirrors the basic distinctions of Gfo in the Cg framework. Consequently, applying
another core ontology could have resulted in another way of enhancing Cgs. For example,
emphasizing the dynamic aspect of roles could have lead to the field of Dynamic Conceptual
Graphs and Actor Models [
There are several aspects which require additional consideration: first, a complete formal semantic foundation of Conceptual Graphs with Relators by introducing an appropriate (power context based) model for roles; second, the given interplay between formal ontology and the modelling language can only be seen as first cycle of a “circulus creativus”[10, p. 129] and would require additional feedback via modelling further examples with this extended graph formalism; third, a comparison to the large field of other Cg based extensions, starting from the above mentioned dynamical extensions.
To conclude, applying Gfo to support the semantic meta-language analysis of the (diagrammatic) modelling language of Conceptual Graphs has proven to be another bread-and-butter task that can be facilitated with the help of formal ontology, and proved to be a first step of combining Gfo and the Cg framework.
First and foremost, I am indebted to the Research Group Onto-Med and H. Herre who allowed
me to write my thesis about the semantic foundation of diagrammatic modelling languages
of which the previous considerations were a minor excerpt [