# Nontransitive vs. intransitive direct part-whole relations in OWL

Confusing is-a with part-of is known to be a common mistake by novice ontology developers. Each time I taught the ontology engineering course, I had included a session of 1-2 hours to explain some basic aspects of part-whole relations and, lo and behold, none of the participants made that mistake in the labs or mini-projects! One awkward thing did pop-up there and at other occasions, though, which had to do with modelling direct parthood that does not go well at the moment, to say the least, for a plethora of reasons. Inclusion of direct parthood is not without philosophical quarrels, and the more I think of it, the more I dislike the relation, but somehow the issue appears often in the context of part-whole relations in ontologies. The observed underlying modelling issue—representing intransitivity versus nontransitivity—holds for any OWL object property anyway, so I will proceed with the general case with an example about giraffes.

Preliminaries

First of all, to clarify terms in the post’s title: INtransitive means that for all x, y, z, if Rxy and Ryz then Rxz does not hold; formally $\forall x, y, z (R(x,y) \land R(y,z) \rightarrow \neg R(x,z)$ and an option to state this in a Description Logic is to use role chaining: $R \circ R \sqsubseteq \neg R$NONtransitive means that we cannot say either way if the property is transitive or intransitive, i.e., in some cases is may be transitive but not in other occasions. Direct parthood is to be understood as follows: if some part x is a direct part of a y, then there is no other object z such that x is a part of z and z is a part of y; formally, $\forall x,y (dpo(x, y) \equiv \neg \exists z (partof(x,z) \land partof(z,y)))$. If direct parthood is in- or non-transitive is beside the point at this stage, so let us look now at what happens with it in an OWL ontology when one tries to model it one way or another.

The OWL ontology and the reasoner

Given that I used the African Wildlife Ontology as a tutorial ontology earlier and the theme appeals to people, I will use it again here. Depending on what we do with the direct parthood relation in the ontology, Giraffe is, or is not, classified automatically as a subclass of Herbivore. Herbivore is a defined class, equivalent to, in Protégé 4.1 notation, (eats only plant) or (eats only (is-part-of some plant)), and Giraffe is a subclass of both Animal and eats only (leaf or Twig). Leaves are part of a twig, twigs of a branch, and branches of a tree that in turn is a subclass of plant. The is-part-of is, correctly according to mereology, included in the ontology as being transitive. Instead of all the is-part-of and is-proper-part-of between plant parts and plants in the AfricanWildlifeOntology1.owl, we model them using direct-part. AfricanWildlifeOntology4a.owl has direct-part as sister object property to is-part-of, AfricanWildlifeOntology4b.owl has it as sub-object property of is-part-of, and neither ontology has any “characteristics” (relational properties) checked for direct-part. Before running the reasoner to classify the taxonomy, what do you think will happen with our Giraffe in both cases?

In AfricanWildlifeOntology4a.owl, Giraffe is still a mere direct subclass of Animal, whereas with AfricanWildlifeOntology4b.owl, we do obtain the (desired) deduction that Giraffe is a Herbivore. That is, we obtain different results depending on where we put the uncharacterized direct-part object property in the RBox. Why is this so?

By not clicking the checkbox “transitive”, an object property is non­-transitive, but not in-transitive. In fact, we cannot represent explicitly that an object property is intransitive in OWL (see OWL guide and related documents). If we put the object property at the top level (or, as in Protégé 4.1, as immediate subproperty of topObjectProperty), then we obtain the behaviour as if the property were intransitive (and therefore Giraffe is not classified as a subclass of Herbivore). However, the direct-part property is really nontransitive in the ontology. When direct-part is put as subproperty of is-part-of, then it inherits the transitivity characteristic from is-part-of and therefore Giraffe is classified as a Herbivore (because now leaf and Twig are part of plant thanks to the transitivity).

Obviously, it holds for any OWL/OWL2 object property that one cannot assert intransitivity explicitly, that an object property’s characteristics are inherited to its subproperties, and this kind of behaviour of nontransitive object properties depends on where you place it in the RBox—whether you like it or not.

How to go forward?

Direct parthood is called isComponentOf in the componency ontology design pattern and is a subproperty of isPartOf. Its inverse is called haspart_directly in the W3C best practices document on Simple Part-Whole relations [1], and is a subproperty of the transitive haspart. The componency.owl notes that isComponentOf is “hasPart relation without transitivity”, the ODP page’s “intent” of the pattern is that it is intended to “represent (non-transitively) that objects either are proper parts of other objects, or have proper parts”, and the W3C best Practices note that, unlike mereological parthood, it is “not transitive”. Hence, if you include either one in your OWL ontology, you will not obtain the intended behaviour. Therefore, I do not recommend using either suggestion.

Setting aside the W3C’s best practices motivation for inclusion of haspart_directly—easier querying for immediate parts, but for the ontology purist this ought not to be the motivation for its inclusion—it is worth digging a little deeper into the semantics of the direct parthood. Maybe a modeller actually wants to represent collections with their members, like each Fleet has as direct parts more than one Ship, or constitution of objects, like clay is directly part of some vase? In both cases, however, we deal with meronymic part-whole relations, not mereological ones (see [2] and references therein); hence, they should not be subsumed by the mereological part-of relation anyway. They can be modelled as sister properties of the part-of relation and have the intended nontransitive behaviour as in, e.g., the pwrelations.owl ontology with a taxonomy of part-whole relations (that can be imported into the wildlife ontology).

Alternatively, there is always the option to choose a sufficiently expressive non-OWL language to represent the direct parthood and the rest of the subject domain and use one of the many first/second order theorem provers.

References

[1] Alan Rector and Chris Welty. Simple Part-Whole relations in OWL ontologies. W3C Editor’s draft, 11 August 2005.

[2] C. Maria Keet and Alessandro Artale. Representing and Reasoning over a Taxonomy of Part-Whole Relations. Applied Ontology, 2008, 3(1-2): 91-110.