Introduction

Part-whole relations are of fundamental importance for how we organize concepts. Consequently, they have been studied in philosophy [1,20,19], linguistics [3,4] geographical information systems (GIS) [2,9,18], to name just a few. Corresponding partonomies or meronomies, i.e. hierarchies built from part-whole relations, are therefore a recurring theme in ontology modeling.

Despite this, however, we have been unable to find a readily available or documented ontology design pattern for part-whole relationships, other than some very minimalistic proposals in the ontologydesignpatterns.org portal. In this paper we want to rectify this by providing such a pattern, together with a contextualized version of it. Our approach to this is to keep things as simple as possible, yet to make sure that the resulting patterns are comprehensive yet general enough to be applied in many contexts.

Concretely, we will follow an approach laid out by Winston in his 1987 landmark paper on "A Taxonomy of Part-Whole Relations" [20]. 3 While this paper was based on linguistic considerations, it also provided for logical characterizations and axiomatics, which will inform our pattern. As such we do not claim much novelty, other than that we cast previous observations by us and others into reuseable ontology design patterns. In fact, the technical content of Section 3 is adapted from [8] by carrying it over to the context of ontology design patterns. The rest of the paper is organized as follows: In Section 2 we briefly review Winston's approach to lay the ground for the technical contributions. In Section 3 we provide the basic Winston-Part-Whole Pattern. In Section 4 we provide the Contextualized Winston-Part-Whole Pattern as an extention of the one presented in Section 3. In Section 5 we describe a usage scenario. In Section 6 we briefly discuss a provenance pattern as an example for contextualization, which is essentially adapted from the core of the PROV-O ontology. Section 7 contains additional release information for the patterns, and Section 8 concludes.

Winston's Approach

Winston in [20] distinguishes six different types of part-whole relationships. His categorization is based on the following three aspects, a different selection of which holds for each of the types.

separable (versus inseparable):

Parts can in principle be physically disconnected from the whole. functional (versus non-funcational): Parts are in specific spatial and temporal position relative to each other which supports their functional role as parts of the whole. homeomerous (versus non-homeomerous): Parts are similar to each other and to the whole.

The six types distinguished by Winston are listed in Table 1. The table also lists which of the just mentioned three aspects holds for each type, and an example from each, taken from [20].

Winston furthermore provides a discussion of logical properties for each type of part-whole relation. E.g., he observes that each type of relation is transitive, however if you mix types, transitivity generally does not hold. E.g., if you have two relations which are both of the component-integral object type, then transitivity holds, as in toe being part of the foot, foot being part of the leg, therefore toe is part of the leg. If you mix types, though, e.g. by mixing a component-integral object relation such as "Derek's nose is part of Derek" and a member-collection relation such as "Derek is part of the Department faculty," then transitivity would result in the nonsensical "Derek's nose is part of the Department faculty."

Rather than going through Winston's observations in detail, let us refer here to the axiomatization which we have drawn from it, and which we give in the next section.

The Winston-Part-Whole Pattern

We are now going to cast Winston's part-whole types into a part-whole ontology design pattern, and that will include the capturing, in OWL, of the logical relationships identified by Winston.

We will use the OWL property names component-integral object: po-component member-collection: po-member potion-mass: po-portion stuff-object: po-stuff feature-activity: po-feature place-area: po-place and we will refer to these as the specific part-whole relations. We also use some other, related, relations identified and discussed by Winston. These are, in paticular, spatially-located-in as the spatial (topological) located-in relation and part-of as the generic part-whole relation of which the specific ones listed above are specializations (i.e., subProperties).

From [20] we can now draw the axioms which together constitute the pattern. They are listed in Figure 1.

Axioms (1) through ( 12) declare transitivity and asymmetry for each of the specific part-whole relations. According to Winston, however, we would also need to declare irreflexivity for each of the specific part-whole relations, which would render each of them a strict partial order. However this is not allowed in OWL 2 DL: according to [15, Section 11] a property cannot be both transitive (and, therefore, non-simple) and irreflexive. 4We believe that dropping the irreflexivity axioms should usually not cause any problems in terms of logical reasoning over the pattern, however as usual it is difficult to formally assess this. A formal declaration of irreflexivity may sometimes be helpful for ontology debugging or data curation, and of course some (correct) inferences will be missed through OWL 2 DL reasoning if the axiom is omitted. Note, though, that due to the open world assumption all inferences drawn from the OWL 2 ontology are still correct with respect to the complete theory (i.e., the one including irreflexivity).

Winston lists a number of additional axioms, however as discussed in [8] they are in fact tautologies, and while they may be informative for a linguistic

po-component • po-component po-component(1)

po-member • po-member po-member

po-portion • po-portion po-portion

po-stuff • po-stuff po-stuff(3)

po-feature • po-feature po-feature

po-place • po-place po-place

AsymmetricObjectProperty(po-component)

AsymmetricObjectProperty(po-member) ( 8)

AsymmetricObjectProperty(po-portion)(9)

AsymmetricObjectProperty(po-stuff) ( 10)

AsymmetricObjectProperty(po-feature)(11)

AsymmetricObjectProperty(po-place) ( 12)

po-component part-of(13)

po-member part-of ( 14)

po-portion part-of(15)

po-stuff part-of ( 16)

po-feature part-of(17)

po-place part-of (18)

spatially-located-in • spatially-located-in spatially-located-in

ReflexiveObjectProperty(spatially-located-in) discussion, they do not really contribute to ontology modeling, and we do not want to include them in the pattern.

Please note that we do not provide a schema diagram for this pattern, as the pattern exists of related properties only.

A Pattern Extension Accounting for Provenance And Other Context Information

Some usages of the Winston-Part-Whole Pattern, such as the one from [8] on which this pattern is based, suggest that it would be helpful to store context information for the part-of relationship. We conceive that this would mostly be in the form of provenance information. For example, in the case of [8], part-of relationships of the various types defined by Winston were generated automatically using Hearst patterns over Web text corpora. In such a case, one may want to store confidence values, or even pointers to the exact algorithm used in each case.

In order to store this information, we now provide a contextualized version of the pattern described in Section 3; it is essentially obtained by "reifying" the properties. It is a known technique, and one could also refer to it as "lifting" or as "typecasting" of properties into classes following [13].

To explain, consider the schema diagram in Figure 2. A triple which according to the pattern in Section 3 would simply be stated as :everglades po:po-place :florida .

would now be expressed using the following set of four triples-note that the original triple is still included. We use cpo as namespace, for "contextualized part-of."

:everglades cpo:po-place :florida ; cpo:isPartOf :everglades-po-place-florida .

:everglades-po-place-florida rdf:type cpo:PO-Place-Type ; cpo:hasWhole :florida .

Additional context information, such as provenance information can then be attached to :everglades-po-place-florida, and we will further elaborate on this in Section 6.

We now show how to derive the axiomatization for the Contextualized Winston-Part-Whole Pattern. First of all, note that all axioms from Figure 1 are fully adopted (with adjusted namespace of course). In the following, let R denote any one of po-component, po-member, po-portion, po-stuff, po-feature, po-place, and C R be the corresponding PO-Component-Type, . . . , PO-Place-Type from Figure 2.

Po

Then we would like to have all of the following axioms, which are here expressed using rules.

isPartOf(x, y) ∧ C R (y) ∧ hasWhole(y, z) → R(x, z)

This rule actually constitutes a generalized role chain which can be cast into OWL using the rolification5 technique described in [12]. The resulting OWL axioms are as follows (please note the lowercase c R , which is the result of typecasting the class C R into a property).

C R ≡ ∃c R .Self (41) isPartOf • c R • hasWhole R(42)

The same axioms would be added for spatially-located-inin place of R.

Note that instead of axioms (42), we would actually have preferred to use

isPartOf • c R • hasWhole ≡ R,

however this is not expressible in OWL. According to [13] use of the latter axiom would be proper typecasting between properties and classes, however this requires right-hand-side property chains, which if added to OWL DL would cause undecidability and are therefore not included in the standard. Please see [13] for a further discussion of this matter. For similar reasons, we are not able to lift most axioms from Figure 1 fully to the contextualized pattern, as they would also result in right-hand-side property chains. In fact, in addtition to the 14 axioms above we have six axioms R part-of, which correspond to axioms ( 13) through (18). The asymmetry declarations from Figure 1 cannot be fully lifted to the contextualized version: to the best of our abilities, they cannot be expressed in OWL, and the same holds for the reflexivity axiom. For axioms ( 1) to ( 6), ( 21) through (32), and ( 19), partial liftings could be given. However, they would be redundant, i.e., inferrable through OWL DL reasoning from the axioms already given. We thus refrain from adding them.

∀isPartOf.RelationInstance (43)

∀hasWhole.RelationInstance (44)

Finally, we give the range and domain for isPartOf and hasWhole, (43) and (44), respectively. In total, we have 32 axioms inherited from the non-contextualized pattern, plus 30 new ones, for a total of 62 axioms for the Contextualised Winston-Part-Whole Pattern.

Usage Scenario

We give a usage scenario for the presented patterns, from the domain of Materials Science. Materials Science is an interdisciplinary field which focuses on the discovery and design of new or enhanced materials. Of central importance to the field is the determination of materials properties using experiment or modeling and simulation. Examples of such properties include ultimate tensile strength and crack growth rate. More data than ever is being generated as the materials science and engineering domain seeks to enhance throughput through the automation of sequential experiments and greater use of modeling and simulation [16]. At the same time, there is no widely accepted ontology we are aware of to facilitate the digital exchange and integration of data in this fast-growing and very active discipline. To start filling this gap, we have begun to investigate core ontology design patterns needed for such an ontology, and this in fact prompted our development of the Winston Part-Whole Patterns based on earlier mentioned work.

The important role of part-whole relations in this context comes from the fact that engineered products are usually created by combining previously created engineered products-and that includes engineered materials. For example, fiberglass and epoxy (glue) are part of a composite material.

Product designers seek materials which possess specific properties (e.g. color, strength) to enable a function (e.g. be atheistically pleasing, resist deformation due to mechanical loads). These properties are established by combining specific materials in a particular way to achieve a certain microstructure. Once the processing is complete, the characteristic properties of the material are "locked-in." If the composition and structure of a material are described completely, a unique set of properties can be inferred. Additionally, since the processing can be associated with the composition and microstructure, it can also be associated with the unique set of properties. Thus, the recording of the parts or components of an engineered material is of importance.

Eventually, one would like to record the whole Part-Whole chain from a complex engineered product down to a very fine granularity. Examples for such relations could be the following.

-A radar system is part of a boat. -component-integral object -An antennae radome is part of a radar system. -component-integral object -Some composite material is part of an antennae radome. -stuff-object -Epoxy is part of this composite material. -stuff-object -Glass fiber is part of this composite material. -stuff-object -Some composite material cure is part of some composite manufactoring.feature-activity -Some damaged area is part of some composite material surface. -place-area -Some broken fiber is part of this damaged area. -component-integral object It becomes apparent from these examples, that a naive approach, i.e., encoding all of these relationships using part-of only, is inferior to using a model based on Winston's work. E.g., in the former it would be incorrect, as duscussed, to declare part-of to be transitive, while our Winston Part-Whole Pattern allows for corresponding inferences where appropriate, e.g., from the above we could infer that An antennae radome is part of a boat (component-integral object) and that Glass fiber is part of an antennae radome (stuff-object).

A Provenance Pattern Derived From PROV-O

Provenance information is arguably among the most prominent types of context information for all kinds of data. We show in the following, how the Contextualized Winston-Part-Whole Pattern can be extended using a Provenance pattern which is derived from the core of PROV-O [5]. In a very similar way, other context information such as confidence values could be added.

The three core classes of PROV-O are Entity, Activity, and Agent. Briefly, an Entity is simply an item that has provenance. Entities are generated by Activities, which are the execution of some algorithm or method. The Activity or Entity may be performed by or attributed to some Agent which may be, for examples, a person or a script.

However, for use in the context of pattern-based modular ontology modeling [11], it is more convenient to have a dedicated pattern-rather than a full-blown ontology-at our disposal, although the pattern we provide is, essentially, the We add some explanations of these axioms, they follow the standard templates of scoped domain and range restrictions. Of course, pairs of different entities with provenance, or different agents, or different provenance activities, may in turn carry part-whole relationships, which could be expressed using Contextualized Winston-Part-Whole Pattern.

Pattern Release Information

We have released the Winston-Part-Whole Pattern, 6 the Contextualized Winston-Part-Whole Pattern, 7 , and the Provenance Pattern8 in OWL/XML syntax on the ontologydesignpatterns.org portal.

In addition, we have annotated the patterns with the appropriate annotations following the OPLa ontology which serves as ontology design pattern representation language [6]. The annotations were generated using the OPLa plugin for Protégé [17].

Conclusion

Part-whole relations are omnipresent and are fundamental to how we organize information and perceive the world. Thus, it is necessary to have a firm understanding of how to model these partonomies or meronomies. To do so, we have followed Winston's approach, as discussed in [20] and as a result, have developed two patterns: the Winston-Part-Whole Pattern and the Contextualized Winston-Part-Whole Pattern. Additionally, we provide a mechanism for augmenting the pattern with provenance.