=Paper=
{{Paper
|id=Vol-2050/foust-paper8
|storemode=property
|title=Parts Unknown: Mereologies for Solid Physical Objects
|pdfUrl=https://ceur-ws.org/Vol-2050/FOUST_paper_8.pdf
|volume=Vol-2050
|authors=Yi Ru,Michael Gruninger
|dblpUrl=https://dblp.org/rec/conf/jowo/RuG17
}}
==Parts Unknown: Mereologies for Solid Physical Objects
==
https://ceur-ws.org/Vol-2050/FOUST_paper_8.pdf
Parts Unknown:
Mereologies for Solid Physical Objects
Yi RU and Michael GRUNINGER
Mechanical and Industrial Engineering, University of Toronto, Canada
Abstract. When describing a mereological relationship, the use of the word “part”
to whole hides the intended semantics of the parthood relations. Existing studies in
mereology usually present part of as a general top level mereology relation, and
all other mereological relations are sub-relations under it. We present an alterna-
tive approach of mereological pluralism, in which each mereological relation cor-
responds to one characteristic of the solid physical object, and the characteristics
are formalized in different modules within an upper ontology. A general suite of
upper ontology modules is proposed in this paper to support the specification of the
parthood relations for solid physical objects. We name the union of these modules
Ontology of Solid Physical Objects (OPO).
Keywords. Upper Ontology, Parthood, Mereology, Solid Physical Objects
1. Introduction
The seminal work of Winston et al. [21] identified multiple relations that capture intu-
itions about parts and wholes, we follow this approach of mereological pluralism1 to
suggest a new relationship among the mereological relations. Our claim is that there is
no super-relation of the mereological relations, but each mereological relation is compre-
hensive by its own. In addition, we propose that each parthood relation corresponds to a
generic module of the Ontology of Solid Physical Objects (OPO). We adopt the sideways
approach to upper ontologies [12], and focus on those modules that axiomatize intuitions
about solid physical objects (although we will briefly discuss parthood relations for other
upper ontology modules such as time and activity). In our ontology of the representation
to the parthood of solid physical objects, our objective focuses on the correctness and
completeness of the ontologies with respect to the semantic requirements.
The Social Needs Marketplace (SNM), as the motivation of this paper, is an online
marketplace for the population who live under the poverty line to exchange, offer and
receive used goods. The match between supply and demand of goods requires a detailed
representation of physical objects. [8] Such a representation faces several daunting chal-
lenges. First, the terminology that people use for these descriptions is rife with ambigu-
ity. Second, it is often unclear whether or not the terminology used is adequate to pro-
vide complete descriptions of a given range of goods. A third challenge is that many of
the terms that people use to describe physical objects are ad hoc and arbitrary. It is thus
essential to develop an ontology for the SNM.
1 Mereological pluralism denotes that there are multiple distinct parthood relations.
In some scenarios of SNM, we need to describe partially damaged object (i.e. a bed
with the headboard missing, a mug with the handle broken or a chipped dish), where
the ”partially” here might have different meanings: partially missing, partially broken,
or material partially removed. To represent these different relationships of the part to the
whole, the specification of damage condition requires a comprehensive description to
parthood relationships beyond mereological monism. In this paper we will focus on the
different notions of parthood of physical objects that have arisen within SNM, but which
are really within the scope of an upper ontology. In particular, we propose Ontology of
Solid Physical Objects (OPO), a general suite of upper ontology modules that support the
specification of the parthood relations for solid physical objects and used goods within
the scope of SNM.
Our approach can be seen as a development along the lines mentioned in Artale’s
work [3]: ”The particular behaviour of the different part-whole relations may lie, among
other things, in the ontological nature of both the whole and the part”, namely, the on-
tological nature of the whole and part are captured by the generic modules of the upper
ontology.
Definition 1.1. Let R be a binary relation, and let MR be a class of structures with
signature hRi.
The relation R is a parthood relation iff T h(MR ) is synonymous with a theory in the
H mereology Hierarchy2 .
2. Previous Work
Existing studies in mereology usually either have a single part-of relation to summarize
all parthood relations, or adopt a taxonomy of parthood relations that all other parthood
relations are specializations or sub-relations of a general top level part-of relation.
2.1. Mereological Monism
Approaches based on classical mereology [20] tend to use a single parthood relation to
specify parthood relationships, an approach known as mereological monism. This is not a
viable approach if we are to support application domains such as SNM. The relationship
between the detachable leg of a table and the table itself is fundamentally distinct from
the relationship between the table and a portion of the table that has been chipped off.
One possible way of maintaining mereological monism while also allowing multiple
parthood relations is to say that all other parthood relations are definable with respect to
a single primitive parthood relation. For example, one can define the restriction of part
to specific classes of elements within the domain. For example, in DOLCE [14] one can
restrict the parthood relation P to the class of time intervals to define a notion of temporal
parthood, as distinct from arbitrary elements.
McDaniel [15] used this approach to define causally integrated parthood, functional
parthood and immediate parthood. Causally integrated parthood applies to cases in which
each of the object’s parts is robustly related to every other part of the object; functional
parthood indicates the functional role of the part in the production of the whole, and im-
2 colore.oor.net/mereology
mediate parthood defines the largest part of the whole. However, all of these approaches
require the new parthood relations to be definable using the single part-of relation, and
we will see that in the case of SNM, this is not possible, that is, the different parthood
relations are not definable with respect to each other.
2.2. Taxonomy of Parthood Relations
Mereological pluralism [15] is based on the idea that there are indeed multiple distinct
parthood relations. The earliest work in this area was by Winston [21], who presented a
taxonomy of part-whole relations as specializations of a general part-of relation. Win-
ston’s approach was informal, and was based on a series of examples that motivated
the types of parthood relationships: phonology-linguistics as component-integral object;
tree-forest as member-collection; slice-pie as portion-mass; steel-bike as stuff-object;
grasping-stacking as feature-activity and oasis-desert as place-area.
Later, Odell [18] proposed six kinds of aggregation relationships: component – in-
tegral object; material – object; portion – object; place – area; member – bunch; and
member – partnership. They are determined by the combination of three basic proper-
ties: configuration (the parts bear a particular functional or structural relationship to one
another or to the object they constitute), homeomerous (the parts are the same kinds of
things as the whole), and invariance (the parts can be separated from the whole).
However, neither Winston nor Odell provided axiomatizations of their different part-
hood relations. For axioms, we need to look at actual parthood ontologies. The upper
ontology SUMO [17] contains the relation part as a spatial relation and a set of other re-
lations that specialize it. In more recent work, Keet [13] introduced a taxonomy as sum-
marization of Odell’s approach to types of part-whole relations [18], and also provided
OWL axiomatizations of the taxonomy.
One of our primary claims is that the notion of a taxonomy of parthood relations is
not correct3 . We will see that each distinct parthood relation has its own axiomatization,
and hence is synonymous with a different mereology ontology within the set of pos-
sible mereologies. The taxonomy approach cannot adequately capture the relationships
between the different axiomatizations.
3. Ontology of Solid Physical Objects (OPO)
The design of the Ontology of Solid Physical Objects (OPO)4 follows the principle that
each module of OPO axiomatizes necessary conditions for solid physical objects. We de-
fine a solid physical object as an object that is self-connected5 , has some shape, is made
of some material, and occupies some space. OPO is built to expand the definition with
four modules: structure module, shape module, constitution module and spatial module.
Each module features a characteristic ontology in the upper ontology, and gives rise to a
corresponding parthood relation: componentOf, pieceOf, portionOf, and containedIn, re-
3 Of course, one can always define a trivial taxonomy by the disjunction of other relations, but the question
is about the relationships between the various parthood relations themselves.
4 colore.oor.net/opo/opo.clif
5 Self-connected is axiomatized in mereology as (∀x)SC(x) ≡ (∀y, z)((∀w) ⊃ (overlaps(w, x) ≡
(overlaps(w, y) ∨ overlaps(w, z))) ⊃ C(y, z))
spectively. The current design of the Ontology of Solid Physical Objects is shown below
in Figure 1.
Figure 1. Ontology of Solid Physical Objects (OPO)
With respect to the modules of OPO, the componentOf relation is axiomatized as
a primitive relation within a theory that is synonymous with MT in mereotopology. On
the other hand, pieceOf, portionOf and containedIn relations are given conservative def-
initions in their respective generic ontology modules, and hence are treated as defined
relations. Examples of each parthood relation are shown in Figure 2.
Figure 2. Physical Examples of Parthood Relations of Objects
3.1. Structure Module
The Structure Module identifies the external connections of an integrated physical ob-
ject and the criteria for the atomicity of a solid physical object. The basic intuition here
is that a physical object is self-connected, the external connection axioms and corre-
sponding parthood ontology are axiomatized by an ontology that is synonymous with
the mereotopology MT 6 [7]. Future extensions on the Structure Module will incorpo-
rate process ontology PSL [9] to represent the assembly and disassembly activities for
furnitures and other integrated objects.
3.1.1. componentOf
If the physical object whole is an integrated object, and can be broken down into smaller
disconnected integrated or atomic self-connected solid physical objects by only breaking
the external connections from the whole, we use atomic component to describe the com-
ponents that cannot be further broken down by only breaking the external connections.
The primitive componentOf relation is synonymous with MT in mereotopology as:
(∀x, y)componentO f (x, y) ⊃ (∀z)(EC(x, z) ⊃ EC(z, y))
where EC is the external-connected relation defined with the primitive C (connected) in
mereotopology. A typical example that will use componentOf to describe its parthood
relationships is a furniture with all components assembled together (i.e. wooden children
bed in Figure 2).
3.2. Shape Module
The Shape Module needs an extension to the BoxWorld7 ([10], [11]), which is an ontol-
ogy for shapes composed of surfaces and edges. As such, it can represent aspects of an
object that have different dimensionality, for example the 2D surfaces of a 3D object and
one-dimensional features (i.e. edges and ridges). We extend it with a featureOf parthood
relation to represent the parthood relationship between a shape feature and the whole. A
shape feature is defined as an individual or sum of adjacent ridges in the whole shape.
Ultimately, we want to categorize sufficient types of shape to identify common atomic
shapes (i.e. cylinder, box, curved cylinder, ball) and differentiate the pieces of one object
and some pieces as atomic shapes.
3.2.1. pieceOf
The pieceOf relation describes the shape-related parthood relationship and is a defined
relation:
(∀x, y)pieceO f (x, y) ≡ (∃ f1 , f2 )boundedBy(x, f1 )∧boundedBy(y, f2 )∧ f eatureO f ( f1 , f2 )
where boundedBy means that the object is partially or fully bounded by the shape feature,
and featureOf is a reflexive parthood relation in the extended BoxWorld Ontology. The
corresponding example in Figure 2 shows the handle of a coffee mug can be identified
as a piece by its distinct shape from the mug cylinder body, as there are two ridges of the
piece. Of course, the pieceOf relation is able to describe the shape identified parthood
beyond atomic but also integrated self-connected objects, for instance, the back piece of
the chair could include the upper piece of the frame and the whole back cushion (the
cushion is a component to the whole chair) with multiple specific ridges.
6 colore.oor.net/combined_mereotopology/mt.clif
7 colore.oor.net/boxworld/boxworld.clif
3.3. Constitution Module
The Constitution Ontology8 defines the constitution between matter and material objects,
independent of types of materials. And the Matter Ontology9 contains the corresponding
parthood ontology for the Constitution Module.
3.3.1. portionOf
Portions of a solid physical object and the whole are all material objects, the boundary
between them is arbitrary. The whole here can be an atomic self-connected object, a
component, or an integrated object. An example is a portion of a broken dish as shown in
Figure 2, or more deliciously, a slice of a pizza. One can divide a pizza in arbitrary ways;
there is no physical boundary or visible standard on how the pizza can be divided. The
portionOf relation is the parthood relation in the Matter Module of the ontology, and it
is defined with the constitutes relation in Constitution Ontology:
(∀x, y)
portionO f (x, y)
≡ (∃m1 , m2 )constitutes(m1 , x) ∧ constitutes(m2 , y) ∧ chunkO f (m1 , m2 )
where chunkOf here is the parthood relation in Constitution Ontology. Borgo [6] dis-
tinguished physical boundary for matter discontinuity, and imaginary boundary for ad-
jacent parts of a body. Following the approach, we indicate that physical boundary lies
for components to the whole and imaginary boundary lies for piece or portion to the
whole; while the boundaries for pieces are explicit and identifiable in shape, boundaries
for portions have some arbitrariness.
3.4. Spatial Module
The Spatial Module represents the location of physical objects within abstract space to-
gether with the spatial relationship between the regions occupied by objects. Incorporat-
ing with the representation of enclosure in the Occupy Ontology10 [2], the Containment
Ontology defines the containment parthood relation.
3.4.1. containedIn
The containedIn relation is defined as the containment relation between the occupied
regions of two self-connected physical objects: the space occupied by one solid physical
object is enclosed by the space occupied by another solid physical object. It is given by
the following conservative definition:
(∀x, y) containedIn(x, y) ≡ (∃r1 , r2 ) occupies(x, r1 ) ∧ occupies(y, r2 ) ∧ region part(r1 , r2 )
where region part is the parthood relation over regions of abstract space within the loca-
tion ontology Toccupy . As the example in Figure 2, the space an individual egg occupied
8 colore.oor.net/constitution/constitution.clif
9 colore.oor.net/matter/matter.clif
10 colore.oor.net/occupy/occupy_root.clif
is a region of the enclosed space occupied by the outer container, therefore we can say
the containedIn relationship holds between an egg and the container.
4. Relationship between Previous Work and Our Work
Looking at each of the parthood relations, the component-integral object and portion-
object parthood relations in Odell [18] are similar with our proposals of componentOf
and portionOf as relationships for integration and mass. The portion-mass in Winston et
al. [21] also shares the definition with our portionOf relation. Despite the lack of axiom-
atization, the three basic properties that these relations were built upon (as discussed in
Section 1) are not specific to physical objects and not suitable for our practice for SNM.
In Borgo’s approach, the concrete existence of physical object is determined by the
material object that is a piece of matter and occupies a region of space. [6] He defined
a notion of contingent part for physical objects with location, he also restricted a bi-
nary predicate P to hold between the substrate matter and another P to hold between the
other substrate region. One could say that OPO is following the definition of stratified
ontology11 Borgo adopted, where the classes of parthood relations correspond to differ-
ent identity criteria of characteristics of solid physical objects, such classes of parthood
relations are disjoint and the ontological dependencies among the criteria of characteris-
tics are explicit. However, resulting from the difference in scope, SNM requires further
mereological pluralism beyond matter and location.
Bittner and Donnelly [5] have also presented an axiomatization that follows mereo-
logical pluralism, and which does not strictly adhere to a taxonomy of parthood relations.
In particular, their axiomatization of CmpOf (component of), and CntIn (contained in)
are similar to what we presented in Section 3. Nevertheless, they still use a general PP
relation which does not itself correspond to any generic ontology for objects, and the
other two parthood relations are not grounded in a generic ontology.
Our approach falls into what is often called the family of 3D representation of phys-
ical objects, in which all of an object’s parts exist at any point in time. This approach
can also be seen in the BFO [4] and DOLCE [14] upper ontologies, although these upper
ontologies are based on a time-indexed version of mereological monism.
5. Beyond Physical Objects
Recall that the original paper of Winston et al. [21] identified parthood relations that
did not apply to physical objects, such as the relationship between feature and activity.
Within the PSL Ontology [9], there are three relations of interest – the subactivity relation
over arbitrary activities, the atomic subactivity relation over concurrent activities, and
the subactivity occurrence relation over occurrences of activities. Each of these relations
corresponds to a different module of the PSL Ontology – Tsubactivity , Tatomic , and Tactocc ,
respectively. Furthermore, the axiomatization of each of these relations is synonymous
with a different ontology in the Hmereology Hierarchy, and hence is a parthood relation.
11 Stratified Ontology is denoted as ”an ontology where classes corresponding to different identity criteria are
kept carefully disjoint and represent the roots of separate hierarchies called strata, and where the ontological
dependencies among strata are made explicit.” [6]
Another distinct parthood relation corresponds to generic time ontologies that in-
clude time intervals within the domain. As shown in [16], the temporalPart relation of
SUMO can be axiomatized as a definitional extension of the time module within SUMO,
and since this extension faithfully interprets the ontology Tcem G within the Hmereology
Hierarchy, the temporalPart relation is also a parthood relation.
6. Conclusion and Future Research
We have proposed a novel approach to the identification and axiomatization of parthood
relations for physical objects as they arise from the generic ontologies required to for-
malize the intuitive definition of a physical object – an object that is self-connected,
has some shape, is made of some material, and occupies some space. We use structure,
shape, constitution, and spatial modules to capture four characteristics of physical ob-
jects to support the definition of proposed parthood relations. The four parthood relations
we identified are componentOf, pieceOf, portionOf and containedIn. Each parthood re-
lation corresponds to one characteristic module in the Ontology of Solid Physical Ob-
jects (OPO). In this sense, the characteristic ontologies in OPO are modules of an upper
ontology that is strong enough to represent any solid physical object.
Given the axiomatizations of these parthood relations shown in Section 3, it is clear
that they are all distinct relations and that there is no priori subproperty taxonomy among
these relations. Furthermore, given the correspondence between these relations and the
generic ontology modules of an upper ontology, we anticipate that new parthood rela-
tions will be recognized as additional modules of upper ontologies are considered. We
have covered structure, geometric and material characteristic in this version of OPO.12
The current framework is able to represent different conditions of partially damaged ob-
jects, especially the examples from SNM shown in Figure 2: children’s bed with missing
component indicates an incomplete structure, mug with fractured handle piece indicates
broken shape, and the dish fragmented into portions indicates some matter is divided.
Future work on OPO will include a surface module ontology to capture color and sur-
face damage such as stains and scratches, as well as a function ontology to represent
functional characteristic of physical object.
A generic ontology of state and change is also needed to be incorporated into part-
hood relations. As it stands, the upper ontologies BFO and DOLCE axiomatize the no-
tion of temporary parthood, in which a physical object (perdurant in BFO and physical
endurant in DOLCE) can have different parts at different times. Using the methodology
from Aameri [1], we can specify particular ontology as the domain state ontology for cor-
responding module in OPO. Thus, an ontology for assembly and disassembly specifies
how the components of an object change, an ontology of production/consumption speci-
fies how portions of the matter that constitutes an object can change, and an ontology for
kinematics would specify how the location of parts within an object can change.
Given the motivation comes from the Social Needs Marketplace (SNM) project, the
OPO supports the description of any solid physical objects and will be used to represent
damages and to automatically categorize household goods such as furniture and home
12 As Sanfilippo stated in [19], features of physical objects can be categorized into functional, geometric,
material, structure and other types.
appliances. This domain will also serve as a testbed for identifying new concepts in upper
ontologies required for the representation of everyday objects.
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