The representation of damage to solid physical objects is complex in nature. Beginning with real world usecases of partially damaged used goods, we derive the description of damage through the approach of representing partial abnormality. We propose four predicates for partial abnormality corresponding to the fundamental characteristics and parthood relationships within solid physical objects: ab portion; ab piece; ab component, and ab containment. Such parthood relations and mereological pluralism concepts are drawn from the Ontology of Solid Physical Objects (SoPhOs), a general suite of upper ontology modules we proposed and axiomatized in First Order Logic. This paper reviews the SoPhOs ontologies that axiomatize parthood, spatial parthood and parthood connection theories and the foundational matter and shape characteristic ontologies that support them. We map the foundational solid physical object ontology to solving the problem of representing damaged goods in various domains including manufacture and e-commerce.
An interesting challenge in conceptual modelling is the representation of some notion
of ideal objects and their relationship to actual objects whose properties diverge from
the specification of the ideal. Such a distinction plays a role in a wide range of
applications, from manufacturing quality to commonsense reasoning with everyday objects. As
such, the distinction between ideal and actual objects can refer to an object that is simply
created in a way that violates the specification of the ideal, or it may refer to objects
that have been changed in damaged through use. The notion of damage we present in
this paper concretely arises from the representation of used goods in The Social Needs
Marketplace (SNM)[
From the motivating scenarios listed in the second section, we narrow our scope to the description of partially damaged goods. The specification of damage condition requires a comprehensive description to parthood relationships beyond mereological monism1. 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. In this paper we will focus on the representation of damage from the approach of pluralism to abnormality of solid physical objects, based on the multiple parthood relations raised from Ontology of Solid Physical Objects (SoPhOs). SoPhOs is 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 falls into what is often called the family of three-dimensional
representation of physical objects, in which all of an object’s parts exist at any point in
time. This approach can also be seen in the BFO [
Below are some scenarios that happened in SNM, and served as the motivation for the approach taken in this paper.
Expectant parents are looking for children’s furniture for their soon to arrive baby. They log into the SNM portal and search for a children’s bed with complete safety measures. The supply of children’s beds on the portal is limited returning only two partial matches to this family’s needs. A family whose children are now of school age is offering a single child’s bed with headboard missing, while another family is offering a bunk style bed for twins but it is convertible into a single bed. A furniture wholesale store is looking to relocate its warehouse due to the increasing rent, the store owner decides to mark down the selling prices heavily on most items stored in the warehouse to reduce the moving cost. However, some items are damaged because of improper storage or transportation, some tables have the corners cut out, some chairs have dents on the legs, and a sofa is also broken with filling coming out. For these items with minor damage, the store owner requests the warehouse manager to list them up on the SNM for donation.
A retired couple is looking to clean up their garage and donate some of their old time favorite tablewares. Among them are a ceramic coffee mug with a handle broken but that has interesting pattern on it, a crystal wine decanter with the detachable stand missing, an emptied cookie can that could be used as a daily container, and a dish with a chipped corner. They log in the portal and listed these items with detailed description.
The ontological commitments are semantic requirements we recognized from the motivating scenarios.
1. Damage to solid physical objects is determined comparing to the design. 1Mereological monism denotes that there is a single parthood relation. Approaches based on classical mereology [24] tend to use a single parthood relation to specify parthood relationships.
2. A partially damaged object can be presented in precise ways in terms of characteristics of solid physical objects. 3. There are multiple parthood relations of solid physical objects, and they are all distinct relations, each of which is synonymous with a mereology theory. Furthermore, there is no taxonomy of parthood relations. 4. There is different mereology for each part, and each mereology has different spatial ontology (radical pluralism). 5. In the scope of solid physical objects, each parthood relation is independently axiomatized with different characteristic module ontologies. 6. Each abnormality parthood relation is associated with a distinct parthood relation of solid physical object.
The seminal work of Winston et al. [25] identified multiple relations that capture
intuitions about parts and wholes, we follow this approach of mereological pluralism and
propose each parthood relation corresponds to a generic module of upper ontologies. We
adopt the sideways approach to upper ontologies [
The design of the Ontology of Solid Physical Objects (SoPhOs) follows the principle that each module of SoPhOs axiomatizes necessary conditions for solid physical objects. We define a solid physical object as an object that is made of some material, has some shape, and occupies some space. SoPhOs is built to expand the definition with two modules: matter module and shape module. Each module features a characteristic ontology in the upper ontology and gives rise to corresponding parthood relations: [portionOf, pieceOf, componentOf ] for parthood on solid physical objects, and [confinedIn, containedIn] for spatial parthood, respectively. The current design of the Ontology of Solid Physical Objects is shown below in Figure 1 and Figure 2. SoPhOs is fully axiomatized in First Order Logic2.
Our approach can be seen as a development along the lines mentioned in Artale’s
work [
Definition 4.1. 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 Hmereology Hierarchy3.
All of the parthood relations are given conservative definitions in their respective generic ontology modules, and hence are treated as defined relations. Real world physical examples of some parthood relation are shown below in Figure 3.
A solid physical object is a material object. Matter constitutes solid physical objects and
is one of the prime characteristics we determine for solid physical objects. Under the
matter module we have The Material Object Ontology, which axiomatizes the
constitution relation constitutes between Mat (matter) and MaterialObject, and defines chunkOf
as the parthood relation within matter. The Portion Ontology contains the
corresponding continuous parthood relation portionOf for the material objects. An example of
portionOf a material object is a one person portion of pizza in a whole pizza, or a bite of
an apple is a portion of the apple. The Attach Ontology features the connection relation
attach between material objects. Incorporating the spatial parthood relation region part
from Occupy Ontology[
The shape module of SoPhOs starts from the Feature Ontology which is an extension to
the BoxWorld[
The Bounds Ontology describes the relationship between the shaped object and the shape feature that bounds it. For the multidimensional parthood of solid physical objects regarding shape, we have Piece Ontology and Component Ontology based on dimensionality. We use pieceOf for shaped objects in different dimensions below or at three-dimension, and componentOf between three-dimensional shaped objects and a set of three-dimensional shaped objects. For instance the mug example in Figure 5, if we say the shape feature that bounds a non-detachable handle of a mug is the two dimensional cylinder surface s1 and the whole mug is bounded by a three dimensional box b1, then the handle of the mug is a piece of the mug in terms of shape. However, if we are talking about the three dimensional box b2 bounded cap and the whole mug with cap which is bounded by a poly4 p1, we use componentOf to define the parthood between the cap and the whole mug with cap. The cube in Figure 5 shows three instances of featureOf : f eatureO f ( f1; b1) ^ f eatureO f ( f2; b1) ^ f eatureO f ( f3; b1).
We use affix to denote the connection between pieces, and components are joint together, where joint is synonymous with ec (external connection) from classic mereotopology. Last but not least, we also have containedIn to describe the spatial parthood when the space occupied by the shape of one solid physical object is fully enclosed in the convex space occupied by the shape of another solid physical object (e.g. an egg is contained in the lunch box).
We find it is ambiguous to take ”ideal object” as the standard for determining abnormality. In real world, a chair can be designed in multiple different styles, it can have four legs, three legs, no legs like sofa, or one leg with four to six wheeled arms like an office chair; it is difficult and inaccurate to determine one specific design as a general ”ideal chair”. In contrast, we define the abnormality in terms of the intended properties instead of type or class, and differentiate abnormality with characteristics of the object.
Reiter [
M2, but in terms of solid physical objects, the spatial relationship between M1 and M2 cannot be changed unless the shape of M2 is altered. We use the notion :Ab to describe the condition when an object coincides with its design, that is, it is not abnormal, or ”ideal”.
Of course, some change/abnormality of characteristics will lead to inevitable change of other characteristics. The correlation between different characteristics are shown in Table 1. From the table we can conclude that the occurrence of ab portion, ab component or ab containment would all result in one or more ab piece, but an occurrence of ab piece might not cause any other partial abnormalities. The occurrence of either ab component or ab containment would result in both some ab portion and ab piece, but the former two are isolated from any subsequent occurrences.
It is important to distinguish our use of abnormality predicates from their earlier use in nonmonotonic reasoning. We are not using abnormality to represent typicality (e.g. a typical chair has four legs, a typical dining room table is made of wood, a typical bookshelf does not have wheels). Instead, we are using abnormality predicates to specify the intended properties of an (ideal) object, so that any divergence from the specification of these properties (e.g. missing or spurious parts) are indications of damage. If a particular class of chairs is designed to have three legs, then it is not considered to be damaged when an instance of this class has three legs, even if a typical chair has four legs. On the other hand, if a class of chairs is designed to have four legs, then an instance of this class is indeed damaged if it is missing a leg.
In the motivating scenarios from SNM specified in Section 2, 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 damaged” here might have different meanings: partially missing, partially broken, or material partially removed. We apply abnormality to scenarios of damages. A standard design of My Chair with three legs is represented in Axiom (1)-(3) as in Figure 6 below, showing the intended properties of components, pieces and portions respectively.
Axiom (4)-(8) in Figure 7 below showed example representations of damaged chairs. C1 denotes an replaced leg resulting an abnormal component; C2 denotes a dent in one leg of a three-leg chair resulting an abnormal piece; C3 denotes one leg missing; C4 denotes when there is an extra leg; last but not least, C5 denotes some additional matter is added to the chair.
Each class of solid physical objects is axiomatized by sentences of the form seen in Figure 6. By using the abnormality predicate Ab, we allow the existence of objects
(9y1; y2; y3)Leg(y1) ^ Leg(y2) ^ Leg(y3) ^ (y1 6= y2) ^ (y1 6= y3) ^ (y2 6= y3) ^componentO f (y1; x) ^ componentO f (y2; x) ^ componentO f (y3; x) ^((8z)componentO f (z; x) z = y1 _ z = y2 _ z = y3))
Leg(x) (:Ab(x) (9y1; y2; y3)bottom(y1) ^ side(y2) ^ top(y3) ^ (y1 6= y2) ^ (y1 6= y3) ^ (y2 6= y3) ^pieceO f (y1; x) ^ pieceO f (y2; x) ^ pieceO f (y3; x) ^((8z)pieceO f (z; x) z = y1 _ z = y2 _ z = y3))
^ab component(L1;C1) ^ componentO f (L2;C1) ^ componentO f (L3;C1) Ab(C1) (4)
MyChair(C2) ^ Leg(L4) ^ Leg(L5) ^ Leg(L6) ^component(L4;C2) ^ componentO f (L5;C2) ^ componentO f (L6;C2) ^Dent(P1) ^ Bottom(P2) ^ Side(P3) ^ Top(P4) ^ab piece(P1; L4) ^ pieceO f (P2; L4) ^ pieceO f (P3; L4) ^ pieceO f (P4; L4) Ab(C2)
Ab(C3) Ab(C4) (1) (2) (3) (5) (6) (7)
MyChair(C4) ^ Leg(L9) ^ Leg(L10) ^ Leg(L11) ^ Leg(L12) ^ab component(L9;C4) ^ componentO f (L10;C4) ^ componentO f (L11;C4) ^ componentO f (L12;C4)
Ab(C5) (8) in a class even if they do not satisfy the conditions for the ideal object in that class; inconsistency is avoided since such an object is simply an abnormal instance of the class. Furthermore, we can use the parthood relations in SoPhOs to identify the nature of the abnormality – missing matter vs. spurious matter, missing shape feature vs. unintended shape features, missing components vs. extra components.
Recall the motivating scenarios mentioned in Section 2, we can now help the SNM users to describe the damages in used goods (the actual terminology can be more userfriendly and easier to interpret by daily users, depends on the wording selection of the application designer):
The expectant parents can eliminate the single children’s bed with missing headboard by adding a filter to their request that map into the category with :ab components.
The warehouse manager can list the tables with corners cut out and the chairs with dents on the legs in the category of ab piece, and the sofa is broken with some ab portion forms came out.
The kind retired couple can donate the ceramic coffee mug with a handle broken as an entry with ab piece, the crystal wine decanter with the detachable stand missing as with ab component, the emptied cookie can as having ab containment, and the dish with a chipped corner as havng either ab portion or ab piece.
Our representation of damage follows Reiter’s[
Many representations of solid physical objects involve time as a fourth
dimension. Bennett[
Some three-dimensional representations to physical objects define by matter and
space. 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. [
Existing studies in mereology usually either have a single part-of relation to
summarize all parthood relations as mereological monism following the classical
mereology[24], 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. This is
5Stratified 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.” [
This paper continues the approach of characteristic upper ontologies modules supported multiple distinct parthood relations from Ru and Gru¨ninger[23].
Ideally, we want to incorporate the Process Specification Language[
What’s the damage? We can now represent both the chair with a dent in one leg and the mug with a handle broken with having abnormal pieces comparing to their intended properties. Starting from the motivating scenarios of partially damaged furnitures from Social Needs Marketplace, we have proposed a representation of damage deriving from partial abnormality. We have described the abnormality of solid physical objects based on the Ontology of Solid Physical Objects (SoPhOs), a general suite of upper ontology modules we proposed and axiomatized in First Order Logic. SoPhOs is a complete system of ontologies featuring the characteristic parthood, spatial parthood, parthood connection theories, and the foundational matter and shape characteristic ontologies that support them. We employ the approach of mereological pluralism, in which each mereological relation corresponds to one characteristic of the solid physical object, and the characteristics are formalized in different modules within an upper ontology. Damage of used goods can be represented in terms of the partial abnormal characteristics. We have introduced four predicates for partial abnormality corresponding to the fundamental characteristics and parthood relationships within solid physical objects: ab portion; ab piece; ab component, and ab containment.
The current framework presented in this paper is able to represent different conditions of partially damaged objects and solve the scenarios raised in SNM. This domain will also serve as a testbed for identifying new concepts in upper ontologies required for the representation of damages in everyday objects.