=Paper=
{{Paper
|id=Vol-2518/paper-SHAPES3
|storemode=property
|title=More than just One Box
|pdfUrl=https://ceur-ws.org/Vol-2518/paper-SHAPES3.pdf
|volume=Vol-2518
|authors=Yi Ru,Michael Grüninger
|dblpUrl=https://dblp.org/rec/conf/jowo/RuG19
}}
==More than just One Box==
https://ceur-ws.org/Vol-2518/paper-SHAPES3.pdf
More than just One Box
Yi RU a and Michael GRÜNINGER a
a Mechanical and Industrial Engineering, University of Toronto, Canada
Abstract. To support the representation of furniture assembly, we need an ontology
to describe the shapes of integrated three-dimensional objects. There are few exist-
ing formal axiomatizations in this domain. MWorld Ontology, with nine modules,
is a first order logic ontology proposed in this paper which allows the description
of topological shapes composed of boxsets, boxes, surfaces, edges and points. We
introduced boxset as the class for the shapes of integrated three-dimensional ob-
jects, componentOf as the proper parthood relation between the components and
the whole, as well as semicomplements to capture the relationship between the dis-
joint components of the same whole. We proposed terms for coincident shapes in
adjacent components: joint point, joint edge and joint surface. We reused Card-
World and BoxWorld, as an extension, we introduced featureOf as a new part-
hood relation to represent the relationship between substructure or basic shapes and
their superstructure. As such, these theories can represent aspects of shapes with
different dimensions following multidimensional and mereological pluralism ap-
proaches. All concepts and relationships are axiomatized and demonstrated with
examples.
Keywords. Shape Ontology, Multidimensional, Integrated Objects, Component,
Solid Physical Objects, Parthood, Topology, Mereological Pluralism
Use Case Expecting parents Alice and Bob were looking for children’s furniture for
their soon to arrive baby, they logged into the SNM portal and made a request. A
family whose children are now in university was offering a crib, but it was disassem-
bled and had been stored in garage for years. Bob accepted the kind offer and wanted
to give a try with assembling the bed. Following the instruction manual, he started
with the headboard section. First step was to insert support bars into the headboard
bottom cross and push down until firmly in place. Then he slid the headboard top
cross onto the support bars. At last he inserted the wood dowels two at a time into
ends of both headboard top and bottom crosses to connect with the two headboard
posts.
1. Introduction
The above use case is drawn from Social Needs Marketplace (SNM)[16]. To support the
representation of such an assembly process, as demonstrated in Figure 1, we of course
need a process ontology to specify states, activities, and different orderings on the oc-
currences of activities. We also need an ontology to describe the different components
of the three-dimensional physical object at each stage of the assembly process, as well
Copyright c 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution
4.0 International (CC BY 4.0).
as the shapes of these components. Within this paper, we will focus on an ontology to
logically specify such components and their topological shapes.
Figure 1. Simulated Instruction of Headboard Assembly
Following the principle of ontology lifecycle, we reuse two closely related existing
first-order ontologies for surfaces and boxes, CardWorld[8] and BoxWorld1 [6]. There
are four disjoint categories of entities within a model domain of the CardWorld and
BoxWorld Ontologies: points, edges, surfaces, and boxes, corresponding to the shapes
of zero-, one-, two-, and three-dimensional objects, respectively. For two-dimensional
objects, CardWorld captures the relationship between points, edges, and surfaces. For
three-dimensional objects, BoxWorld describes properties of a single box and its parts.
The set of edges in a surface, the set of border edges, and the set of edges that meet
at the same vertex, each forms a cyclic ordering. Both ontologies only use the notions
of incidence and betweenness, rather than the Euclidean geometry as axiomatized by
Hilbert and Tarski. And they have their basic ontological commitments based on a binary
relation, part, which describes the incidence relations between different categories of
objects.
But how about the shapes of objects that are composed by more than one box?
In this paper, we introduce MWorld Ontology, which is a first order logic ontology
for shapes of integrated three-dimensional objects. Axioms of MWorld are categorized
into nine modules. The relationships between these theories, the BoxWorld and the Card-
World Ontology is shown in Figure 2 below. We also extended the BoxWorld with a
featureOf parthood relation to represent the parthood relationship between a shape sub-
structure and the whole, for example, holes and voids can be features of a box. There are
three parthood relations in MWorld Ontology. We reuse part as the basic parthood rela-
tion between basic shapes of different dimensions in a weak tripartite incidence struc-
ture, an example is that edge is a part of some surface. In MWorld, we named compo-
nentOf as the proper parthood relationship between the composing box or boxset and
composed boxset. Corresponding modules of these parthood ontologies are axiomatized
and synonymous with the classic mereology.
2. Extension to BoxWorld
We introduce featureOf as the general parthood relation between shapes in different
dimensions in BoxWorld. Following Axiom (1), a feature in the BoxWorld is an enclosed
1 colore.oor.net/boxworld/boxworld.clif
Figure 2. Relationships between the modules in the CardWorld, BoxWorld2 and MWorld Ontologies. Solid
lines denote conservative extension and dashed lines denote nonconservative extension.
shape or its substructure. It can be a basic atomic shape in its dimension: a point in zero-
dimension, an edge in one-dimension, a surface in two-dimension and a box in three-
dimension. A feature can also be a hole, the shape of a void, and a corner of a table with
three edges meet at the same vertex. featureOf is the parthood relation between feature
and the whole. The extended BoxWorld has one new module boxworld f eature and one
updated closure module boxworld2.
(∀x, y) f eatureO f (x, y) ⊃ (∃z) f eatureO f (z, y) ∧ ¬ f eatureO f (z, x) (1)
f1 to f5 in Figure3 below are examples of features of a cube.
Figure 3. Examples of features to a cube
Linking back to our motivation scenario, Bob can now indicate the wear at the top
of both posts, where most corners are already rounded and the straight corner features
have disappeared, showing the signs of age.
3. MWorld
We need representation of a collection of connected three-dimensional shapes to de-
scribe the structure of furniture, for example the assembled children’s bed. Thus, we in-
troduce MWorld as an extension to BoxWorld that was able to represent shapes that are
composed of multiple boxes. The axioms of MWorld are decomposed into nine mod-
ules: mworld part, mworld edge, mworld border, mworld peak, mworld component,
mworld joint, mworld complement, mworld f eature and mworld.
The module with basic ontological commitments mworld part extends boxworld part
with the new class of boxset, elements of which are composed of boxes. The mod-
ule mworld edge is an adjustment from boxworld edge with violating axioms re-
moved, for instance, ”Every border meets another unique border at a vertex.” is no
longer true in MWorld, as in Figure 5(v), two borders meet at a joint edge. The mod-
ules mworld border and mworld peak are adjustments from boxworld border and
boxworld peak with updated importing statements from MWorld. As in the updated
BoxWorld above, featureOf in boxsets also represents incomplete shapes - it can be a
hole in the boxset, a void, or a boxset with some surfaces missing. The sole axiom in the
module mworld is a closure axiom, so that all objects are either points, edges, surfaces,
boxes or boxsets.
3.1. Boxset and its components
In MWorld, we introduce a new parthood relation componentOf, which is a primitive
relation to describe the proper parthood between the boxes or boxsets composing a boxset
and the whole boxset, as axiomatized in Axiom (2) and (3) from Figure 4. Axioms (4)
and (5) ensure that a boxset is composed by at least two components and every box in
a boxset meets another distinct box in that boxset. Axiom (6) shows that two boxes or
boxsets are semicomplements of each other when they are both components of the same
boxset and they don’t overlap in features or boxes.2
One thing to note is that boxsets are composed of boxes, but a box is not always
part of a boxset. This nature is unlike the basic ontological commitments of CardWorld
and BoxWorld, where the part relation forms a weak tripartite incidence structure over
the three disjoint sorts of objects: every point is part of an edge, every edge is part of a
surface, and every surface is part of a box.
3.2. Joints that coincide
A joint as defined in Axiom (7), either coincides with a part of a component or is created
to be coincident with the intersection between two components. We say that two shapes
coincide when they are congruent and there is no distance between them. Axiom (8)
makes sure that if there is a part of a component that coincides with a joint, then there
must be another component of the same whole that meets this component at the joint.
Examples of how boxsets can be composed by boxes are listed in Figure 5. Examples
5(i) to 5(iv) show the scenarios of a joint point: 5(i) shows that point p1 of cone b2 meets
surface s1 of cube b1 at joint point j p1 which coincides with p1 ; 5(ii) shows that point
p1 of cube b1 meets point/vertex p2 of cone b2 at joint point j p1 which coincides with
p1 and p2 ; 5(iii) shows that point p1 of cone b2 meets edge/boundary e1 of cube b1 at
joint point j p1 which coincides with p1 ; and 5(iv) shows that edge/boundary e1 of cube
2 In Lattice Theory, the definition to semicomplement is: in a lattice L bounded below an element y is called
complement of x if x ∧ y = 0; and L is said to be semicomplemented (SC) if each x ∈ L (with x 6= 1 if 1 exists
in L) admits at least one nonzero semicomplement.[17]
(∀x, y)componentO f (x, y) ⊃ (Box(x) ∨ Boxset(x)) ∧ Boxset(y) (2)
(∀x, y)componentO f (x, y) ⊃ (∃z)componentO f (z, y) ∧ ¬componentO f (z, x) (3)
(∀b1 , x)componentO f (b1 , x) ⊃ (∃b2 )componentO f (b2 , x) ∧ (b1 6= b2 ) (4)
(∀b1 , x)componentO f (b1 , x) ∧ Box(b1 ) ⊃
(∃b2 , j)componentO f (b2 , x) ∧ Box(b2 ) ∧ (b1 6= b2 ) ∧ meets(b1 , b2 , j) (5)
(∀x, y)semicomplements(x, y) ⊃ ¬componentO f (x, y) ∧ ¬componentO f (y, x) ∧ (x 6= y)∧
((¬∃z)componentO f (z, x) ∧ componentO f (z, y))∧
((∃b)Boxset(b) ∧ componentO f (x, b) ∧ componentO f (y, b)) (6)
(∀ j)Joint( j) ⊃ (∃b1 , b2 , y1 , y2 )semicomplements(b1 , b2 ) ∧ box(b1 ) ∧ box(b2 )∧
part(y1 , b1 ) ∧ part(y2 , b2 ) ∧ (coincides( j, y1 ) ∨ coincides( j, y2 )∨
(coincides( j, y1 ) ∧ coincides( j, y2 )) ∨ ((∃i)intersection(i, y1 , y2 ) ∧ coincides( j, i))) (7)
(∀b1 , x, j)(∃y)componentO f (b1 , x) ∧ Joint( j) ∧ part(y, b1 ) ∧ coincides( j, y) ⊃
(∃b2 )componentO f (b2 , x) ∧ meets(b1 , b2 , j) (8)
Figure 4. Selected Axioms from Tcomponent , T joint and Tsemicomplement in MWorld Ontology
b1 meets the surface s1 of sphere b2 at a new joint point j p1 . The shape of sphere is a box
consisting of one sole surface, and it does not have any edge or point.
Example 5(v) to 5(vii) show the scenarios of a joint edge: 5(v) shows that edge e1
of cube b1 meets edge e2 of cube b2 at joint edge je1 which coincides with e1 and e2 ,
similarly, we also have p1 of b1 and p3 of b2 coincide at j p1 and p2 of b1 and p4 of
b2 coincide at j p2 ; 5(vi) shows that surface s1 of cube b1 meets surface s2 of cube b2
at a joint area js1 , while the boundaries of js1 coincide with e1 and e2 as je1 and je2
respectively, in addition, the four circled joint points of js1 also coincide with all four
vertices of both je1 and je2 ; and 5(vii) shows that surface s1 of cube b1 meets surface s2
of cylinder b2 at the new joint edge je1 , and the two vertices of je1 , which are also the
interceptions between s2 and e1 and e2 respectively, are also created as j p1 and j p2 .
Example 5(viii) and 5(ix) show the scenarios of a joint surface/area: in example
5(viii), two cubes are stacked together horizontally, where surface s1 of cube b1 meets
surface s2 of cube b2 at the new joint area js1 , and similarly as in 5(vi), boundaries/edges
and vertices/points of js1 coincide with the boundaries/edges and vertices/points of s1
and s2 ; last but not least, 5(ix) shows that surface s1 of cube b1 meets surface s2 of
Figure 5. Examples of Joints
cylinder b2 at joint area js1 which coincides with s2 , and similarly, the boundary/edge e1
of s2 and b2 is coincided with joint edge je1 .
In a similar case as example 5(viii), if the two cubes are melted in with each other in-
stead of stacked together, the surface s1 of cube b1 and the surface s2 of cube b2 will dis-
appear with no new joint surface created. Same situation goes for the boundaries/edges
and vertices/points that are melted in.
Figure 6. Illustrations of Headboard Assembly
Back to our story at the beginning of this paper, in the first step of assembling the
headboard, Bob inserted the support bars into the headboard bottom cross-piece. Figure
6(a) simulates the scenario where the headboard bottom cross and one of the bars are
placed separately. The assembled shape is demonstrated in 6(aa), where the middle joint
edge je1 , bottom joint edge je2 , bottom joint area js1 and round joint surface js2 are
specified with the original shapes they coincide with. Bob then slid the headboard top
cross onto the support bars, similar to the example 5(ix) where the joint area js1 is on the
bottom surface s1 of the headboard top cross b1 , and coincides with the top surface s2
of one of the support bars b2 . The representation of relationships in 6(aa) are shown in
Axiom (9) below.
Boxset(bs1 )
∧ componentO f (b1 , bs1 ) ∧ componentO f (b2 , bs1 ) ∧ semicomplements(b1 , b2 )
∧ Box(b1 ) ∧ Sur f ace(s3 ) ∧ Sur f ace(s5 ) ∧ Edge(e1 ) ∧ Edge(e3 )
∧ f eatureO f (s3 , b1 ) ∧ f eatureO f (s5 , b1 ) ∧ f eatureO f (e1 , b1 ) ∧ f eatureO f (e3 , b1 )
∧ Box(b2 ) ∧ Sur f ace(s2 ) ∧ Sur f ace(s4 ) ∧ Edge(e2 )
∧ f eatureO f (s2 , b2 ) ∧ f eatureO f (s4 , b2 ) ∧ f eatureO f (e2 , b2 )
∧ JointEdge( je1 ) ∧ coincides( je1 , e1 ) ∧ f eatureO f ( je1 , bs1 )
∧ JointEdge( je2 ) ∧ coincides( je2 , e2 ) ∧ coincides( je2 , e3 ) ∧ f eatureO f ( je2 , bs1 )
∧ JointSur f ace( js1 ) ∧ coincides( js1 , s2 ) ∧ coincides( js1 , s3 ) ∧ f eatureO f ( js1 , bs1 )
∧ JointSur f ace( js2 ) ∧ coincides( js2 , s5 ) ∧ f eatureO f ( js2 , bs1 )
∧ Sur f ace(s6 ) ∧ f eatureO f (s6 , bs1 ) (9)
The last step Bob performed was inserting the wood dowels into the ends of crosses,
two dowels at a time. This scenario is captured by Figure 6(b) and 6(bb), while the
assembly of each individual dowel is similar with the first step featured in Figure 6(aa),
the adjacency of two dowels creates another joint edge marked as je3 .
4. Shapes and Objects
For objects with shapes, we also have pieceOf from SoPhOs[15] as the parthood relation
in shaped objects. The pieceOf relation is a defined relation:
(∀x, y)pieceO f (x, y) ≡ (∃ f1 , f2 )bounds( f1 , x) ∧ bounds( f2 , y) ∧ f eatureO f ( f1 , f2 )
(10)
where bounds is a primitive relation in SoPhOs captures the relationship between an
enclosed or non-enclosed shape feature and the object that is partially or fully bounded
by the shape feature. featureOf is a reflexive parthood relation in the extended BoxWorld
Ontology as mentioned above in Section 2. One corresponding example is that the handle
object of a coffee mug is a piece of the mug object. Of course, the pieceOf relation
is able to describe shape identified parthood relationship of both atomic shaped objects
and integrated shaped objects. For instance, the back piece of an assembled dining chair
includes the back cushion and the upper piece of the wooden frame.
4.1. Shape Spatial Structure - MultiDimensional Occupy
Pieces share boundaries with the whole, while containment does not. Shape of a solid
physical object encloses some physical space, it is represented with the multidimensional
Occupy Ontology[7]. The parthood relationship of the physical spaces occupied by the
shapes of physical entities is denoted by containedIn:
(∀x, y)containedIn(x, y) ≡ (∃r1 , r2)occupies(x, r1 )∧occupies(y, r2 )∧region part(r1 , r2 )
(11)
5. Relationship to Assembly Processes
We began the paper with a motivating scenario that described the assembly of a baby’s
crib. How is an ontology for assembly processes related to the shape ontology described
within this paper?
Using the PSL Ontology as the underlying generic process ontology, the notion of
state is represented by reified fluents. Intuitively, a change in state is captured by fluents
that are either achieved or falsified by an activity occurrence. The prior relation is used
to specify the fluents that are intuitively true prior to an activity occurrence and the holds
relation specifies the fluents that are intuitively true after an activity occurrence. Further-
more, a fluent can only be changed by the occurrence of activities. Thus, if some fluent
holds after an activity occurrence, but after an activity occurrence later along the branch
it is false, then an activity must occur at some point between that changes the fluent. This
also leads to the requirement that the fluents holding after an activity occurrence will
be the same fluents that are prior to any successor occurrence, since there cannot be an
activity occurring between them.
Using the methodology introduced by [1], we classify all possible activities in a do-
main by characterizing possible all changes in the domain. We translate a domain ontol-
ogy to a domain state ontology. Activity occurrences correspond to mappings between
models of the domain ontology. Finally, we classify activities with respect to possible
changes.
For example, the mereology on material objects leads to a classification of ma-
terial removal and addition activities. A mereology of components leads to activi-
ties that change the corresponding componentO f fluent. Assembly activities achieve
componentO f , while disassembly activities falsify the componentO f fluent.
6. Previous Research
There are limited existing formal axiomatizations in first order logic for shape represen-
tations. Shape grammars[18] have been proposed as a way of modelling the shapes of ob-
jects; however, such approaches do not provide a logical theory, and hence do not support
automated reasoning through deduction or model construction. We therefore focus only
on ontological approaches. In [2], Aameri proposed the Shape Ontology as an extension
to CardWorld and BoxWorld, with an extended module describing relationships between
multiple boxes. In her Shape Ontology, she considers the entity that consists of multiple
boxes as one dimension higher than the three-dimensional box which is a different ap-
proach than that of this paper. We took the perspective that the integration of multiple
three-dimensional shapes is still three-dimensional shape, and we introduce features and
joints to represent substructures and superstructure of boxes which allow description to
more shape forms. The approach of multidimensional mereotopology are also adopted
in GFO-Space theory[4] and CODIB[9], which deal with an arbitrary mereotopology
instead of focusing on shapes of objects.
Our multidimensional approach falls into what is often called the family of 3D rep-
resentation 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 continuants of BFO[3] and the endurants
of DOLCE[12] upper ontologies, although these upper ontologies are based on a time-
indexed version of mereological monism.
The term component is commonly adapted in works of mereology, the meaning we
give is similar to the previous works, but the application domain is different. We fol-
low the approach of mereological pluralism[14]. In one of the earliest works in the area,
Winston[19] presented a taxonomy of part-whole relations, and included component-
integral object as a parthood relationship but for abstract concepts like phonology to lin-
guistics. Later, Odell[13] also included component as one of his six proposed kinds of
aggregation relationships. However, neither Winston nor Odell provided axiomatizations
of their different parthood relations. In more recent work, Keet[10] introduced a tax-
onomy as summarization of Odell’s approach to types of part-whole relations[13], and
also provided OWL axiomatizations of the taxonomy. Bittner and Donnelly[5] have also
presented an axiomatization of CmpOf in biological ontology.
Koslicki lists material components and formal components as kinds of proper parts
in her book[11]. She defines that material components are intuitively from which these
wholes come into existence, and formal components act as a sort of recipe in specifying
the range and configuration of material components eligible to compose a whole of this
kind. Our concept of componentOf is more aligned with the definition of the formal
component concept, but in the domain of shape representation.
7. Conclusion
There are few existing first order logic axiomatizations in describing shapes of integrated
three-dimensional objects. With Bob’s assembly scenario in the use case as an moti-
vation, we proposed MWorld Ontology and reused existing topological shape ontolo-
gies CardWorld and BoxWorld. MWorld is a first order logic ontology with nine mod-
ules, which allows the description of topological shapes from zero-dimension to three-
dimension, including boxsets, boxes, surfaces, edges and points. We named boxset as the
class for the shapes of integrated three-dimensional objects, componentOf as the proper
parthood relation between the composing components and the whole, as well as semi-
complements as the relation between disjoint components of the same whole. We also
proposed terms in MWorld for the coincident shapes in adjacent components: joint point,
joint edge and joint surface. Furthermore, as an extension to BoxWorld, we introduced
featureOf as a new general parthood relation to capture the relationship between sub-
structure or basic shape and its superstructure. As such, these theories can represent as-
pects of the shapes with different dimensions following multidimensional and mereolog-
ical pluralism approaches. Future researches to MWorld include discussions to convex-
ity, granularity and shape orientation. In addition, to better follow the mereological plu-
ralism approach, we suggest that CardWorld and BoxWorld be revised with a different
terminology other than part for the parthood relationship between basic shape entities
and the whole. What’s more, with incorporation of process and motion ontologies, we
can then describe Bob’s full day of fun assembly.
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