More than just One Box

Abstract

To support the representation of furniture assembly, we need an ontology to describe the shapes of integrated three-dimensional objects. There are few existing 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 objects, componentOf as the proper parthood relation between the components and the whole, as well as semicomplements to capture the relationship between the disjoint components of the same whole. We proposed terms for coincident shapes in adjacent components: joint point, joint edge and joint surface. We reused CardWorld and BoxWorld, as an extension, we introduced featureOf as a new parthood 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 approaches. All concepts and relationships are axiomatized and demonstrated with examples.

Keywords

• Shape Ontology Multidimensional Integrated Objects Component Solid Physical Objects Parthood Topology Mereological Pluralism

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 occurrences 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 as the shapes of these components. Within this paper, we will focus on an ontology to logically specify such components and their topological shapes.

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 CardWorld 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 substructure 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 relation between basic shapes of different dimensions in a weak tripartite incidence structure, an example is that edge is a part of some surface. In MWorld, we named componentOf 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 1colore.oor.net/boxworld/boxworld.clif shape or its substructure. It can be a basic atomic shape in its dimension: a point in zerodimension, an edge in one-dimension, a surface in two-dimension and a box in threedimension. 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.

(8x; y) f eatureO f (x; y) (9z) f eatureO f (z; y) ^ : f eatureO f (z; x) (1) f1 to f5 in Figure3 below are examples of features of 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 describe the structure of furniture, for example the assembled children’s bed. Thus, we introduce 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 modules: 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 module mworld edge is an adjustment from boxworld edge with violating axioms removed, 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 modules 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 jp1 which coincides with p1; 5(ii) shows that point p1 of cube b1 meets point/vertex p2 of cone b2 at joint point jp1 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 jp1 which coincides with p1; and 5(iv) shows that edge/boundary e1 of cube 2In 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 2 L (with x 6= 1 if 1 exists in L) admits at least one nonzero semicomplement.[ 17 ]

(8x; y)componentO f (x; y) (Box(x) _ Boxset(x)) ^ Boxset(y) (8x; y)componentO f (x; y) (9z)componentO f (z; y) ^ :componentO f (z; x) (8b1; x)componentO f (b1; x) (9b2)componentO f (b2; x) ^ (b1 6= b2)

(8b1; x)componentO f (b1; x) ^ Box(b1) (9b2; j)componentO f (b2; x) ^ Box(b2) ^ (b1 6= b2) ^ meets(b1; b2; j) (8x; y)semicomplements(x; y)

:componentO f (x; y) ^ :componentO f (y; x) ^ (x 6= y)^ ((:9z)componentO f (z; x) ^ componentO f (z; y))^ ((9b)Boxset(b) ^ componentO f (x; b) ^ componentO f (y; b)) (8 j)Joint( j) (9b1; 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)) _ ((9i)intersection(i; y1; y2) ^ coincides( j; i))) (8b1; x; j)(9y)componentO f (b1; x) ^ Joint( j) ^ part(y; b1) ^ coincides( j; y) (9b2)componentO f (b2; x) ^ meets(b1; b2; j) (2) (3) (4) (5) (6) (7) (8) b1 meets the surface s1 of sphere b2 at a new joint point jp1. 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 jp1 and p2 of b1 and p4 of b2 coincide at jp2; 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 jp1 and jp2.

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 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 instead of stacked together, the surface s1 of cube b1 and the surface s2 of cube b2 will disappear with no new joint surface created. Same situation goes for the boundaries/edges and vertices/points that are melted in.

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: (8x; y)pieceO f (x; y) (9 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: (8x; y)containedIn(x; y) (9r1; 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. Furthermore, 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 domain by characterizing possible all changes in the domain. We translate a domain ontology 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 material removal and addition activities. A mereology of components leads to activities 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 representations. Shape grammars[ 18 ] have been proposed as a way of modelling the shapes of objects; 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 approach 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 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 continuants of BFO[ 3 ] and the endurants of DOLCE[ 12 ] upper ontologies, although these upper ontologies are based on a timeindexed 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 follow 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 componentintegral object as a parthood relationship but for abstract concepts like phonology to linguistics. 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 taxonomy 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 motivation, we proposed MWorld Ontology and reused existing topological shape ontologies CardWorld and BoxWorld. MWorld is a first order logic ontology with nine modules, which allows the description of topological shapes from zero-dimension to threedimension, 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 semicomplements 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 substructure or basic shape and its superstructure. As such, these theories can represent aspects of the shapes with different dimensions following multidimensional and mereological pluralism approaches. Future researches to MWorld include discussions to convexity, granularity and shape orientation. In addition, to better follow the mereological pluralism 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.

References

[1] Aameri , B. ( 2012 ). Using Partial Automorphisms to Design Process Ontologies . Formal Ontology in Information Systems: Proceedings of the Seventh International Conference (FOIS 2012 ), pages 309 - 322 .

[2] Aameri , B. and Gruninger , M. ( 2017 ). Encountering the Physical World . Eighth Conference on Formal Ontology Meets Industry. In: Proceedings of the Joint Ontology Workshops 2017 Episode 3 : The Tyrolean Autumn of Ontology, Bozen-Bolzano, Italy, September 21-23 , 2017 .

[3] Arp , R. , Smith , B. , Spear , A. D. ( 2015 ). Building Ontologies with Basic Formal Ontology .

[4] Baumann , R. , Loebe , F. and Herre , H. ( 2016 ). Towards an ontology of space for GFO . In Ninth International Conference on Formal Ontology in Information Systems , pages 53 - 66 . Nancy, France.

[5] Bittner , T. and Donnelly , M. ( 2005 ). Computational ontologies of parthood, componenthood, and containment . In Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI 2005 ), pages 382 - 387 .

[6] Gruninger , M. ( 2011 ). Thinking outside (and inside) the box . In SHAPES 1.0 Conference , Karlsruhe, Germany.

[7] Gruninger , M. , Aameri , B. , Chui , C. , Hahmann , T. , Ru , Y. ( 2018 ). Foundational Ontologies for Units of Measure . Proceedings of the 10th International Conference on Formal Ontology in Information Systems. Cape Town , South Africa.

[8] Gruninger , M. and Delaval , A. ( 2009 ). A first order cutting process ontology for sheet metal parts . In Brender, J., Christensen , J. P. , Scherrer , J.R. , and McNair , P., editors, Proceedings of the 2009 conference on Formal Ontologies Meet Industry , pp. 22 - 33 .

[9] Hahmann , T. ( 2013 ). A reconciliation of logical representations of space: from multidimensional mereotopology to geometry . PhD thesis , PhD thesis , Univ. of Toronto, Dept. of Comp. Science.

[10] Keet , M. C. ( 2006 ). Introduction to part-whole relations: Mereology, conceptual modelling and mathematical aspects . KRDB Research Centre Technical Report KRDB06-3.

[11] Koslicki , K. ( 2008 ). The Structure of Objects. The Dichotomous Nature of Wholes , pp. 176 - 191 .

[12] Masolo , C. , Borgo , S. , Gangemi , A. , Guarino , N. , Oltramari , A. ( 2003 ). Ontology library . WonderWeb Deliverable D18 (ver. 1.0 , 31 - 12 - 2003 ). http://wonderweb.semanticweb.org.

[13] Odell , J., editor ( 1998 ). Six different kinds of composition, Advanced Object-Oriented Analysis and Design using UML . Cambridge University Press.

[14] Ru , Y. , and Gruninger , M. ( 2017 ). Parts Unknown: Mereologies for Solid Physical Objects . Eighth Conference on Formal Ontology Meets Industry. In: Proceedings of the Joint Ontology Workshops 2017 Episode 3 : The Tyrolean Autumn of Ontology, Bozen-Bolzano, Italy, September 21-23 , 2017 .

[15] Ru , Y. , and Gruninger , M. ( 2018 ). What's the Damage? Abnormality in Solid Physical Objects . Sixth International Workshop on Ontologies and Conceptual Modelling , Cape Town, South Africa.

[16] Rosu , D. , Aleman , D. M. , Beck , J. C. , Chignell , M. , Consens , M. , Fox , M. S. , Gruninger , M. , Liu , C. , Ru , Y. , Sanner , S. ( 2017 , March). Knowledge-Based Provisioning of Goods and Services: Towards a Virtual Social Needs Marketplace . In 2017 AAAI Spring Symposium Series.

[17] Stern , M. ( 1999 ). Semimodular lattices: theory and applications (Vol. 73 ). Cambridge University Press.

[18] Thaller , W. , Krispel , U. , Zmugg , R. , Haveman , S. , Fellner , D. ( 2013 ) Shape gammars on convex polyhedra , Computers and Graphics 37 : 707 - 717 .

[19] Winston , M. , Chan , R. , and Herrmann , D. ( 1987 ). A taxonomy of part-whole relations . Cognitive Science , 11 : 417 - 444 .