=Paper=
{{Paper
|id=Vol-2518/paper-FOUST3
|storemode=property
|title=Properties Defined on the Basis of
Coincidence in GFO-Space
|pdfUrl=https://ceur-ws.org/Vol-2518/paper-FOUST3.pdf
|volume=Vol-2518
|authors=Ringo Baumann,Frank Loebe,Heinrich Herre
|dblpUrl=https://dblp.org/rec/conf/jowo/BaumannLH19
}}
==Properties Defined on the Basis of
Coincidence in GFO-Space==
https://ceur-ws.org/Vol-2518/paper-FOUST3.pdf
Properties Defined on the Basis of
Coincidence in GFO-Space
Ringo BAUMANN a , Frank LOEBE a,1 and Heinrich HERRE b
a
Computer Science Institute, University of Leipzig, Germany
b
IMISE, University of Leipzig, Germany
Abstract. The General Formal Ontology (GFO) is a top-level ontology that in-
cludes theories of time and space, two domains of entities of fundamental charac-
ter. In this connection the axiomatic theories of GFO are based on the relation of
coincidence, which can apply to the boundaries of time and space entities. Two co-
incident, but distinct boundaries account for distinct temporal or spatial locations
of no distance. This is favorable for modeling or grasping certain situations. At
a second glance and especially concerning space, in combination with axioms on
boundaries and mereology coincidence also gives rise to a larger variety of entities.
This deserves ontological scrutiny.
In the present paper we introduce a new notion (based on coincidence) referred
to as ‘continuousness’ in order to capture and become capable of addressing an
aspect of space entities that, as we argue, is commonly implicitly assumed for basic
kinds of entities (such as ‘line’ or ‘surface’). Carving out that definition first leads
us to adopting a few new axioms for GFO-Space. Together with the two further
properties of connectedness and ordinariness we then systematically investigate
kinds of space entities.
Keywords. top-level ontology, ontology of space, coincidence, GFO
1. Introduction
The General Formal Ontology (GFO) [1] is one of those top-level ontologies that aims at
accounting for entities of time and space by means of axiomatic theories about these do-
mains. In line with philosophical work of Franz Brentano [2], firstly GFO considers time
and space as manifestations of the more abstract notion of a continuum. Furthermore,
there is another idea by Brentano with deep impact on the theories of time and space
developed for GFO: It is the idea that certain time or space entities can coincide / are
coincident. Two entities coincide, if, intuitively speaking, they are compatible and they
yield temporal or spatial co-locations (for entities that are in time and/or space). Put
differently, they must be “congruent” and there is no distance between them.
In the case of time, the motivating idea for this relationship is that if a time interval
is partitioned into two sub intervals that meet, then both of these intervals are equipped
1 Corresponding Author: Frank Loebe, University of Leipzig, Computer Science Institute, P.O. Box 100920,
04009 Leipzig, Germany. E-mail: frank.loebe@informatik.uni-leipzig.de Copyright © 2019 for this paper by
its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
with their own two time points that start and end the respective sub interval. For example,
if the period of an hour is split into two halves of 30 minutes each, each half is claimed
to have a first and a last time point. In contrast to classical topology, the last point of the
first 30 minutes and the starting point of the subsequent 30 minutes are understood as
two distinct time entities, whereas measuring time yields zero distance between them –
these two time points coincide.
The notion of coincidence is intimately tied to the notion of boundaries, which de-
rives from the relation of one entity being a boundary of another one. Indeed, time points
in GFO are understood as the boundaries of time intervals. Boundaries existentially de-
pend on the entities they are boundaries of.
Regarding space entities, matters become substantially more complex. Not only two
points on a line may coincide (in analogy to time intervals and time points), but lines
themselves may be coincident. In that case they are boundaries of a partitioned surface.
The possibility to coincide extends further to surfaces, two of which may coincide if they
result from “slicing” a three-dimensional space entity, say, a sphere into an upper and a
lower part. Again, each part is conceived to be fully bounded. Where the upper and the
lower part touch each other, those (parts of the maximal) boundaries coincide – they are
congruent, but distinct and have no measurable distance.
Corresponding theories for GFO have been developed and are most recently ex-
pounded in [3] for the domain of time (also referred to as “Brentano Time”, thus abbrevi-
ated as BT ). Another work [4] presents an initial version of a GFO space theory BS (for
“Brentano Space”). As it turns out, there are still kinds of entities that are conceivable
in the theory BS, e.g., based on common mereological assumptions in combination with
the availability of coincidence, which should be scrutinized ontologically.
The current paper reports on work in progress to extend BS, focusing on properties
that are derived from coincidence and which shall allow for a finer-grained classification
of space entities. Sect. 2 motivates our research on continuousness, as a newly deter-
mined property in the context of GFO-Space. The subsequent sect. 3 presents a solution
exemplified at the level of zero- and one-dimensional entities, by defining continuous
line entities on the basis of directionally compatible points. Then we gather systemat-
ically and illustrate the kinds of line entities that can be distinguished on the basis of
continuousness, ordinariness and connectedness, in sect. 4. Besides concluding the pa-
per, the final sect. 5 points to related work and indicates the continuation of the work
presented herein.
We lack space in this paper in order to recapitulate all relevant content of the the-
ory BS and thus we mainly refer to [4] for this purpose. However, the most uncom-
mon notion of the coincidence of spatial boundaries, which is a primitive in BS, is al-
ready introduced above. For all remaining notions of major relevance, we believe that
GFO-Space is rather close to an intuitive-commonsensical understanding of phenomenal
space. This concerns the remaining three primitive notions of space region, of being a
spatial boundary of an entity and of spatial parthood, i.e., the latter two are relations.
All other categories and relations are defined. Space entities are understood to be of dis-
tinct dimensions, referred to as either zero- to three-dimensional entities or as point, line
and surface entities for lower dimensions and space region for three-dimensional ones.
We mention in addition only the relation of being a hyper part of a space entity. Given
that spatial parthood applies only among entities of the same dimension, the term ‘hyper
part’ is used to refer to “parts” of entities with co-dimension ≥ 1. For example, a point
or a line segment “within” a space region are called hyper parts of that region. A final,
technical note concerns the numbering of formulas: we presuppose the identifiers of all
definitions, axioms and theorems as in [4]. Thus, formula identifiers herein do not start
at 1, but rather continue or refer to the counting in accordance with [4].
2. Motivation – The Quest for Continuousness
On the one hand, coincidence is clearly a useful “tool” for modeling certain situations
of space entities, such as allowing for fully bounded entities that meet / touch each other.
This applies all the more in connection with material entities that occupy space entities,
cf. [4, sect. 3 and 5]. On the other hand, coincidence in combination with mereological
principles leads to a richer typology of entities compared to an ontology that lacks coin-
cidence. In the theory BS there are two central aspects of mereological summation. First,
summation is restricted to equidimensional entities, which follows from the stronger re-
quirement that already spatial parthood implies equidimensionality [4, A7]. Secondly,
the formation of mereological sums is not only unconstrained for equidimensional enti-
ties, but it is even enforced as a principle of existence.
A16. eqdim(x, y) → ∃z sum(x, y, z) (existence of sum)
In combination with coincidence, this leads easily to kinds of entities that are uncommon
compared to intuitive notions such as points or lines, for instance. Imagine two lines,
which happen to be coincident with one another. Each of these lines is a one-dimensional
entity, hence by the axiom just given there is the mereological sum of those two lines. In
contrast to a “common” line entity, there are distinct parts of the entity under considera-
tion that coincide. This kind of entities has already been singled out in BS (as presented
in [4]) by distinguishing ordinary from extraordinary entities.
D22. ExOrd(x) ∶= ∃yz (spart(y, x) ∧ spart(z, x) ∧ ¬sov(y, z) ∧ scoinc(y, z))
(x is an extraordinary space entity)
D23. Ord(x) ∶= ¬ExOrd(x) (x is an ordinary space entity)
In the course of further studies another case of entities occurred that exhibits features
that are similarly surprising compared to common intuitions, that is not covered by the
notion of extraordinariness and that turns out to be substantially harder to grasp – ideally
in the form of a definition, but at least by an axiomatic characterization. We refer to this
notion as continuous or, nominalized, as continuousness. Our prototypical example that
is not continuous or discontinuous in the realm of one-dimensional entities is the line
entity consisting of the components y ′ and x′ in Fig. 1. Notably, speaking of lines or
drawing lines such as y ′ and x′ (each considered on its own), we argue that these are
implicitly assumed to be continuous in the sense that we intend to understand and believe
to have captured in sect. 3 below.
In its four ‘scenes’ (or sections) identified by the numbered scrolls, Fig. 1 provides
material in order to illustrate and intuitively explain the discontinuous line entity in
scene 3. In slight abuse of formal notation, we may use names of entities in illustrations
in argument positions of predicates of BS, such as writing 1D(y ′ ) for the fact that the
entity y ′ in Fig. 1 stands for a line, i.e. (so far), a one-dimensional entity that is connected
and ordinary.
Figure 1. “Constructing” an extraordinary (2) and a discontinuous (3) line entity.
Scene 0 shows two unconnected spatial rectangles x and y. Each is a fully bounded
entity, but the relation sb of being a spatial boundary of an entity neither assumes nor
entails maximality. Hence (and among other segments of the maximal boundaries), the
two lines x′ and y ′ are spatial boundaries of x and y, respectively. These lines, in turn,
have their own zero-dimensional boundaries, such that sb(z1, y ′ ), sb(z2, y ′ ), sb(z3, x′ )
and sb(z4, x′ ) apply.
Next, consider scene 1 with two alike rectangles x and y (and their boundaries x′
and y ′ ), but which are in contact with / touch each other. x′ and y ′ are still distinct and
do not even overlap, but now they coincide (which in BS accounts for the fact that x and
y touch). Accordingly, the sum of x′ and y ′ is an extraordinary entity. The latter remains
the case in scene 2, where x′ and y ′ are laterally offset, albeit x′ and y ′ themselves do
not coincide. But they have parts that do so, namely those parts directly opposed to each
other in the area where x and y still touch. (Note that x and y are not displayed in scene 2
any more, but are still there, not at least because boundaries must have an entity that they
are a boundary of.)
Eventually, in scene 3 (x and y and thus) x′ and y ′ are offset in such a way that
(1) there are no more parts of x′ and y ′ that coincide, but (2) it is the case that z2 coincides
with z3. The mereological sum of x′ and y ′ is no longer extraordinary, but ordinary,
due to (1). Moreover, it is a connected entity because of (2). Up to this point, there
is no notion in BS that would allow us to distinguish that mereological sum from an
“intuitively proper” line, such as x′ in itself.
We refer to x′ and y ′ as continuous lines, whereas the sum of the two does not exhibit
such continuousness. The sum contains a kind of “crack” due to switching from one part
originating from a surface above it (y) to another part originating from a surface below it
(x). Considering the level of zero-dimensional entities, we perceive a difference between
an arbitrary pair of coincident points within the line x′ (or within the line y ′ ) and the
special situation of z2 and z3. The latter two points do coincide, but they are boundaries
of parts of the sum of x′ and y ′ that originate from surfaces that differ in the specific way
just indicated. Insofar we say that z2 and z3 are not directionally compatible, whereas
pairs of coincident points within, say, x′ are called directionally compatible.
Besides in itself raising this problem, the main aims of this paper are to capture these
intuitive notions of continuousness and directional compatibility formally precisely and
to apply continuousness together with extraordinariness in classifying line entities.
3. Directional Compatibility and Continuousness
The development of the definitions of directional compatibility and continuousness can-
not reasonably be reflected here in terms of various intermediate versions that failed in
one way or another prior to reaching the results presented in this section. Therefore, the
section gathers only newly required preliminaries for those definitions, actually intro-
duces and discusses them and concludes with a few new axioms and theorems that played
a role in dealing with continuousness and that serve as an extension of the theory BS as
in [4].
3.1. Distinguishing Strict and Weak Boundaries of Entities
The primitive relation sb of being a spatial boundary turned out to be not sufficiently
discriminating in order establish the targeted definitions. Returning to Fig. 1, note that
in all four scenes it is the case that (1) sb(z1, y ′ ), sb(z2, y ′ ), sb(z3, x′ ) and sb(z4, x′ )
as well as (2) each of the points z1, . . . , z4 is a boundary of the mereological sum of x′
and y ′ . However, referring to scenes 2 and 3, we need to distinguish between boundaries
such as z1 and z4, on the one hand, and z2 and z3 on the other hand. The former are
called strict boundaries (of the sum), the latter weak boundaries.
D34. strictsb(x, y) ∶= sb(x, y) ∧ ∀x′ (hypp(x′ , y) ∧ scoinc(x, x′ ) → x = x′ )
(x is a strict spatial boundary)
D35. weaksb(x, y) ∶= sb(x, y) ∧ ¬strictsb(x, y) (x is a weak spatial boundary)
We illustrate the effects of the definitions a bit further with Fig. 2. Let us denote by
s′ the sum of x′ and y ′ . Then, again, z1, . . . , z4 are boundaries of s′ . z2 is a weak spatial
boundary of s′ , because there are exactly two distinct zero-dimensional hyper parts h′1
and h′2 (of x′ and thus also of s′ ) which coincide with z2 (and are also each distinct from
z2). Analogously, there are two hyper parts of y ′ that coincide with z3, the second weak
boundary of s′ . The boundaries z1 and z4 are indeed strict boundaries, since all points
coincident with one of the two are “outside” of x′ and y ′ , respectively, i.e. they are no
hyper parts of them.
Figure 2. Weak and Strict Boundaries.
3.2. Definitions of Directional Compatibility and Continuousness
Let us first state the definitions of directional compatibility and continuousness formally.
Herein, we focus on continuousness of one-dimensional entities (considered as embed-
ded in a surface), which requires 1-directional compatibility of zero-dimensional entities,
but confer the outlook in sect. 5 for remarks on the general case.
D36. 1dircomp(x, y) ∶= scoinc(x, y) ∧
∃x′ y ′ z ′ ( Ord(x′ ) ∧ Ord(y ′ ) ∧ sb(x, x′ ) ∧ sb(y, y ′ ) ∧ ¬sov(x′ , y ′ ) ∧
(SB(x′ ) ∧ SB(y ′ ) →
∃x′′ y ′′ z ′′ g ′ ( sb(x′ , x′′ ) ∧ sb(y ′ , y ′′ ) ∧ ¬sov(x′′ , y ′′ ) ∧
sum(x′′ , y ′′ , z ′′ ) ∧ grsb(g ′ , z ′′ ) ∧ 0dhypp(x, g ′ ) ∧ 0dhypp(y, g ′ ) ∧
∀z(0dhypp(z, g ′ ) →
((scoinc(x, z) ∧ ¬sov(x, z) → y = z) ∧
(scoinc(y, z) ∧ ¬sov(y, z) → x = z) )))))
(x and y are 1-directionally compatible)
D37. 1cont(x) ∶= 1DE(x) ∧
∀y ( (0dhypp(y, x) ∧ ∀y ′ (spart(y ′ , y) → ¬strictsb(y ′ , x))) →
∃z(0dhypp(z, x) ∧ ¬sov(z, y) ∧ 1dircomp(y, z)))
(x is 1-continuous)
A space entity x is 1-continuous iff it is a 1-dimensional entity and for each of
its zero-dimensional hyper parts y that (*) has no strict boundary of x among its parts,
there is another zero-dimensional hyper part that (**) does not overlap with y, but is 1-
directionally compatible with it. Note that in the case of y being a point, (*) boils down
to not being a strict boundary of x, while (**) then just means that there must be another,
1-directionally compatible point within x. Rephrased, except for its strict boundaries,
all points within a continuous line entity must be complemented by a 1-directionally
compatible point that is also within the line entity. This is the gist of the definition, while
the more general formulations of (*) and (**) take mereological sums of points into
account.
Returning to Fig. 2, point z2 witnesses that the sum entity s′ is not 1-continuous.
The points h′1 and h′2 within x′ (and s′ ) are mutually 1-directionally compatible, but
none of the two is 1-directionally compatible with z2. Since h′1 and h′2 are the only
points within s′ that coincide with z2 , but are distinct from it, there is no complementary,
1-directionally compatible point for z2 in s′ – thus s′ is discontinuous. An analogous
argument applies to z2 in scene 3 of Fig. 1, where the only candidate within the sum
entity that is distinct from z2, but coincides with it, is z3 – which is not 1-directionally
compatible with z2.
Admittedly, the definition of (1-)directional compatibility is the more challenging
one, based on the following key ideas. First, only coincident points x and y can be 1-
directionally compatible. In order to verify the latter, in addition there must be ordinary,
non-overlapping line entities x′ and y ′ as well as non-overlapping surface entities x′′ and
y ′′ ,2 such that x “derives” from x′ and thus x′′ (i.e, sb(x, x′ ), sb(x′ , x′′ )) and likewise
for y, y ′ and y ′′ . The next idea is to consider the mereological sum of those two surfaces
and inspect it’s greatest spatial boundary g ′ for additional points that coincide with x and
y. If there is no such point, then x and y are indeed 1-directionally compatible, whereas
if there is a third coincident point within g ′ , they are not.
Reconsider first scene 1 of Fig. 2 to see where we drew the inspiration for this
definition from. The boundaries h′1 and h′2 are (intuitively) directionally compatible. We
2 The precondition of x′ and y ′ being spatial boundaries is satisfied by any ordinary line entity. That condition
becomes relevant if the definitional scheme is applied to define 3-directional compatibility of surfaces, cf.
sect. 5.
can select x′1 and x′2 with h′1 and h′2 as boundaries, and then select two non-overlapping,
but touching surfaces below x′1 and x′2 , respectively, such that the sum of those surfaces
has x′ as part of its greatest boundary. Within x′ , h′1 and h′2 remain the only points
that coincide. There are further points that coincide with the two, for example one on a
boundary (of the assumed left-hand surface), which leads downwards from h′1 . However,
since those surfaces below x′1 and x′2 touch, all those points do not belong to x′ (nor the
greatest boundary of the sum of the surfaces), but are inside them.
In contrast, returning to scene 3 of Fig. 1, where z2 and z3 are (intuitively) not di-
rectionally compatible, there it is not possible, starting from z2 and z3, to find corre-
sponding lines and (non-overlapping) surfaces, such that those surfaces touch (in order
to achieve the same effect as above, that additional points coinciding with z2 and z3
“disappear” within the sum of those surfaces). The reason for that impossibility is that
any such surface at the side of z2 extends upwards from y ′ (or a part of y ′ ), whereas
any surface at the side of z3 extends downwards from x′ . The greatest boundary of the
sum of such surfaces will therefore have four line segments as parts, all of which meet
at (also) z2 and z3, which means that z2 and z3 coincide with two further points within
that greatest boundary.
Sect. 4 comprises additional examples that demonstrate the effects of the definitions
here given. Beforehand, we present new axioms introduced to the theory BS together
with further theorems derived in the course of determining those definitions.
3.3. New Axioms and New Theorems
The claim near the end of the previous sub section, that “any surface at the side of z3
extends downwards from x′ ” in scene 3 of Fig. 1 is centrally based on one of three
new axioms adopted for BS (namely A31). Indeed, we stipulate that boundaries are not
boundaries of arbitrary objects. To the contrary, there is “local information” inherent in
each boundary about the entities that it is a boundary of.
A31. sb(x, y) ∧ sb(x, z) → ∃u(sb(x, u) ∧ sppart(u, y) ∧ sppart(u, z))
(entities with a common boundary share a proper part at that boundary)
A32. sb(x, y) ∧ sb(u, v) ∧ x ≠ u →
∃y ′ v ′ ( spart(y ′ , y) ∧ sb(x, y ′ ) ∧ spart(v ′ , v) ∧ sb(u, v ′ ) ∧ ¬sov(y ′ , v ′ ))
(entities with distinct boundaries have parts at those boundaries that do not overlap)
A33. LDE(x) ∧ Ord(x) → SB(x)
(ordinary lower-dimensional entities are spatial boundaries)
A33 is added to exclude intuitively strange cases of ordinary entities that could not
be derived from – eventually – space regions by the sb relation. But A31 and A32 are
much more insightful regarding the nature of boundaries and coincidence, in that the
dependence of boundaries on the entities they are boundaries of involves a kind of ex-
clusiveness concerning the question which entities they actually can be boundaries of.
Interestingly, we further observe that A31 can be seen as a generalization of A28 and
A29 of the time theory BT C in [3].
The remainder of this section lists new sentences that emerged while working on
the definitions of continuousness and directional compatibility, which can be shown to
follow from BS (including A31–A33). For each theorem, Table 1 gathers a set of axioms
and definitions from which it can be proved.
T17. SReg(x) → ¬SB(x) (space regions are no spatial boundaries)
T18. SReg(x) → Ord(x) (space regions are ordinary)
T19. LDE(x) → (Ord(x) ↔ SB(x))
(for lower-dimensional ent. no difference between spatial boundary and ordinariness)
T20. scoinc(x, y) → Ord(x) ∧ Ord(y) (coincidence requires ordinariness)
T21. 1dircomp(x, y) → 1dircomp(y, x) (symmetry of 1-directional compatibility)
T22. ¬1dircomp(x, x) (irreflexivity of 1-directional compatibility)
Theorem Sub Theory
T17 D4, A2, A26
T18 D22, A2, A22
T19 A2, A22, A33, T18
T20 A22, T19
T21 D36
T22 D36, A31
Table 1. Theorems and sub theories from which they are provable.
4. Classification of Line Entities
In this section we distinguish line entities w.r.t. the properties of ordinariness, continu-
ousness and connectedness, which implies that the section also demonstrates further ef-
fects of the definition of continuousness as formally introduced above. It turns out that all
three properties are independent, i.e., any combination of them has witnessing examples.
The subsequent illustrations provide for examples all of which are connected, while we
discuss disconnected variants below.
All sub figures in this section are constructed in a way that is meant to support the
verification of continuousness or discontinuousness of the sum entity that is shown in
scence 3 of each sub figure, i.e., on the right-hand side. The scenes to the left allow one
to explain from which line entities (x′ , y ′ ) that sum can be derived (scenes 1 and 2), as
well as which surface entities have those lines as boundaries (x, y; scene 0). The analyses
of those cases yield the classifications specified in the figure captions.
Figure 3. Two cases, where x′ , y ′ and their sum are each ordinary, continuous
and connected.
Figure 4. Two cases in which the sum of x′ and y ′ is extraordinary, continu-
ous and connected.
Considering the lower part of Fig. 4, imagine that we punch a circle out of a square
(scene 0). The two resulting objects y, the negative of the circle, and x, the circle itself,
possess coinciding spatial boundaries y ′ and x′ (scenes 1 and 2). The mereological sum
of them is an example of an extraordinary and continuous line entity (scene 3), where the
extraordinariness should be obvious. In order to see that the sum is continuous, it suffices
to realize that any single point located on y ′ or x′ possesses exactly one directionally
compatible counterpart on y ′ or x′ , respectively.
Disconnected versions of all cases above can be obtained by considering space enti-
ties that correspond to “copies” of each other, but which do neither overlap nor share any
hyper parts. Such “copies” are insensible to ordinariness and continuousness, i.e., these
properties apply likewise to the sum of two such copies as they do to each “copy”.
Figure 5. Two cases in which the sum of x′ and y ′ is ordinary, discontinuous
and connected.
Figure 6. Two cases in which the sum of x′ and y ′ is extraordinary,
discontinuous and connected.
5. Conclusions
This paper is a work-in-progress report on dealing with specific issues identified in con-
nection with the relation of coincidence in space ontologies, exemplified by the develop-
ment of the space ontology [4] of GFO, the General Formal Ontology [1].
Concerning closely related work, we must admit not to be aware of other works
that deal with a space ontology that involves coincidence, and even less the problem of
continuousness. With a wider focus, space is clearly an important topic, cf. e.g. [5,6,7],
albeit we see no closer connection to continousness, as well.
Our work continues to generalized definitions of directional compatibility as well as
of continuousness, of which the definitions D36 and D37 form instances. Currently, we
test these definitions further with entities of higher dimension, i.e., with lines, surface
and space regions, preparing a classification extended to those types of entities.
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