=Paper=
{{Paper
|id=Vol-2977/paper12
|storemode=property
|title=Connecting Granular and Topological Relations through Description Logics (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2977/paper12.pdf
|volume=Vol-2977
|authors=Elio Hbeich,Ana Roxin,Nicolas Bus
|dblpUrl=https://dblp.org/rec/conf/esws/HbeichRB21
}}
==Connecting Granular and Topological Relations through Description Logics (short paper)==
https://ceur-ws.org/Vol-2977/paper12.pdf
Connecting Granular and Topological Relations through
Description Logics
Elio Hbeich 1, 2, Ana Roxin2, and Nicolas Bus1
1
Université of Bourgogne Franche-Comté– LIB EA7534, Dijon 21000, France
2
Information System and Applications Division, CSTB, Sophia Antipolis 06560, France
*email:elio.hbeich@cstb.fr, ana-maria.roxin@u-bourgogne.fr,
nicolas.bus@cstb.fr
Abstract. Granularity deals with organizing in greater or lesser detail data, in-
formation, and knowledge that resides at a granular level. This organization is
carried out according to certain criteria, which thereby provide a context view or
dimension also called granular perspective. Topological relations express spatial
associations among geospatial features (points, polylines, and polygons); they
represent a horizontal spatial analysis. The two domains allow scientists to con-
ceive different perspectives of the world. In this article, we aim to combine the
two representations through Description Logics (DL) rules to relate granular
(vertical representation) and geospatial topological (horizontal representation) re-
lations. The following consequences are thus noted: (1) geospatial features be-
come granules, (2) geospatial features are grouped into different levels of granu-
larity and different granules, and finally, (3) granular construction and decompo-
sition operations are integrated into the spatial domain.
Keywords: Geospatial Data, GeoSPARQL, Description Logic, Topological Re-
lations, Granular Computing, Granular Relations.
1 Introduction
Scientists are continually endeavoring to structure their perception of the environ-
ment and the world, i.e., geospatial and building data. With the recent advances in Ar-
tificial Intelligence (AI), there is a growing need to analyze and reason over such data
in the context of numerous use cases, i.e., disaster management, compliance checking
(Bus et al., 2018). Granular Computing (GrC) has been recognized as a promising ap-
proach for representing human reasoning and problem solving through the levels of
granularity (Keet, 2008). It considers for modelling a specific domain of knowledge or
a worldview. While not fully implemented in ontology languages (such as OWL), GrC
relationships structure knowledge into multi-level hierarchies by identifying parts of
such knowledge, their relations, and their connections to the whole. In the present arti-
cle, we seek to identify logical relations between GrC principles and existing topolog-
ical relations as defined for existing geographical datasets. The overall goal is to use
such logical rules to help automatically build perspectives or granular levels for
knowledge in a specific area. Like the Level of Detail (LoD) concept used in CityGML
“Copyright c 2021 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0).”
2
(Gröger et al., 2012), such logical rules would facilitate different computing perspec-
tives of the geospatial data available for a considered area. We aim to combine the two
representations using Description Logics (DL) rules to relate granular and geospatial
topological relations. We consider geospatial features as granules, grouped into differ-
ent levels of granularity and different granules. The article is organized as follows: Sec-
tion 2 introduces geospatial topological relations; section 3 presents the granular com-
puting perspective, notions, and relationships; section 4 introduces related work; sec-
tion 5 presents DL rules that relate topological and granular relationships, section 6
discusses our use case and future work, and finally we conclude in section 7.
2 Geospatial Topological Relations
Geospatial data contains geospatial features with geographic aspects, e.g., objects,
locations. Geospatial features are connected through two types of geospatial relations:
geometrical and topological (Zlatanova, 2015). This article focuses on topological con-
nections, as they provide a general structure linking geospatial features. Geo-
SPARQL topological relations are used to link data among different geospatial re-
sources. (Perry & Herring, 2012) GeoSPARQL is an Open Geospatial Consortium
(OGC) standard used to represent and query geospatial data on the Web. Furthermore,
it provides a vocabulary for asserting topological relations between geospatial features
such as simple feature, RCC8 (Region Connection Calculus), and Egenhofer (see table
below).
Table 1. GeoSPARQL topological relations (GeoSPARQL Ontology, 2012)(Avail-
able online at http://schemas.opengis.net/geosparql/1.0/geosparql_vocab_all.rdf# ).
Simple Feature Egenhofer RCC8
Equals sfEquals Equals ehEquals Equals rcc8eq
Disjoint sfDisjoints Disjoint ehDisjoint Disconnected rcc8dc
Intersects sfIntersects Meets ehMeet Externally connected rcc8ec
Touches sfTouches Overlaps ehOverlap Partially overlaps rcc8po
Within sfWithin Covers ehCovers Tangential proper part inverse rcc8tppi
Contains sfContains Covered by ehCoveredBy Tangential proper part rcc8tpp
Overlaps sfOverlaps Inside ehInside Non-tangential proper part rcc8ntpp
Crosses sfCrosses Contains ehContains Non-tangential proper part inverse rcc8ntppi
The vocabulary described above illustrates connectivity, adjacency, and enclosure
relations among geospatial features. For example, sfTouches describes whether two ge-
ospatial features are next to each other.
3 Granular Computing
(Yao, 2007) explains that GrC is studied from three perspectives philosophical,
methodological, and computing. In this article, we are interested in the philosophical
perspective as it explores the compositing of parts, their relations, and their connections
3
to the whole. GrC’s philosophical perspective exploits structures in terms of granules,
levels, and hierarchies based on multilevel representations. Consequently, a granule can
be considered part of another granule or may include a family of granules, creating a
hierarchical structure. (J. T. Yao et al., 2013) details three basic notions of GrC: (1)
Granule: defined as a small particle among numerous particles forming a large unit. It
represents classes, objects, data, elements, or any sort of real or virtual information. The
partition of a granule into smaller ones results in subgranules. (2) Granulation: presents
construction or decompose operations. The construction process forms high-level gran-
ules from lower-level subgranules; the decomposition process splits high-level granules
into lower-levels subgranules. (3) Granular relationships: used as a foundation to
gather lower-level granules into higher-level ones (interrelationship) or to split high-
level granules into low-level granules (intrarelationship). Note that high-level granules
represent abstract concepts, and lower-level granules represent specific concepts. In the
presented work, we consider the DL formalisms for modelling granules and the granu-
lar relations between them. The notation granular-relation(x, y) thus represents the
granular relation between granules x and y. We also consider granules x and y as con-
cepts (or concept instances) in DL. Based on these assumptions, the table below pre-
sents granular relationships as defined in (J. T. Yao et al., 2013) :
Table 2. Granular relationships mathematical definition and DL notation
Relation Definition DL notation
𝑥𝑖 ∈ 𝑋, 𝑦𝑗 ∈ 𝑌 refine(x, y)
Refine
∀𝑥𝑖 ∈ 𝑋 ⇒ 𝑥𝑖 ⊂ ∃ 𝑦𝑗 ∈ 𝑌
Coarse 𝑥𝑖 ∈ 𝑋, 𝑦𝑗 ∈ 𝑌 coarse(x, y)
∀𝑦𝑗 ∈ 𝑌 ⇒ 𝑦𝑗 ⊃ ∃ 𝑥𝑖 ∈ 𝑋
Partial fine 𝑥𝑖 ∈ 𝑋, 𝑦𝑗 ∈ 𝑌 prefine(x, y)
∃𝑥𝑖 ∈ 𝑋 ⇒ 𝑥𝑖 ⊄ ∀𝑦𝑗 ∈ 𝑌
Partial coarse 𝑥𝑖 ∈ 𝑋, 𝑦𝑗 ∈ 𝑌 𝑝𝑐𝑜𝑎𝑟𝑠𝑒(𝑥, 𝑦)
∃𝑦𝑗 ∈ 𝑌 ⇒ 𝑦𝑗 ⊅ ∀𝑥𝑖 ∈ 𝑋
Partition 𝑖𝑓 𝑈 𝑎 𝑓𝑖𝑛𝑖𝑡𝑒 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛 𝜋 = {𝑋𝑖 |1 ≤ 𝑖 ≤ 𝑚}, 𝑡ℎ𝑒𝑛 partition(x, y)
𝑚
𝑋𝑖 ≠ ∅, ∀𝑖 ≠ 𝑗, 𝑋𝑖 ∩ 𝑋𝑗 = ∅, ⋃ 𝑋𝑖 = 𝑈
1
Covering 𝑖𝑓 𝑈 𝑎 𝑓𝑖𝑛𝑖𝑡𝑒 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛 𝜋 = {𝑋𝑖 |1 ≤ 𝑖 ≤ 𝑚}, 𝑡ℎ𝑒𝑛 covering(x, y)
𝑚
𝑋𝑖 ≠ ∅, ∀𝑖 ≠ 𝑗, ⋃ 𝑋𝑖 = 𝑈
1
𝑚,𝑛
Similar 1 similar(x, y)
𝑆𝑖𝑚 (𝑥, 𝑦) = ∑ 𝑆𝑖𝑚(𝑥𝑖 , 𝑦𝑗 )
𝑚 × 𝑛
𝑖=1 𝑗=1
In GrC domain, the term subgranule is used to differentiate granules that exist on two
different levels, where the high-level contains granules and the low-level contains sub-
granules. In other words, refine (g1, g2) indicates that g1 and g2 are distinct granules
located on two different granular levels, and that g1 is located on a higher-level than
g2. As a result, g1 is considered a granule, while g2 is considered a subgranule. To
incorporate GrC notions (granules, levels, and hierarchies) into OWL, we need to ad-
dress certain differences between them. While OWL differentiates concepts and in-
stances, GrC represents them both using granules. While OWL uses subclass relation
4
to refer to hierarchy, this relation does not represent the complexities of the granular
world (high-level and lower-level granule). At the same time, OWL uses different types
of relations to represent connection between concepts, instances, etc. GrC only uses
granular relationships. Therefore, the below DL rules allow building granular levels
that do not exist in OWL.
4 Related Work
Inspired by geospatial topological relations, the authors in (Dube & Egenhofer,
2009) created coarse topological relations to analyze data in different contexts. For ex-
ample, inside and coveredBy are grouped in a single relation called IN to indicate that
one region is a proper subset of the other. In addition, the authors mapped both topo-
logical relations to address the zonal representation of the relations neighborhoods be-
tween spatial entities. (Fent et al., 2005) proposes an extension of GeoGraph called
Granular GeoGraph that supports spatial and semantic granularity by adding granular
notion (aggregation and generalization) to geospatial conceptual models. These notions
allow modification in geometry description and topological relation between spatial
objects. The authors in (Khamespanah et al., 2016) propose a reliable model for earth-
quake vulnerability assessment to manage the uncertainty associated with the experts’
opinions. To achieve their objective geospatial data were integrated using Dempster-
Shafer theory, and granular-tree was applied to extract rules with minimum incompat-
ibility from the information table provided by the experts.
5 Connecting Topological and Granular Relations through
Logical Rules
Topological relations describe the interactions among geospatial features horizon-
tally; they consider that all geospatial features are situated on the same level or layer.
GrC builds knowledge in a hierarchical structure by exploring the composition/decom-
position of parts, their relations, and connections. Nevertheless, both domains represent
information from orthogonal perspectives. We have noted the potential connection be-
tween granular and topological relations. To apply the philosophical perspective of GrC
to the spatial domain, we develop a set of DL rules to build granular levels from geo-
spatial data and their topological relationships, thus bringing a multilevel interpretation
to the spatial domain. Before connecting granular and topological relations, we noticed
the following (equivalent: ≡, not: ¬, Union:∪, Intersection ∩):
𝑠𝑓𝐸𝑞𝑢𝑎𝑙𝑠 ≡ 𝑒ℎ𝐸𝑞𝑢𝑎𝑙𝑠 ≡ 𝑟𝑐𝑐8𝑒𝑐
𝑠𝑓𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡𝑠 ≡ 𝑒ℎ𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡 ≡ 𝑟𝑐𝑐8𝑑𝑐
𝑠𝑓𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑠 ≡ ¬𝑒ℎ𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡 ≡ ¬𝑟𝑐𝑐8𝑑𝑐
𝑠𝑓𝑇𝑜𝑢𝑐ℎ𝑒𝑠 ≡ 𝑒ℎ𝑀𝑒𝑒𝑡 ≡ ¬𝑟𝑐𝑐8𝑑𝑐
𝑠𝑓𝑊𝑖𝑡ℎ𝑖𝑛 ≡ (𝑒ℎ𝐼𝑛𝑠𝑖𝑑𝑒 ∪ 𝑒ℎ𝐶𝑜𝑣𝑒𝑟𝑒𝑑𝐵𝑦) ≡ (𝑟𝑐𝑐8𝑛𝑡𝑝𝑝 ∪ 𝑟𝑐𝑐8𝑡𝑝𝑝𝑖)
𝑠𝑓𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠 ≡ (𝑒ℎ𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠 ∪ 𝑒ℎ𝐶𝑜𝑣𝑒𝑟𝑠) ≡ (𝑟𝑐𝑐8𝑛𝑡𝑝𝑝𝑖 ∪ 𝑟𝑐𝑐8𝑡𝑝𝑝)
𝑠𝑓𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑠 ≡ 𝑒ℎ𝑂𝑣𝑒𝑟𝑙𝑎𝑝 ≡ 𝑟𝑐𝑐8𝑝𝑜
5
The above equivalences imply that if one topological relation is associated with a
granular relationship, its equivalent vocabulary is also linked. For example, if sfCon-
tains is connected to refine relation and sfContains ≡ (ehCoveredBy U ehInside), then
(ehCoveredBy U ehInside) is connected to the refine relation. However, GrC (granules
and granular relationships) has not been implemented as an ontology, such as Geo-
SPARQL [7]. Hence, the tables below represent DL rules that relate topological and
granular relationships.
Table 3. DL rules relating granular and Simple Feature topological relations
DL rules
1 𝑠𝑓𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑦) → 𝑟𝑒𝑓𝑖𝑛𝑒(𝑥, 𝑦)
2 𝑠𝑓𝑊𝑖𝑡ℎ𝑖𝑛(𝑦, 𝑥) → 𝑐𝑜𝑎𝑟𝑠𝑒(𝑦, 𝑥)
3 𝑠𝑓𝐶𝑟𝑜𝑠𝑠𝑒𝑠(𝑥, 𝑦) → 𝑝𝑟𝑒𝑓𝑖𝑛𝑒(𝑥, 𝑦) ∩ 𝑝𝑐𝑜𝑎𝑟𝑠𝑒(𝑦, 𝑥)
4 𝑠𝑓𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑠(𝑥, 𝑦) → 𝑝𝑟𝑒𝑓𝑖𝑛𝑒(𝑥, 𝑦) ∩ 𝑝𝑐𝑜𝑎𝑟𝑠𝑒(𝑦, 𝑥)
5 𝑠𝑓𝐸𝑞𝑢𝑎𝑙𝑠(𝑥, 𝑦) → 𝑠𝑖𝑚𝑖𝑙𝑎𝑟(𝑥, 𝑦)
6 𝑠𝑓𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑦) ∩ 𝑠𝑓𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠 (𝑥, 𝑧) ∩ 𝑠𝑓𝑇𝑜𝑢𝑐ℎ𝑒𝑠(𝑦, 𝑧) → 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛(𝑥, 𝑦) ∩ 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛(𝑥, 𝑧)
7 𝑠𝑓𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑦) ∩ 𝑠𝑓𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠 (𝑥, 𝑧) ∩ 𝑠𝑓𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡(𝑦, 𝑧) → 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛(𝑥, 𝑦) ∩ 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛(𝑥, 𝑧)
8 𝑠𝑓𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑦) ∩ 𝑠𝑓𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠 (𝑥, 𝑧) ∩ 𝑠𝑓𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑠(𝑦, 𝑧) → 𝑐𝑜𝑣𝑒𝑟𝑖𝑛𝑔(𝑥, 𝑦) ∩ 𝑜𝑣𝑒𝑟𝑖𝑛𝑔(𝑥, 𝑧)
9 𝑠𝑓𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑦) ∩ 𝑠𝑓𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠 (𝑥, 𝑧) ∩ 𝑠𝑓𝐶𝑟𝑜𝑠𝑠𝑒𝑠(𝑦, 𝑧) → 𝑐𝑜𝑣𝑒𝑟𝑖𝑛𝑔(𝑥, 𝑦) ∩ 𝑜𝑣𝑒𝑟𝑖𝑛𝑔(𝑥, 𝑧)
Table 4. DL rules relating granular and Egenhofer topological relations
DL rules
10 𝑒ℎ𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑦) ∪ 𝑒ℎ𝐶𝑜𝑣𝑒𝑟𝑠(𝑥, 𝑦) → 𝑟𝑒𝑓𝑖𝑛𝑒(𝑥, 𝑦)
11 𝑒ℎ𝐼𝑛𝑠𝑖𝑑𝑒(𝑦, 𝑥) ∪ 𝑒ℎ𝐶𝑜𝑣𝑒𝑟𝑒𝑑𝐵𝑦(𝑦, 𝑥) → 𝑐𝑜𝑎𝑟𝑠𝑒(𝑥, 𝑦)
12 𝑒ℎ𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑠(𝑥, 𝑦) → 𝑝𝑟𝑒𝑓𝑖𝑛𝑒(𝑥, 𝑦) ∩ 𝑝𝑐𝑜𝑎𝑟𝑠𝑒(𝑦, 𝑥)
13 𝑒ℎ𝐸𝑞𝑢𝑎𝑙𝑠(𝑥, 𝑦) → 𝑠𝑖𝑚𝑖𝑙𝑎𝑟(𝑥, 𝑦)
14 (𝑒ℎ𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑦) ∪ 𝑒ℎ𝐶𝑜𝑣𝑒𝑟𝑠(𝑥, 𝑦)) ∩ ((𝑒ℎ𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑧) ∪ (𝑒ℎ𝐶𝑜𝑣𝑒𝑟𝑠(𝑥, 𝑧)) ∩ 𝑒ℎ𝑀𝑒𝑒𝑡(𝑦, 𝑧)
→ 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛(𝑥, 𝑦) ∩ 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛(𝑥, 𝑧)
15 (𝑒ℎ𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑦) ∪ 𝑒ℎ𝐶𝑜𝑣𝑒𝑟𝑠(𝑥, 𝑦)) ∩ ((𝑒ℎ𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑧) ∪ (𝑒ℎ𝐶𝑜𝑣𝑒𝑟𝑠(𝑥, 𝑧))
∩ 𝑒ℎ𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡(𝑦, 𝑧) → 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛(𝑥, 𝑦) ∩ 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛(𝑥, 𝑧)
16 (𝑒ℎ𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑦) ∪ 𝑒ℎ𝐶𝑜𝑣𝑒𝑟𝑠(𝑥, 𝑦)) ∩ ((𝑒ℎ𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑧) ∪ (𝑒ℎ𝐶𝑜𝑣𝑒𝑟𝑠(𝑥, 𝑧))
∩ 𝑒ℎ𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑠(𝑦, 𝑧) → 𝑐𝑜𝑣𝑒𝑟𝑖𝑛𝑔(𝑥, 𝑦) ∩ 𝑐𝑜𝑣𝑒𝑟𝑖𝑛𝑔(𝑥, 𝑧)
17 (𝑒ℎ𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑦) ∪ 𝑒ℎ𝐶𝑜𝑣𝑒𝑟𝑠(𝑥, 𝑦)) ∩ ((𝑒ℎ𝐶𝑜𝑛𝑡𝑎𝑖𝑛𝑠(𝑥, 𝑧) ∪ (𝑒ℎ𝐶𝑜𝑣𝑒𝑟𝑠(𝑥, 𝑧))
∩ ¬𝑒ℎ𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡 → 𝑐𝑜𝑣𝑒𝑟𝑖𝑛𝑔(𝑥, 𝑦) ∩ 𝑐𝑜𝑣𝑒𝑟𝑖𝑛𝑔(𝑥, 𝑧)
Table 5. DL rules relating granular and RCC8 topological relations
DL rules
18 𝑟𝑐𝑐8𝑛𝑡𝑝𝑝𝑖(𝑥, 𝑦) ∪ 𝑟𝑐𝑐8𝑡𝑝𝑝(𝑥, 𝑦) → 𝑟𝑒𝑓𝑖𝑛𝑒(𝑥, 𝑦)
19 𝑟𝑐𝑐8𝑛𝑡𝑝𝑝(𝑦, 𝑥) ∪ 𝑟𝑐𝑐8𝑡𝑝𝑝𝑖(𝑦, 𝑥) → 𝑐𝑜𝑎𝑟𝑠𝑒(𝑥, 𝑦)
6
20 𝑟𝑐𝑐8𝑝𝑜(𝑥, 𝑦) → 𝑝𝑟𝑒𝑓𝑖𝑛𝑒(𝑥, 𝑦) ∩ 𝑝𝑐𝑜𝑎𝑟𝑠𝑒(𝑦, 𝑥)
21 𝑟𝑐𝑐8𝑒𝑞(𝑥, 𝑦) → 𝑠𝑖𝑚𝑖𝑙𝑎𝑟(𝑥, 𝑦)
22 (𝑟𝑐𝑐8𝑛𝑡𝑝𝑝𝑖(𝑥, 𝑦) ∪ 𝑟𝑐𝑐8𝑡𝑝𝑝(𝑥, 𝑦)) ∩ ((𝑟𝑐𝑐8𝑛𝑡𝑝𝑝𝑖(𝑥, 𝑧) ∪ (𝑟𝑐𝑐8𝑡𝑝𝑝(𝑥, 𝑧)) ∩ 𝑟𝑐𝑐8𝑑𝑐(𝑥, 𝑧)
→ 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛(𝑥, 𝑦) ∩ 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛(𝑥, 𝑧)
23 (𝑟𝑐𝑐8𝑛𝑡𝑝𝑝𝑖(𝑥, 𝑦) ∪ 𝑟𝑐𝑐8𝑡𝑝𝑝(𝑥, 𝑦)) ∩ ((𝑟𝑐𝑐8𝑛𝑡𝑝𝑝𝑖(𝑥, 𝑧) ∪ (𝑟𝑐𝑐8𝑡𝑝𝑝(𝑥, 𝑧)) ∩ 𝑟𝑐𝑐8𝑝𝑜(𝑦, 𝑧)
→ 𝑐𝑜𝑣𝑒𝑟𝑖𝑛𝑔(𝑥, 𝑦) ∩ 𝑐𝑜𝑣𝑒𝑟𝑖𝑛𝑔(𝑥, 𝑧)
24 (𝑟𝑐𝑐8𝑛𝑡𝑝𝑝𝑖(𝑥, 𝑦) ∪ 𝑟𝑐𝑐8𝑡𝑝𝑝(𝑥, 𝑦)) ∩ ((𝑟𝑐𝑐8𝑛𝑡𝑝𝑝𝑖(𝑥, 𝑧) ∪ (𝑟𝑐𝑐8𝑡𝑝𝑝(𝑥, 𝑧)) ∩ ¬𝑟𝑐𝑐8𝑑𝑐(𝑦, 𝑧)
→ 𝑐𝑜𝑣𝑒𝑟𝑖𝑛𝑔(𝑥, 𝑦) ∩ 𝑐𝑜𝑣𝑒𝑟𝑖𝑛𝑔(𝑥, 𝑧))
In addition, we have noted that refine is the inverse of coarse and that prefine is the
inverse relation of pcoarse. Even when granular relationships connect two levels of
granularity, one can infer granular relationships to accommodate multiple scales. For
example, 𝑟𝑒𝑓𝑖𝑛𝑒(𝑥, 𝑦) ∩ 𝑟𝑒𝑓𝑖𝑛𝑒 (𝑦, 𝑧) → 𝑟𝑒𝑓𝑖𝑛𝑒 (𝑥, 𝑧).
6 Use Case
Our scope of work focuses on creating a multiscale semantic checker that verifies
the compliance of construction at urban and building levels. After creating our
knowledge base that integrates and connects urban and building concepts, and divided
French urban regulations into several scales: building, district, city, and region (Hbeich
et al., 2019). We will apply the GrC notion and relationships on our knowledge base
to produce a multiscale structure, enabling us to connect the urban regulation to the
appropriate level of the knowledge base. The reason behind the implementation of GrC
notion and relations refers to the limitations of OWL language to represent the com-
plexity of the GrC philosophical perspective. For example, OWL uses subclass to high-
light hierarchy, e.g., A subclassOf B. This relation implies that (1) all the instances of
A are instances of B, (2) A inherent all relations and restriction from B, and finally (3)
A inherent all properties of B. While granular relation such as A refine B indicates (1)
A is at a lower level than B, (2) A and B could represent a concept or instance, (3) A
and B are different concepts or instances, and finally (4) both granules don’t inherit any
relations or restriction from one another. In this article, we have investigated the rela-
tions between topological and granular relationships. Our future work will apply the
same methodology (philosophical perspective) to the Building Information Model
(BIM), more specifically to IFC relations, in order to create a hierarchical structure for
building models. By connecting the two hierarchical structures (geospatial and build-
ing), we will then be able to generate a multiscale knowledge base ranging from City
to building elements.
7 Conclusion and Future Work
As mentioned above, topology structures information horizontally, whereas GrC
creates hierarchies of information. Our work combines the two representations using
7
DL rules to relate granular and geospatial topological relations. Thus, we consider ge-
ospatial features as granules, grouped into different levels of granularity and into dif-
ferent granules. In this way, geospatial data is presented as granular multiscale hierar-
chies. Our future work will specify an OWL vocabulary for granular relations and fur-
ther apply Linked Data principles along with the DL rules elaborated here. The goal is
to create an ontology similar to GeoSPARQL for GrC and use it to structure existing
geospatial datasets. In doing so, the rules presented above can be adapted either into
SHACL rules (Shapes Constraint Language (SHACL), s. d.) or SPARQL queries
(SPARQL Query Language for RDF, 2019), thus structuring geospatial knowledge
(pertaining to the considered datasets) into multiscale, granular knowledge. This would
enable multiscale compliance checking rules, for example considering a building's en-
vironment when checking specific regulations.
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