78
5 Applications of the Metadata Standards
I viprs:n_loc_cor_rnoc J ≡ ∃K I , K J ∈ I : (K I C I ) ∧ (K J C J )
∧
∃I
, J
: (K I C I
) ∧ (K J C J
) ∧ (occ[J
] → occ[I
])
. (5.6)
Relations from the EMMO are mereological (or, more properly, mereotopological [10, 18, 19]), represented here at the highest level by proper parthood
P ≡ viprs:is_proper_part_of ≡ emmo-mereotopology:hasProperPart
−
, (5.7)
and semiotic, represented at the highest level by the sign-to-object reference relation
6
S ≡ viprs:is_sign_for ≡ emmo-semiotics:hasSign
−
,
(5.8)
cf. Expressions (3.6) and (3.7). To facilitate ontology alignment, which is discussed in
Sects. 5.3 and 5.4, VIPRS also contains mereosemiotic chain products of these fundamental relations, i.e. elements of the free semigroup R
+
ms over R ms = {P, S, P
−
, S
−
},
with the product defined by concatenation. The mereosemiotic relations for which
there is an explicit definition in VIPRS are limited to R ms ∪ R
2
ms ∪ R
3
ms , i.e. relations
generated by a sequence of up to three fundamental relations which are not redundant
(P ◦ P and its inverse),
7 complete (or almost complete), i.e. relating everything to
everything, except possibly for a single “universe” entity,
8 as it is the case for P ◦ P
− ,
or consist of three elements from the same category, e.g. S
−
◦ S ◦ S
− is excluded,
because all three constituent elements are semiotic. In the nomenclature employed
6 It will require an explanation why the shorthand symbols P (“is proper part of”) and S (“is sign
for”) are here assigned a meaning corresponding to the inverse of relations defined by the EMMO,
which only include “has proper part” and “has sign,” respectively. While making the correspondence
with the EMMO slightly more indirect, we find this to be more in line with common conventions.
First, concerning mereology, parthood is a partial ordering relation; such relations are conventionally defined in terms of the operator meaning “is smaller than,” not “is greater than,” e.g. in
description logic, where subsumption () rather than inclusion () is employed as a primitive. The
seminal papers on mereotopology by Smith [18], Varzi [10], as well as Smith and Varzi [19] unsurprisingly all define “is part of” as fundamental, which they denote by P. We rely on proper rather
than improper parthood here based on our empirical assessment that proper parthood is more frequently the most useful choice for ontology alignment, cf. Sect. 5.3, when the EMMO is considered
as a target ontology, cf. Sect. 5.4.
Second, concerning semiotics, when drawing a knowledge graph using the relation “is sign for”,
the arrow points from the sign to the object, which is intuitive; with “has sign”, the object would
have to point to the sign. In view of this, to avoid a counterintuitive notation that would encourage
the misinterpretation of diagrams, the symbol S is here employed for “is sign for”.
7 In terms of the 4D spatiotemporal entities considered within the EMMO, P and P − are idempotent,
since for any I P J there is an I such that (I P I ) ∧ (I P J ) due to the continuum nature of
spacetime. The EMMO explicitly permits items to be “void,” i.e. not to contain any physical matter,
so that continuum nature can be assumed for EMMO spacetime even concerning properties that are
subject to quantization. Hence, chains that contain P ◦ P or P − ◦ P − can be excluded.
8 There is a 4D spatiotemporal entity (“trajectory of the universe”) that encloses everything that
exists within any given knowledge base; therefore, “I is a proper part of something (namely, )
that has J as a proper part” holds for all EMMO individuals I, J = from the knowledge base.
Hence, P ◦ P − is (almost) complete, and any chains that contain it can be excluded.
5 Applications of the Metadata Standards
I viprs:n_loc_cor_rnoc J ≡ ∃K I , K J ∈ I : (K I C I ) ∧ (K J C J )
∧
∃I
, J
: (K I C I
) ∧ (K J C J
) ∧ (occ[J
] → occ[I
])
. (5.6)
Relations from the EMMO are mereological (or, more properly, mereotopological [10, 18, 19]), represented here at the highest level by proper parthood
P ≡ viprs:is_proper_part_of ≡ emmo-mereotopology:hasProperPart
−
, (5.7)
and semiotic, represented at the highest level by the sign-to-object reference relation
6
S ≡ viprs:is_sign_for ≡ emmo-semiotics:hasSign
−
,
(5.8)
cf. Expressions (3.6) and (3.7). To facilitate ontology alignment, which is discussed in
Sects. 5.3 and 5.4, VIPRS also contains mereosemiotic chain products of these fundamental relations, i.e. elements of the free semigroup R
+
ms over R ms = {P, S, P
−
, S
−
},
with the product defined by concatenation. The mereosemiotic relations for which
there is an explicit definition in VIPRS are limited to R ms ∪ R
2
ms ∪ R
3
ms , i.e. relations
generated by a sequence of up to three fundamental relations which are not redundant
(P ◦ P and its inverse),
7 complete (or almost complete), i.e. relating everything to
everything, except possibly for a single “universe” entity,
8 as it is the case for P ◦ P
− ,
or consist of three elements from the same category, e.g. S
−
◦ S ◦ S
− is excluded,
because all three constituent elements are semiotic. In the nomenclature employed
6 It will require an explanation why the shorthand symbols P (“is proper part of”) and S (“is sign
for”) are here assigned a meaning corresponding to the inverse of relations defined by the EMMO,
which only include “has proper part” and “has sign,” respectively. While making the correspondence
with the EMMO slightly more indirect, we find this to be more in line with common conventions.
First, concerning mereology, parthood is a partial ordering relation; such relations are conventionally defined in terms of the operator meaning “is smaller than,” not “is greater than,” e.g. in
description logic, where subsumption () rather than inclusion () is employed as a primitive. The
seminal papers on mereotopology by Smith [18], Varzi [10], as well as Smith and Varzi [19] unsurprisingly all define “is part of” as fundamental, which they denote by P. We rely on proper rather
than improper parthood here based on our empirical assessment that proper parthood is more frequently the most useful choice for ontology alignment, cf. Sect. 5.3, when the EMMO is considered
as a target ontology, cf. Sect. 5.4.
Second, concerning semiotics, when drawing a knowledge graph using the relation “is sign for”,
the arrow points from the sign to the object, which is intuitive; with “has sign”, the object would
have to point to the sign. In view of this, to avoid a counterintuitive notation that would encourage
the misinterpretation of diagrams, the symbol S is here employed for “is sign for”.
7 In terms of the 4D spatiotemporal entities considered within the EMMO, P and P − are idempotent,
since for any I P J there is an I such that (I P I ) ∧ (I P J ) due to the continuum nature of
spacetime. The EMMO explicitly permits items to be “void,” i.e. not to contain any physical matter,
so that continuum nature can be assumed for EMMO spacetime even concerning properties that are
subject to quantization. Hence, chains that contain P ◦ P or P − ◦ P − can be excluded.
8 There is a 4D spatiotemporal entity (“trajectory of the universe”) that encloses everything that
exists within any given knowledge base; therefore, “I is a proper part of something (namely, )
that has J as a proper part” holds for all EMMO individuals I, J = from the knowledge base.
Hence, P ◦ P − is (almost) complete, and any chains that contain it can be excluded.