3.1 Ontologies and Formal Notation
33
In TTL notation, ending the first triple with a semicolon implies that the subject (here,
ex:D) is reused for the second triple, the full representation of which would therefore be ex:D vov:shares_value_with osmo:ZERO_VECTOR_3D. The DL notation for
Expression (3.2) is
ex:D : vov:electric_dipole_moment,
(ex:D, osmo:ZERO_VECTOR_3D) : vov:shares_value_with.
(3.3)
Operators can be applied to elementary names, yielding composites such as
3
C C
[owl:intersectionOf (CC’)],
C C
[owl:unionOf (CC’)],
¬C
[owl:complementOf C],
R
−
[owl:inverseOf R],
∃R.C
[a owl:Restriction; owl:onProperty R; owl:someValuesFrom C],
∀R.C
[a owl:Restriction; owl:onProperty R; owl:allValuesFrom C],
C
•
[rdfs:domain C],
• C
[rdfs:range C],
(3.4)
where C and C
are concepts; R is a relation; and , and ¬ denote the intersection,
union and complement of concepts, respectively, and R
− denotes the inverse relation
to R. For an individual I ∈ I, the assertion I : ∃R.C entails that there is a J ∈ I with
J :C
and (I, J ) : R, whether explicitly asserted or not, while I : ∀R.C entails J :C
for all J ∈ I with (I, J ) : R, whether explicitly asserted or not. The relation C
• holds
whenever its subject (i.e. its first argument) is an individual that instantiates C, i.e. C
•
relates all C individuals to all individuals;
• C holds whenever its object (i.e. its second
argument) instantiates C, i.e.
• C relates all individuals to all C individuals.
Rules can include subsumption and equivalence for concepts and relations
4
3 In the last two rows, for C • and • C, the TTL version is an approximation, since in OWL, the
use of domain and range composites is restricted to what would best be represented formally as a
subsumption rule, cf. Expression (3.5). The subsumption R C • then becomes R rdfs:domain
C, and R • C becomes R rdfs:range C. However, it is impossible to use this construction for
any other purpose, e.g. to state R ≡ C • .
4 The use of the article “a” is a possible source of misunderstandings between communities due to the way in which it is treated by two different notations, namely, TTL,
which is used for the VIMMP ontologies and throughout this book, as opposed to the
Open Biomedical Ontologies (OBO) format [9]. TTL uses a for instantiation (rdfs:type)
such as in macro:VIMMP_MARKETPLACE a macro:digital_marketplace, signifying “the
VIMMP Marketplace is a digital marketplace.” OBO format denotes conceptual subsumption
(rdfs:subClassOf) by the keyword is_a, such as in id: macro:digital_marketplace
[…] is_a: macro:bidirectional_channel, signifying “every digital marketplace is a
bidirectional channel,” since macro:digital_marketplace macro:bidirectional_channel.
Motivated by that standard, the OWLViz tool [10] automatically labels all arrows with “is-a” when
visualizing a taxonomy. Where this occurs here (Figs. 3.2, 3.3, etc.), it should be understood as “is
subclass of”.
33
In TTL notation, ending the first triple with a semicolon implies that the subject (here,
ex:D) is reused for the second triple, the full representation of which would therefore be ex:D vov:shares_value_with osmo:ZERO_VECTOR_3D. The DL notation for
Expression (3.2) is
ex:D : vov:electric_dipole_moment,
(ex:D, osmo:ZERO_VECTOR_3D) : vov:shares_value_with.
(3.3)
Operators can be applied to elementary names, yielding composites such as
3
C C
[owl:intersectionOf (CC’)],
C C
[owl:unionOf (CC’)],
¬C
[owl:complementOf C],
R
−
[owl:inverseOf R],
∃R.C
[a owl:Restriction; owl:onProperty R; owl:someValuesFrom C],
∀R.C
[a owl:Restriction; owl:onProperty R; owl:allValuesFrom C],
C
•
[rdfs:domain C],
• C
[rdfs:range C],
(3.4)
where C and C
are concepts; R is a relation; and , and ¬ denote the intersection,
union and complement of concepts, respectively, and R
− denotes the inverse relation
to R. For an individual I ∈ I, the assertion I : ∃R.C entails that there is a J ∈ I with
J :C
and (I, J ) : R, whether explicitly asserted or not, while I : ∀R.C entails J :C
for all J ∈ I with (I, J ) : R, whether explicitly asserted or not. The relation C
• holds
whenever its subject (i.e. its first argument) is an individual that instantiates C, i.e. C
•
relates all C individuals to all individuals;
• C holds whenever its object (i.e. its second
argument) instantiates C, i.e.
• C relates all individuals to all C individuals.
Rules can include subsumption and equivalence for concepts and relations
4
3 In the last two rows, for C • and • C, the TTL version is an approximation, since in OWL, the
use of domain and range composites is restricted to what would best be represented formally as a
subsumption rule, cf. Expression (3.5). The subsumption R C • then becomes R rdfs:domain
C, and R • C becomes R rdfs:range C. However, it is impossible to use this construction for
any other purpose, e.g. to state R ≡ C • .
4 The use of the article “a” is a possible source of misunderstandings between communities due to the way in which it is treated by two different notations, namely, TTL,
which is used for the VIMMP ontologies and throughout this book, as opposed to the
Open Biomedical Ontologies (OBO) format [9]. TTL uses a for instantiation (rdfs:type)
such as in macro:VIMMP_MARKETPLACE a macro:digital_marketplace, signifying “the
VIMMP Marketplace is a digital marketplace.” OBO format denotes conceptual subsumption
(rdfs:subClassOf) by the keyword is_a, such as in id: macro:digital_marketplace
[…] is_a: macro:bidirectional_channel, signifying “every digital marketplace is a
bidirectional channel,” since macro:digital_marketplace macro:bidirectional_channel.
Motivated by that standard, the OWLViz tool [10] automatically labels all arrows with “is-a” when
visualizing a taxonomy. Where this occurs here (Figs. 3.2, 3.3, etc.), it should be understood as “is
subclass of”.