OWL 2 Web Ontology Language:RDF-Based Semantics

Definition 3.14.1 (OWL 2 Full Datatype Map): Let D be a datatype map as defined in Section 5.1 of the RDF Semantics. D is ana OWL 2 Full datatype map, if it contains at least all datatypes listed in Table 2.2.

Definition 3.24.2 (OWL 2 Full Interpretation): Let D be ana OWL 2 Full datatype map, and let V be a vocabulary that includes the RDF and RDFS vocabularies, and the OWL 2 Full vocabulary together with all the datatype and facet names listed in Section 2. An OWL 2 Full interpretation, I = ⟨ IR, IP, IEXT, IS, IL, LV ⟩, of V with respect to D, is a D-interpretation of V that meetssatisfies all the extra semantic conditions given in Section 45.

Definition 3.34.3 (OWL 2 Full Consistency): Let K be a collection of RDF graphs, and let D be ana OWL 2 Full datatype map, and let V be amap. K is OWL 2 Full consistent with respect to D iff there is some OWL 2 Full interpretation I with respect to D of some vocabulary V that includes the RDF and RDFS vocabularies, and the OWL 2 Full vocabulary together with all the datatype and facet names listed in Section 2 . K is OWL 2 Full consistent with respect to D iff there is some OWL 2 Full interpretation of V with respect to D that, where I satisfies everyall the RDF graphgraphs in K.

Definition 3.44.4 (OWL 2 Full Entailment): Let K and Q be collections of RDF graphs, and let D be ana OWL 2 Full datatype map, and let V be amap. K OWL 2 Full entails Q with respect to D iff every OWL 2 Full interpretation I with respect to D of any vocabulary V that includes the RDF and RDFS vocabularies,vocabularies and the OWL 2 Full vocabulary together with all the datatype and facet names listed in Section 2 . K OWL 2 Full entails Q with respect to D iff every OWL 2 Full interpretation of V with respect to D that, and where I satisfies everyall the RDF graphgraphs in KK, then I also satisfies everyall the RDF graphgraphs in Q.

The semantic conditions presented here haveare those specific to the OWL 2 Full vocabulary. This set of semantic conditions is not self-contained, but has to be regarded in conjunction with the semantic conditions given for Simple Entailment, RDF, RDFS and D-Entailment in the RDF Semantics.

The denotations of RDF triples, which occur in the semantic conditions, are closely related to triple sets given in the RDF Mapping .Table 4.15.1 on "Basic Sets" enumerates the different parts of the OWL 2 Full universe, such as the sets of classes, properties, etc., which are referred to by many semantic conditions in this section. Table 4.25.2 lists several "Convenient Abbreviations" for sets that are often used within semantic conditions. Table 4.35.3 and Table 4.45.4 list basic semantic conditions for the classes and the properties of the OWL 2 Full vocabulary, which can be regarded as the OWL 2 Full "axiomatic triples" (this is further explained by the introduction of these tables). The remaining tables in this section specify the OWL 2 Full semantic conditions for the different language features of OWL 2.

Most semantic conditions are "iff" conditions, which exactly specify the semantics of the respective language feature. Some language features only have "if-then" conditions in order to avoid certain semantic paradoxes and other problems with the semantics. Several language features with "iff" conditions, namely Sub Property Chains in Table 4.105.10, N-ary Axioms in Table 4.125.12, and Negative Property Assertions in Table 4.165.16, have a multi-triple representation in RDF, where the different triples share a common "root node" x. In order to treat this specific syntactic aspect technically, the "iff" conditions of these language features are split into two "if-then" conditions, and the right-to-left "if" condition contains an additional premise of the form "∃x ∈ IR", which has the single purpose to provide the needed "root node" x.

Conventions used in this section:

Several conventions are used when presenting logic expressions in the below tables. Having a comma between two entriesstatements in a semantic condition, as in

means a logical "and". If no scope is given for a variable x, as in "∀x:…" or in "{x|…}", then x is unconstrained, which means that x ∈ IR. An expression of the form

means that u₁ ∈ S,… , u_n ∈ S, and that there exist x₁ ∈ IR,…, x_n ∈ IR, such that

The following names for sets are used in addition to those given in Section 4:

• ISEQ: The set of all sequences. This set equals the class extension of rdf:List.
• INNI: The set of all non-negative integers. This set equals the value space of xsd:NonNegativeInteger, but is also contained in the value spaces of other datatypes, such as xsd:integer.

The semantic conditions in the tables below sometimes do not explicitly list typing statements in their consequent that one would normally expect. This may be the case if these statements can already be deduced by means of the semantic conditions given in Table 4.35.3 and Table 4.45.4, occasionally in connection with Table 4.15.1. For example, the semantic condition for owl:allValuesFrom restrictions in Table 4.75.7 does not have the statement x ∈ ICEXT(IS(ICEXT(I(owl:Restriction)) on its right hand side. The reason is that this result can already be obtained from the third column of the entry for owl:allValuesFrom in Table 4.45.4, which determines that IEXT(IS(owl:allValuesFrom))IEXT(I(owl:allValuesFrom)) ⊆ ICEXT(IS(owl:Restriction))ICEXT(I(owl:Restriction)) × IC.

Table 4.15.1 presents basic sets used in OWL 2 Full, their relationship to other sets, and properties of their instances.

Table 4.25.2 provides abbreviations for additional sets, which are used throughout this document.

Table 4.35.3 lists the classes of the OWL 2 Full vocabulary (and certain classes from RDF and RDFS), together with their relationship to other classes. Not included in this table are the different datatypes, as given in Table 2.2. For a datatype URI D, IS(D)I(D) ∈ IDC, and ICEXT(IS(D))ICEXT(I(D)) ⊆ LV.

Informative Note: The semantic conditions given here can be regarded as the OWL 2 Full "axiomatic triples" for classes: For each URI U occurring in the first column of the table, if the second column contains an entry "IS(U)"I(U) ∈ S" for some set S, then this entry corresponds to some RDF triple of the form "U rdf:type C", where C is the URI of some class with ICEXT(IS(C))ICEXT(I(C)) = S. In this table, S will always be either the set IC of all classes, or a subset of IC. Hence, in a corresponding RDF triple the URI C will typically be one of "rdfs:Class" or "owl:Class" (S=IC in both cases), or "rdfs:Datatype" (S=IDC). Further, for each URI U in the first column, if the third column contains an entry "ICEXT(IS(U))"ICEXT(I(U)) ⊆ S" ("ICEXT(IS(U))("ICEXT(I(U)) = S") for some set S, then this entry corresponds to some RDF triple of the form "U rdfs:subClassOf C" ("U owl:equivalentClass C"), where C is the URI of some class with ICEXT(IS(C))ICEXT(I(C)) = S. Additionally, the conditions on the sets given in Table 4.15.1 have to be taken into account. In particular, if an entry of Table 4.15.1 states S₁ ⊆ S₂ for some sets S₁ and S₂, then this corresponds to some RDF triple C₁ owl:subClassOf C₂, where C₁ and C₂ are the URIs of some classes with ICEXT(IS(CICEXT(I(C₁)) = S₁ and ICEEXT(IS(CICEXT(I(C₂)) = S₂, respectively, according to Table 4.35.3. Note that some of the RDF triples received in this way already follow from the RDFS Semantics. These axiomatic triples are "simple" in the following sense: For every set S mentioned in the second and the third column of the table there exists a URI C of some class in the vocabularies for RDF, RDFS or those given in Section 2, for which S = ICEXT(IS(C))ICEXT(I(C)).

Table 4.45.4 lists the properties of the OWL 2 Full vocabulary (and certain properties from RDF and RDFS), together with their domain and range. Not included in this table areNote that the ranges of the properties owl:deprecated and owl:hasSelf are not restricted to be boolean values, in order to allow assertions with one of these properties to have objects of arbitrary type.

Not included in this table are the different datatype facets, as given in Table 2.3. For a facet URI F, IS(F)I(F) ∈ IP, and IEXT(IS(F))IEXT(I(F)) ⊆ IR × LV.

Informative Note: The semantic conditions given here can be regarded as the OWL 2 Full "axiomatic triples" for properties: For each URI U occurring in the first column of the table, if the second column contains an entry "IS(U)"I(U) ∈ S" for some set S, then this entry corresponds to some RDF triple of the form "U rdf:type C", where C is the URI of some class with ICEXT(IS(C))ICEXT(I(C)) = S. In this table, S will always be either the set IP of all properties, or a subset of IP. Hence, in a corresponding RDF triple the URI C will typically be one of "rdf:Property" or "owl:ObjectProperty" (S=IP in both cases), "owl:DatatypeProperty" (S=IODP), "owl:AnnotationProperty" (S=IOAP), or "owl:OntologyProperty" (S=IOXP). Further, for each URI U in the first column, if the third column contains an entry "IEXT(IS(U))"IEXT(I(U)) ⊆ S₁ × S₂" for some sets S₁ and S₂, then this entry corresponds to some RDF triples of the forms "U rdfs:domain C₁" and "U rdfs:range C₂", where C₁ and C₂ are the URIs of some classes with ICEXT(IS(CICEXT(I(C₁)) = S₁ and ICEEXT(IS(CICEXT(I(C₂)) = S₂, respectively. Exceptions are the semantic conditions "IEXT(IS(owl:TopObjectProperty))"IEXT(I(owl:topObjectProperty)) = IR × IR" and "IEXT(IS(owl:TopDataProperty))"IEXT(I(owl:topDataProperty)) = IR × LV", for which there are no corresponding domain and range triples. These axiomatic triples are "simple" in the following sense: For every set S mentioned in the second column of the table, and as the left or right hand side of a Cartesian product in the third column of the table there exists a URI C of some class in the vocabularies for RDF, RDFS or those given in Section 2, for which S = ICEXT(IS(C))ICEXT(I(C)).

Table 4.55.5 lists the semantic conditions for boolean combinations,class expressions, which exist for building class and datatypecomplements, intersections, and unions of classes. An intersection or union of a set of datatypes is itself a datatype. While a "normal" complement of a class is created w.r.t. to the whole domain, a datatype complement is created for a datatype w.r.t. the set of data values only, and results itself in a datatype.

Table 4.65.6 lists the semantic conditions for enumerations, i.e. classes, which have an explicitly defined, finite set of instances. EnumerationsAn enumeration entirely consisting of datadatatype values are datatypes.is a datatype.

Table 4.75.7 lists the semantic conditions for restriction classes, which areproperty restrictions, including self restrictions, value restrictions, and (qualified) cardinality restrictions. Note that the semantic condition for self restrictions does not entail the right hand side of a owl:hasSelf assertion to be a boolean value, in order to allow such assertions having right hand sides of arbitrary type.

Table 4.85.8 lists the semantic conditionconditions for datatype restrictions, which are specified by a set of facets with their facet values applied to a datatype. Note that if no facet is applied to a given datatype, then the resulting datatype will be equivalent to the original datatype. Note further that the semantic condition is specified in a way that applying a facet to a datatype, for which it is not defined, orwill lead to an unsatisfiable ontology. Likewise, adding an illegalinapplicable facet value to a facet/datatypecertain combination of a datatype a facet will lead to an unsatisfiable ontology. Table 4.8:As a consequence, a datatype Restrictions if then l sequence of yrestriction with one or more specified facets will lead to an unsatisfiable ontology when applied to a datatype for which no facets are defined (usually a set of facets only exists for datatypes contained in the datatype map).

Table 4.95.9 extends the semantic conditions for the RDFS vocabulary. The original semantics for RDFS, as specified in the RDF Semantics, only provide "if-then" semantic conditions, while OWL 2 Full specifies stronger "iff" semantic conditions. Note that only the additional semantic conditions are given here and that the other conditions on the RDF and RDFS vocabulary are retained.

Table 4.105.10 lists the semantic conditions for sub property chains.

Table 4.115.11 lists the semantic conditions for equal and different individuals, equivalent and disjoint classes, and equivalent and disjoint properties. Also treated here are axioms to specify disjoint unions.

Table 4.125.12 lists the semantic conditions for the n-ary versions of the axioms for different individuals, disjoint classes, and disjoint properties.

Table 4.135.13 lists the semantic condition for inverse property axioms.

Table 4.145.14 lists the semantic conditions for property characteristics, i.e. functionality and inverse functionality, reflexivity and irreflexivity, symmetry and asymmetry, and transitivity of properties.

Table 4.155.15 lists the semantic condition for Keys.

Table 4.165.16 lists the semantic conditions for negative property assertions.