OWL 2 Web Ontology Language:
Profiles

1 Introduction

The purpose of an OWL 2 profile is to provide a trimmed down version of OWL 2 that trades expressive power for efficiency of reasoning. In logic, a profile is usually called a fragment or a sublanguage. This document describes three important profiles of OWL 2, each of them achieving efficiency in a different way and useful in different application scenarios:

• The EL++ profile enables polynomial time algorithms for the reasoning tasks consistency, classification, and instance checking. Dedicated reasoning algorithms for this profile are available and have been demonstrated to be implementable in a highly scalable way.

• In the DL-Lite profile, conjunctive query answering can be implemented using conventional relational database systems (after rewriting the query). In particular, this requires that conjunctive query answering is in LogSpace regarding data complexity. As in EL++, there are polynomial time algorithms for consistency, classification, and instance checking.

• The OWL-R profile enables the implementation of reasoning algorithms by forward chaining rules applied to triples of the RDF serialization. In particular, such rule approaches can be used to compute consistency, classification, and instance checking in polynomial time. Conjunctive query answering in OWL-R can be implemented by extending a standard relational database system with rules. OWL-R comes in two varieties, OWL-R DL and OWL-R Full.

The profiles EL++, DL-Lite, and OWL-R DL are defined by placing restrictions on the syntax of OWL 2 DL. In contrast, OWL-R Full is defined by placing restrictions on the semantics of OWL 2 Full.

Editor's Note: The working group intends to provide more guidance for users about how to select a profile well suited for their situation. It is not yet determined whether that guidance will be here or in the Primer.

2 EL++

2.1 Feature Overview

EL++ provides the following features:

• existential quantification to a class (ObjectSomeValuesFrom) or a data range (DataSomeValuesFrom) ,
• existential quantification to a nominal (ObjectHasValue) or a constant (DataHasValue) ,
• self quantification (ObjectExistsSelf),
• enumerations involving a single element (ObjectOneOf) or a single constant (DataOneOf) ,
• intersection of classes (ObjectIntersectionOf),
• class inclusion (SubClassOf),
• class equivalence (EquivalentClasses),
• class disjointness (DisjointClasses),
• object property inclusion (SubObjectPropertyOf) possibly involving property chains and data property inclusion (SubDataPropertyOf) ,
• property equivalence (EquivalentObjectProperties and EquivalentDataProperties),
• transitive object properties (TransitiveObjectProperty),
• reflexive object properties (ReflexiveObjectProperty),
• domain restrictions (ObjectPropertyDomain and DataPropertyDomain ),
• range restrictions (ObjectPropertyRange and DataPropertyRange),
• facts (SameIndividual, DifferentIndividuals, ClassAssertion, ObjectPropertyAssertion, DataPropertyAssertion, NegativeObjectPropertyAssertion, and NegativeDataPropertyAssertion),
• functional data properties (FunctionalDataProperty),
• a limited range of datatypes.

The following features of OWL 2 are missing in EL++:

• universal quantification to a class (ObjectAllValuesFrom) or a data range(DatAllaValuesFrom) ,
• counting quantification (ObjectMaxCardinality, ObjectMinCardinality, ObjectExactCardinality),DataMaxCardinality, DataMinCardinality, and DataExactCardinality))
• disjunction (ObjectUnionOf, DisjointUnion),
• negation (ObjectComplementOf),
• enumerations involving multiple elements (ObjectOneOf, DataOneOf),
• disjoint properties (DisjointObjectProperties and DisjointDataProperties),
• irreflexive object properties (IrreflexiveObjectProperty),
• inverse object properties (InverseObjectProperties),
• functional object properties (FunctionalObjectProperty),
• symmetric object properties (SymmetricObjectProperty),
• asymmetric object property (AsymmetricObjectProperty).

2.2 Profile Specification

2.2.1 Entities

2.2.2 Class Expressions

The design principle of EL++ is to focus on the class constructors ObjectIntersectionOf and ObjectSomeValuesFrom, which are used extensively in many large-scale ontologies. It additionally provides for objectExistsSelf, objectHasValue, dataSomeValuesFrom, dataHasValue, and objectOneOf enumerations that contain a single individual (called nominals in the DL literature). Thus, EL++ class descriptions are defined according to the following production:

2.2.3 Property Expressions

Inverse properties are not supported in EL++, and thus object property expressions are restricted to named properties. Data property expressions are as in OWL 2.

2.2.4 Data Range Expressions

A data range expression is restricted in EL++ to the predefined datatypes admitted in EL++ and enumerated datatypes consisting of a single constant.

EL++ does not support DataComplementOf and DatatypeRestriction (i.e., no facet is admitted).

2.2.5 Axioms

The class axioms of EL++ are the same as in full OWL 2, except that DisjointUnion is not allowed.

2.2.6 Global Restriction

3 DL-Lite

DL-Lite is a syntactic profile of OWL 2 that admits sound and complete reasoning in LOGSPACE with respect to the size of the data (facts). DL-Lite includes most of the main features of conceptual models such as UML class diagrams and ER diagrams.

Several variants of DL-Lite have been described in the literature. The variant presented here is called DL-Lite_R since it allows for property inclusion axioms; it therefore contains the intersection between RDFS and OWL 2 DL. Other variants trade property inclusion axioms for functionality and inverse-functionality of object properties.

Note that the profile presented here is asymmetric: it is defined not only in terms of the set of supported constructs, but it also restricts the places in which these constructs can be used.

3.1 Feature Overview

The following constructs can be used to define subclasses in SubClassOf axioms:

• OWL classes, except for owl:Thing
• existential quantification to a class ( ObjectSomeValuesFrom ) where the class is limited to owl:Thing,

The following constructs can be used to define superclasses in SubClassOf axioms:

• OWL classes
• existential quantification to a class ( ObjectSomeValuesFrom ) where the class is limited to owl:Thing,
• negation (ObjectComplementOf)

All class axioms in DL-Lite are constrained in a way that is compliant with these restrictions. For example, the property domain and range axioms are allowed to refer only to the superclasses mentioned above:

• class inclusion (SubClassOf),
• class equivalence (EquivalentClasses),
• class disjointness (DisjointClasses),

Moreover, DL-Lite allows for the following property axioms and facts:

• inverse object properties (InverseObjectProperties),
• property inclusion (SubObjectPropertyOf) not involving property chains,
• property equivalence (EquivalentObjectProperties),
• property domain (ObjectPropertyDomain),
• property range (ObjectPropertyRange),
• disjoint properties (DisjointObjectProperties),
• symmetric properties (SymmetricObjectProperty),
• facts (SameIndividual, DifferentIndividuals, ClassAssertion, ObjectPropertyAssertion)

The following features of OWL 2 are missing in DL-Lite:

• existential quantification to a class (ObjectSomeValuesFrom),
• self quantification (ObjectExistsSelf),
• existential quantification to a nominal (ObjectHasValue),
• nominals (ObjectOneOf),
• universal quantification to a class (ObjectAllValuesFrom),
• counting quantification (ObjectMaxCardinality, ObjectMinCardinality, ObjectExactCardinality),
• disjunction (ObjectUnionOf, DisjointUnion),
• property inclusion (SubObjectPropertyOf) involving property chains,
• transitivity (TransitiveObjectProperty),
• reflexivity (ReflexiveObjectProperty),
• irreflexivity (IrreflexiveObjectProperty),
• asymmetric properties (AsymmetricObjectProperty),
• (inverse)functional properties (FunctionalObjectProperty and InverseFunctionalObjectProperty)

3.2 Profile Specification

3.2.1 Entities

DL-Lite supports all OWL 2 entities apart from data properties. The entities in DL-Lite are defined as follows.

3.2.2 Class Expressions

In DL-Lite, there are two types of class expressions. The subClass production defines the classes that can occur in the antecedents of implications; for example, such classes can occur as subclasses in a SubClassOf axiom. The superClass production defines the classes that can occur in the consequents of implications; for example, such classes can occur as superclasses in a SubClassOf axiom.

3.2.3 Property Expressions

DL-Lite object property expressions are the same as in OWL 2 DL.

3.2.4 Axioms

DL-Lite axioms are defined to exclude membership assertions on data properties.

4 OWL-R

OWL-R is a profile of OWL 2 that allows for scalable reasoning using rule-based technologies. The profile has been designed so as to avoid the need to infer the existence of individuals not explicitly present in the knowledge base. This design goal enables a straightforward translation of OWL’s semantic conditions into rules, on which most rule-based reasoning engines terminate in a finite amount of time.

Another design goal for OWL-R is flexibility. On the one hand, OWL-R can accommodate OWL 2 DL applications that can trade the full expressivity of the language for efficiency; on the other hand, OWL-R can also accommodate RDF(S) applications that need some added expressivity from OWL. For this purpose, this document provides two variants of OWL-R.

4.1 Feature Overview

OWL-R, although just a profile of OWL 2, is quite expressive. An OWL-R DL ontology can use, in a nutshell, most OWL 2 language constructs except owl:cardinality, owl:minCardinality, owl2:NegativeObjectPropertyAssertion, owl2:NegativeDataPropertyAssertion, and owl:complementOf.

Not all constructs of OWL-R DL can be used freely in all places in the axioms. For example, in SubClassOf axioms, the usage of the constructs on the left- and right-hand side of the implication must follow the patterns shown in Table 4.1.

4.2 OWL-R DL

4.2.1 Entities

4.2.2 Data Ranges

4.2.3 Classes

There are three types of classes in OWL-R. The subClass production defines the classes that can occur in the antecedents of implications; for example, such classes can occur as subclasses in a SubClassOf axiom. The superClass production defines the classes that can occur in the consequents of implications; for example, such classes can occur as superclasses in a SubClassOf axiom. Finally, the equivClass production defines the classes that can occur in an EquivalentClasses axiom.

4.2.4 Properties

4.2.5 Axioms

4.3 OWL-R Full

4.3.1 Weakened OWL 2 Full Semantic Conditions

This section defines OWL-R Full by weakening the OWL 2 Full semantic conditions on an interpretation.

Editor's Note: We need to add a reference of OWL 2 Full Semantics

4.3.1.1 Main elements of OWL 2 Full Semantics

Before specifying in more details how the semantic weakining is performed for various features of OWL-R Full, we briefly present here the main elements of OWL 2 Full semantics.

First, a datatype map D is a partial mapping from URI references to datatypes that maps xsd:string and xsd:integer to the appropriate XML Schema datatypes.

Next, the OWL 2 Full model-theoretic semantics defines an interpretation as follows.

From OWL 2 Full Semantics, for V a set of URI references and literals containing the RDF, RDFS and OWL vocabulary and D a datatype map, a D-interpretation of V is a tuple I = < R_I, P_I, EXT_I, S_I, L_I, LV_I >. R_I is the domain of discourse or universe, i.e., a nonempty set that contains the denotations of URI references and literals in V. P_I is a subset of R_I consisting of the properties of I. EXT_I is used to give meaning to properties, and is a mapping from P_I to P(R_I × R_I). S_I is a mapping from URI references in V to their denotations in R_I. L_I is a mapping from typed literals in V to their denotations in R_I. LV_I is a subset of R_I that contains at least the set of Unicode strings, the set of pairs of Unicode strings and language tags, and the value spaces for each datatype in D. The set of all classes in R_I is C_I, and the mapping CEXT_I from C_I to P(R_I) is defined as CEXT_I(c) = { x∈RI | <x,c>∈EXT_I(S_I(rdf:type)) }. CEXT_I(c) maps a class c to its extension. D-interpretations must meet several other conditions, as detailed in the OWL 2 Full semantics.

Finally, the following important sets are used in the definitions of OWL 2 Full semantic conditions. IOOP denotes the set of OWL object properties, and IODP the set of OWL datatype properties. Both are subsets of P_I. IOC, a subset of C_I, denotes the set of OWL classes, and IDC is the set of OWL datatypes. IOR represents the set of OWL restrictions. IOT is the set of OWL individuals.

4.3.1.2 Restrictions defining OWL-R Full

In OWL-R Full, the weakening of the OWL 2 Full semantic conditions on an interpretation is mainly done by weakening some equivalences in the OWL Full semantics to implications.

For example, the semantics of the owl:someValuesFrom restriction is defined in OWL Full using the following restrictions on the RDF interpretation:

In a simplified form, these conditions can be understood as the following two implications:

The first implication captures the notion of existential restrictions occurring in the antecedents of implications, while the second implication captures the notion of existential restrictions occurring in the consequents of implications. In OWL-R Full, the second implication is discarded. Note the parallel with OWL-R DL, where syntactic restrictions prevent existential restrictions occurring in the consequents of implications.

Next, the restrictions that define OWL-R Full are listed. Instead of repeating all the intricate definitions of OWL Full, this section just specifies the difference to the definitions in the OWL Full document. For readers less familiar with OWL Full semantics, the next section provides a more self-contained axiomatization of OWL-R.

• The conditions in the table defining the characteristics of OWL classes, datatypes, and properties are changed as follows:
  • The conditions defining the sets IOT, LV_I, and IX are dropped.
  • The if-and-only-if in the definition of the semantics of owl:FunctionalProperty, owl:InverseFunctionalProperty, owl:SymmetricProperty, and owl:TransitiveProperty is changed into only-if.
• In the table defining the semantics of rdfs:subClassOf, rdfs:subPropertyOf, rdfs:domain, and rdfs:range, the if-and-only-if condition in the third column of the table header is changed into only-if.
• In the table defining the characteristics of OWL vocabulary related to equivalence, the if-and-only-if condition in the second column of the table header is changed into only-if.
• In the table defining the conditions on OWL vocabulary related to boolean combinations and sets, the conditions for owl:complementOf and owl:oneOf are dropped, and the condition for owl:unionOf is changed into only-if. The condition for owl:intersectionOf is left unchanged.

• The table defining the conditions on OWL restrictions is modified as follows.
  • The condifion for owl:allValuesFrom is changed into CEXT_I(x) ⊆ {u ∈ IOT | < u,v > ∈ EXT_I(p) implies v ∈ CEXT_I(y) }.
  • The condifion for owl:someValuesFrom is changed into CEXT_I(x) ⊇ {u ∈ IOT | < u,v > ∈ EXT_I(p) implies v ∈ CEXT_I(y) }.
  • The condition for owl:hasValue is let unchanged.
  • The conditions for cardinality restrictions are changed such that y must be 1, and CEXT_I(x) = ... is changed into CEXT_I(x) ⊆ ....
• The comprehension principles are dropped.

4.3.2 Axiomatization Using First-Order Implications

This section defines OWL-R Full in terms of first-order (material) implications. This definition is intended to be equivalent to the one from the previous section. This definition should provide a useful starting point for the practical implementation using rule-based technologies.

The implications are given as universally quantified first-order implications over a ternary predicate T. This predicate represents RDF triples; thus, T(s, p, o) represents a RDF triple with the subject s, predicate p, and the object o. Variables in the implications are preceeded with the question mark. The semantic conditions are split into several tables for easier navigation. These tables are exhaustive: they specify exactly all the semantic conditions that must hold.

Table 1 axiomatizes the semantics of equality. In particular, it defines the equality relation on resources owl:sameAs as being reflexive, symmetric, and transitive, and it axiomatizes the standard replacement properties of equality for it.

Table 2 specifies the semantic conditions on axioms about properties.

Table 3 specifies the semantic conditions on classes.

Table 4 specifies the semantic conditions on class axioms.

Table 5 specifies the semantic restrictions on the vocabulary used to define the schema.

4.4 Relationship between OWL-R DL and OWL-R Full

Let AXIOMS be a set containing all the implications listed in Section 4.3.2; let O be an OWL-R DL ontology in which no URI is used both as an object and a data property; let F be a set of assertions of the following form:

• ClassAssertion( i C ) where C is an OWL class
• ObjectPropertyAssertion( P i₁ i₂ ) where P is an object property
• DataPropertyAssertion( P i v ) where P is a data property
• SameIndividual( i₁ ... i_n )
• DifferentIndividuals( i₁ ... i_n )

5 Computational Properties

This section describes the computational complexity of important reasoning problems in the described profiles.

Note that in languages that are propositionally closed (i.e. that provide, either implicitly or explicitly, conjunction, union and negation of class descriptions), such as OWL 2 DL and OWL 2 Full, the problems of ontology consistency, concept satisfiability, concept subsumption and instance checking can be reduced to each other in polynomial time. However, none of the described profiles is propositionally closed, and these reasoning problems may thus have different complexity and require diferent algorithmic solutions.

This section describes the computational complexity of the most relevant reasoning problems in the languages introduced so far. The reasoning problems considered here are the following:

• Ontology Consistency: Check whether a given ontology has at least one model.
• Concept Satisfiability: Given an ontology O and a class A, verify whether there is a model of O in which the interpretation of A is a nonempty set.
• Concept Subsumption: Given an ontology O and two classes A, B, verify whether the interpretation of A is a subset of the interpretation of B in every model of O
• Instance Checking: Given an ontology, an individual a and a class A, verify whether a is an instance of A in every model of the ontology.
• Conjunctive Query Answering: Given an ontology O and a conjunctive query q, return the answers of the query with respect to O.

When evaluating the complexity, the following parameters will be considered:

• Data Complexity: the complexity measured with respect to the total size of the facts in the ontology.
• Taxonomic Complexity: the complexity measured with respect to the total size of the axioms in the ontology.
• Query Complexity: the complexity measured with respect to the total size of the query.
• Combined Complexity: the complexity measured with respect to both the size of the axioms and the facts. In the case of conjunctive query answering, the combined complexity also includes the size of the query.

Table 6 summarizes the known complexity results for OWL 2 DL, OWL 1 DL, EL++, DL-Lite, and OWL-R. Whenever the complexity for a given problem is described as Open, with a star, (*), it is meant that its decidability is still an open question; if the star (*) is omitted, then the problem is known to be decidable but precise complexity bounds have not yet been established.

In DL-Lite, instance checking and conjunctive query evaluation can be performed by exploiting relational database technology, i.e., through a translation to SQL queries. The fact that data complexity goes beyond LOGSPACE means that query answering and instance checking require more powerful engines than the ones provided by relational database technologies. PTIME-hardness essentially requires Datalog technologies. For the CoNP cases, Disjunctive Datalog technologies could be adopted.

6 References

7 Appendix: Complete Grammars for Profiles

Editor's Note: This appendix will contain the full grammars of each of the profiles. The grammar will be completed when the technical work on each of the profiles has been finished.

7.1 EL++

7.2 DL-Lite

7.3 OWL-R