OWL 1.1 Web Ontology Language:
Fragments

W3C Editor's Draft 02 April 2008

This version:

http://www.w3.org/2007/OWL/draft/ED-owl11-fragments-20080402/

Latest editor's draft:

http://www.w3.org/2007/OWL/draft/owl11-fragments/

Previous version:

http://www.w3.org/2007/OWL/draft/ED-owl11-fragments-20080326/ (color-coded diff)

Authors:

Bernardo Cuenca Grau, Oxford University

Boris Motik, Oxford University

Zhe Wu, Oracle

Achille Fokoue, IBM

Carsten Lutz, Dresden University of Technology

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Copyright © 2008 W3C^® (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark and document use rules apply.

Abstract

Status of this Document

May Be Superseded

This section describes the status of this document at the time of its publication. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at http://www.w3.org/TR/.

Set of Documents

This document is being published as one of a set of 6 documents:

1. Structural Specification and Functional-Style Syntax
2. Model-Theoretic Semantics
3. Mapping to RDF Graphs
4. XML Serialization
5. Fragments (this document)
6. Primer

Please Comment By 2008-04-03

The OWL Working Group seeks public feedback on these Working Drafts. Please send your comments to public-owl-comments@w3.org (public archive). If possible, please offer specific changes to the text that would address your concern.

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This document describes three important fragments of OWL 1.1. Most fragments are defined by placing restrictions on the syntax of OWL 1.1. These restrictions have been specified by modifying some of the productions of the functional-style syntax [OWL 1.1 Specification].

• In terms of the richness of the expression language (e.g. class expressions: , property expressions) used to describe new entities from existing ones.
• In terms of the set of allowed axioms (axiom).

However, OWL fragments share important features that are presented in this section.

As in [OWL 1.1 Specification], the grammar of an OWL fragment is presented in the standard BNF notation. Nonterminal symbols are written in bold (e.g., owlClassURI), terminal symbols are written in single quotes (e.g. 'ObjectPropertyRange'), zero or more instances of a symbol is denoted with curly braces (e.g., { description }), alternative productions are denoted with the vertical bar (e.g., fact | declaration), and zero or one instances of a symbol are denoted with square brackets (e.g., [ description ]).

Regardless of the considered fragment, an ontology consists of a set of axioms and may include axioms from other, imported, ontologies. The following basic syntax of an ontology is shared by all three fragments of OWL 1.1:

The namespace production defines an abbreviation for namespaces in a document. The details of the Full-IRI and URI, used to uniquely identified a resource, and of annotation, which has no effects on the semantics, are given in section 2.2 of [OWL 1.1 Specification].

All fragments support two types of axioms with no effects on their semantics: declaration and entityAnnotation.

A declaration declares the existence of an entity. It can be used to check the structural consistency of an ontology (e.g. check that all entity URIs used in an ontology corresponds to an declared entity). The grammar for declarations is as follows:

An entityAnnotation provides a mechanism to annotate an entity:

EL++ [EL++] is a syntactic fragment of OWL 1.1 that admits sound and complete reasoning in polynomial time with respect to the size of the ontology. Many large-scale biomedical ontologies, such as SNOMED CT, fall within this fragment.

The productions for EL++ are defined in the following sections. All the nonstructural restrictions on axioms are exactly as in Section 10 of the structural specification [OWL 1.1 Specification].

EL++ does not impose any restrictions on OWL 1.1 Entities. Therefore, entities defined here are the same as in [OWL 1.1 Specification]

All entities may have an associated URI. The syntax for writing entity URIs in EL++ is as follows:

Entities are written in the following way:

EL++ defines the same set of well-known entities as the whole OWL 1.1 language. These entities are identified by the following predefined URIs:

• The class with URI owl:Thing is the set of all objects. (In DL literature this is often called the top concept.)
• The class with URI owl:Nothing is the empty set of objects. (In DL literature this is often called the bottom concept.)
• The unary datatype with URI rdfs:Literal is the set of all concrete objects.
• The datatypes with URIs as mentioned in [OWL 1.1 Semantics].

Of all class expressions in EL++, only objectOneOf is different from the one defined in [OWL 1.1 Specification]. EL++ objectOneOf enumeration contains a single individual:

objectIntersectionOf, which is a conjunction of a set of descriptions, objectSomeValuesFrom (resp. dataSomeValuesFrom), which denotes the set of objects that are connected via the given object (resp. data) property to at least one instance of the given description (resp. at least one literal of the given data range), objectExistsSelf, which denotes the set of objects that are connected to themselves via the given object property, objectHasValue (resp. dataHasValue), which denotes the set of objects that are connected via the given object (resp. data) property to the object denoted by the given individual (resp. to the given constant), are unchanged:

A data range expression defines a range over data values. In EL++, a data range expression is restricted to named atomic datatypes (e.g. the list of datatypes supported by EL++ is the same as the one in [OWL 1.1 Semantics]), enumerated datatypes consisting of a single constant, and datatype restrictions, specified by applying some facets to limit the value space of an pre-existing datatype.

All object property axioms supported by EL++ are the same as in OWL 1.1, with the difference that they use the new description production. For your convenience, their definition is provided below:

EL++ disallows the FunctionalDataProperty axioms. Therefore, data property axioms in EL++ are defined as follows.

All data property axioms supported by EL++ are the same as in OWL 1.1, with the difference that they use the new description production. For your convenience, their definition is provided below:

EL++ fact axioms are almost the same as in OWL 1.1, with the difference that they use the new description and dataRange productions.

Note that the fragment 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.

Editor's Note: Please note that Data Properties are not yet incorporated into DL-Lite in this draft, pending consultation with some DL-Lite experts.

Editor's Note: See ISSUE-80 (DL-Lite) This document currently contains the DL-lite_R version of DL-lite. In future versions of this document, however, this language is likely to be extended with additional constructs that preserve its computational properties, such as Data Properties and suitably restricted Functional Properties.

The productions for DL-Lite are defined in the following sections. No nonstructural restricitons on axioms defined in the structural specification [OWL 1.1 Specification] apply to DL-Lite.

Editor's Note: Please note that Data Properties are not yet incorporated into DL-Lite in this draft, pending consultation with some DL-Lite experts.

DL Lite supports all OWL 1.1 Entities except datatype and dataProperty. All entities may have an associated URI. The syntax for writing entity URIs in EL++ is as follows:

Entities are written in the following way:

The only well-known entity defined in DL Lite us the class with URI owl:Thing, which corresponds to the set of all objects. (In DL literature this is often called the top concept.)

Furthermore, DL-Lite redefines all axioms from the functional-style syntax [OWL 1.1 Specification] that refer to the description production. In particular, it restricts various class axioms to appropriate forms of classes, and it disallows DisjointUnion. DL Lite classAxiom production is as follows:

DL-Lite disallows the use of property chains in property inclusion axioms; moreover, it disallows the use of transitive, asymmetric, reflexive and irreflexive properties. Finally, it redefines the domain and range axioms to use the new class productions.

Editor's Note: I assume that negativeObjectPropertyAssertion is not supported by DL-Lite

Finally, DL-Lite disallows axioms about data properties and negative object property assertion, and class membership assertions in DL-Lite are restricted to only atomic classes. Therefore, the fact axioms of DL-Lite are defined as follows:

OWL-R, although just a fragment of OWL 1.1, is quite expressive. An OWL-R DL ontology can use, in a nutshell, most OWL 1.1 language constructs except owl:cardinality, owl:minCardinality, owl11:NegativeObjectPropertyAssertion, owl11: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.

Unlike OWL-R DL, in OWL-R Full there are no syntactic restrictions on the way language constructs can be used: any RDF graph constitutes a valid OWL-R Full ontology. The semantics of language constructs, however, is weakened in OWL-R Full to mimic the usage patterns of OWL-R DL. For example, in OWL 1.1 Full (or DL), an OWL class C1 is a subclass of C2 if and only if the extension of C1 is a subset of the extension of C2. In OWL-R Full, that "if and only if" condition is weakened to "only if." The principles according to which this weakening has been derived are presented in Section 4.3.1. An equivalent characterization of the weakened semantics by means of first-order implications is given in Section 4.3.2. Table 2 lists the language constructs that are supported in OWL-R Full.

OWL-R DL is a syntactic fragment of OWL 1.1 DL. The fragment is defined not only in terms of a set of supported constructs, but it also restricts the places in which these constructs can be used. It is based on Description Logic Programs [DLP] -- a logic obtained by intersecting description logics with rule-based languages.

OWL-R DL does not impose any restrictions on OWL 1.1 Entities. Therefore, entities defined here are the same as in [OWL 1.1 Specification]

All entities may have an associated URI. The syntax for writing entity URIs in OWL-R DL is as follows:

Entities are written in the following way:

OWL-R defines the same set of well-known entities as the whole OWL 1.1 language. These entities are identified by the following predefined URIs:

• The class with URI owl:Thing is the set of all objects. (In DL literature this is often called the top concept.)
• The class with URI owl:Nothing is the empty set of objects. (In DL literature this is often called the bottom concept.)
• The unary datatype with URI rdfs:Literal is the set of all concrete objects.
• The datatypes with URIs as mentioned in [OWL 1.1 Semantics].

A data range expression defines a range over data values. In OWL-R DL, a data range expression is restricted to either a named atomic datatype (the list of datatypes supported by OWL-R DL is identical to the one in [OWL 1.1 Semantics]) or a datatype restriction, specified by applying some facets to limit the value space of an pre-existing datatype.

OWL-R constructs used to build more complex properties from existing ones are identical to the ones defined in [OWL 1.1 Specification]:

inverseObjectProperty:= 'InverseObjectProperty' '(' objectPropertyExpression ')'

OWL-R redefines all axioms from the functional-style syntax OWL 1.1 Specification that refer to the description production. In particular, it restricts various class axioms to use the appropriate form of class expressions (i.e. one of subClass , superClass, or equivClass), and it disallows the DisjointUnion axiom.

OWL-R property expression language is very similar to OWL 1.1. The only difference is that OWL-R restricts property domain and range axioms to the appropriate form of class expressions as follows:

Therefore, the OWL-R objectPropertyAxiom and dataPropertyAxiom are defined as follows:

Besides objectPropertyDomain, objectPropertyRange, dataPropertyDomain, and dataPropertyRange, all the other non-terminals appearing in the definitions of dataPropertyAxiom and objectPropertyAxiom are identical to those defined in OWL 1.1 Specification. For reader's convenience, we provide here their definition:

OWL-R restricts the facts to a particular type of classes:

Furthermore, it disallows negative property assertions. Therefore, OWL-R fact production is as follows:

Besides classAssertion, which has previously been redefined, the other types of facts are identical to those defined in OWL 1.1 Specification. For your convenience, their definition is presented below:

OWL-R Full is defined by weakening the semantic conditions on an interpretation from OWL 1.1 Full. An equivalent definition is also provided in terms of an "axiomatization" using first order implications. The latter definition should provide a useful starting point for practical implementation using rule-based technologies. It is based on [pD*].

This section defines OWL-R Full by weakening the OWL 1.1 Full semantic conditions on an interpretation.

Editor's Note: We need to add a reference of OWL 1.1 Full Semantics

From OWL 1.1 Full Semantics, for V a set of URI references and literals containing the RDF and RDFS 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 1.1 Full semantics.

Finally, the following important sets are used in the definitions of OWL 1.1 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.

• 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.

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.

Furthermore, let RDF(O) and RDF(F) be the translations of O and F into RDF graphs as specified in the RDF mapping [ OWL 1.1 RDF Mapping ] in which triples are represented using the T predicate. Then, the following relationship between consequences in OWL-R DL and OWL-R Full holds:

F is a consequence of O under the OWL 1.1 DL semantics if and only if RDF(F) is a consequence of RDF(O) AXIOMS under the standard first-order semantics.

Table 6 summarizes the known complexity results for OWL 1.1 DL, OWL 1.0 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.

[OWL 1.1 Specification]

OWL 1.1 Web Ontology Language:Structural Specification and Functional-Style Syntax Boris Motik, Peter F. Patel-Schneider, Ian Horrocks. W3C Editor's Draft, 02 April 2008, http://www.w3.org/2007/OWL/draft/ED-owl11-syntax-20080402/. Latest version available at http://www.w3.org/2007/OWL/draft/owl11-syntax/.

[OWL 1.1 Semantics]

OWL 1.1 Web Ontology Language:Model-Theoretic Semantics Bernardo Cuenca Grau, Boris Motik. W3C Editor's Draft, 02 April 2008, http://www.w3.org/2007/OWL/draft/ED-owl11-semantics-20080402/. Latest version available at http://www.w3.org/2007/OWL/draft/owl11-semantics/.

[OWL 1.1 RDF Mapping]

OWL 1.1 Web Ontology Language: Mapping to RDF Graphs. Bernardo Cuenca Grau and Boris Motik, eds., 2006.

[EL++]

Pushing the EL Envelope. Franz Baader, Sebastian Brandt, and Carsten Lutz. In Proc. of the 19th Joint Int. Conf. on Artificial Intelligence (IJCAI 2005), 2005.

[DL-Lite]

Tractable Reasoning and Efficient Query Answering in Description Logics: The DL-Lite Family. Diego Calvanese, Giuseppe de Giacomo, Domenico Lembo, Maurizio Lenzerini, Riccardo Rosati. J. of Automated Reasoning 39(3):385--429, 2007.

[Complexity]

Complexity Results and Practical Algorithms for Logics in Knowledge Representation. Stephan Tobies. Ph.D Dissertation, 2002

[DLP]

Description Logic Programs: Combining Logic Programs with Description Logic. Benjamin N. Grosof, Ian Horrocks, Raphael Volz, and Stefan Decker. in Proc. of the 12th Int. World Wide Web Conference (WWW 2008), Budapest, Hungary, 2003. pp.: 48--57

[pD*]

Completeness, decidability and complexity of entailment for RDF Schema and a semantic extension involving the OWL vocabulary. Herman J. ter Horst. J. of Web Semantics 3(2--3):79--115, 2005.