OWL 2 Web Ontology Language
Direct Semantics

W3C Recommendation 27 October 2009

This version:

http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/

Latest version (series 2):

http://www.w3.org/TR/owl2-direct-semantics/

Latest Recommendation:

http://www.w3.org/TR/owl-direct-semantics

Previous version:

http://www.w3.org/TR/2009/PR-owl2-direct-semantics-20090922/ (color-coded diff)

Editors:

Boris Motik, Oxford University Computing Laboratory

Peter F. Patel-Schneider, Bell Labs Research, Alcatel-Lucent

Bernardo Cuenca Grau, Oxford University Computing Laboratory

Contributors: (in alphabetical order)

Ian Horrocks, Oxford University Computing Laboratory

Bijan Parsia, University of Manchester

Uli Sattler, University of Manchester

Please refer to the errata for this document, which may include some normative corrections.

This document is also available in these non-normative formats: PDF version.

See also translations.

Copyright © 2009 W3C^® (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark and document use rules apply.

Abstract

Status of this Document

May Be Superseded

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There have been no changes to the body of this document since the previous version. For details on earlier changes, see the change log.

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Please send any comments to public-owl-comments@w3.org (public archive). Although work on this document by the OWL Working Group is complete, comments may be addressed in the errata or in future revisions. Open discussion among developers is welcome at public-owl-dev@w3.org (public archive).

Endorsed By W3C

This document has been reviewed by W3C Members, by software developers, and by other W3C groups and interested parties, and is endorsed by the Director as a W3C Recommendation. It is a stable document and may be used as reference material or cited from another document. W3C's role in making the Recommendation is to draw attention to the specification and to promote its widespread deployment. This enhances the functionality and interoperability of the Web.

Patents

This document was produced by a group operating under the 5 February 2004 W3C Patent Policy. W3C maintains a public list of any patent disclosures made in connection with the deliverables of the group; that page also includes instructions for disclosing a patent.

1 Introduction

This document defines the direct model-theoretic semantics of OWL 2. The semantics given here is strongly related to the semantics of description logics [Description Logics] and it extends the semantics of the description logic SROIQ [SROIQ]. As the definition of SROIQ does not provide for datatypes and punning, the semantics of OWL 2 is defined directly on the constructs of the structural specification of OWL 2 [OWL 2 Specification] instead of by reference to SROIQ. For the constructs available in SROIQ, the semantics of SROIQ trivially corresponds to the one defined in this document.

Since each OWL 1 DL ontology is an OWL 2 ontology, this document also provides a direct semantics for OWL 1 Lite and OWL 1 DL ontologies; this semantics is equivalent to the direct model-theoretic semantics of OWL 1 Lite and OWL 1 DL [OWL 1 Semantics and Abstract Syntax]. Furthermore, this document also provides the direct model-theoretic semantics for the OWL 2 profiles [OWL 2 Profiles].

The semantics is defined for OWL 2 axioms and ontologies, which should be understood as instances of the structural specification [OWL 2 Specification]. Parts of the structural specification are written in this document using the functional-style syntax.

OWL 2 allows ontologies, anonymous individuals, and axioms to be annotated; furthermore, annotations themselves can contain additional annotations. All these types of annotations, however, have no semantic meaning in OWL 2 and are ignored in this document. OWL 2 declarations are used only to disambiguate class expressions from data ranges and object property from data property expressions in the functional-style syntax; therefore, they are not mentioned explicitly in this document.

2 Direct Model-Theoretic Semantics for OWL 2

This section specifies the direct model-theoretic semantics of OWL 2 ontologies.

2.1 Vocabulary

A datatype map, formalizing datatype maps from the OWL 2 Specification [OWL 2 Specification], is a 6-tuple D = ( N_DT , N_LS , N_FS , ⋅ ^DT , ⋅ ^LS , ⋅ ^FS ) with the following components:

• N_DT is a set of datatypes (more precisely, names of datatypes) that does not contain the datatype rdfs:Literal.
• N_LS is a function that assigns to each datatype DT ∈ N_DT a set N_LS(DT) of strings called lexical forms. The set N_LS(DT) is called the lexical space of DT.
• N_FS is a function that assigns to each datatype DT ∈ N_DT a set N_FS(DT) of pairs ( F , v ), where F is a constraining facet and v is an arbitrary data value called the constraining value. The set N_FS(DT) is called the facet space of DT.
• For each datatype DT ∈ N_DT, the interpretation function ⋅ ^DT assigns to DT a set (DT)^DT called the value space of DT.
• For each datatype DT ∈ N_DT and each lexical form LV ∈ N_LS(DT), the interpretation function ⋅ ^LS assigns to the pair ( LV , DT ) a data value ( LV , DT )^LS ∈ (DT)^DT.
• For each datatype DT ∈ N_DT and each pair ( F , v ) ∈ N_FS(DT), the interpretation function ⋅ ^FS assigns to ( F , v ) the set ( F , v )^FS ⊆ (DT)^DT.

The set of datatypes N_DT of a datatype map D is not required to contain all datatypes from the OWL 2 datatype map; this allows one to talk about subsets of the OWL 2 datatype map, which may be necessary for the various profiles of OWL 2. If, however, D contains a datatype DT from the OWL 2 datatype map, then N_LS(DT), N_FS(DT), (DT)^DT, ( LV , DT )^LS for each LV ∈ N_LS(DT), and ( F , v )^FS for each ( F , v ) ∈ N_FS(DT) are required to coincide with the definitions for DT in the OWL 2 datatype map.

A vocabulary V = ( V_C , V_OP , V_DP , V_I , V_DT , V_LT , V_FA ) over a datatype map D is a 7-tuple consisting of the following elements:

• V_C is a set of classes as defined in the OWL 2 Specification [OWL 2 Specification], containing at least the classes owl:Thing and owl:Nothing.
• V_OP is a set of object properties as defined in the OWL 2 Specification [OWL 2 Specification], containing at least the object properties owl:topObjectProperty and owl:bottomObjectProperty.
• V_DP is a set of data properties as defined in the OWL 2 Specification [OWL 2 Specification], containing at least the data properties owl:topDataProperty and owl:bottomDataProperty.
• V_I is a set of individuals (named and anonymous) as defined in the OWL 2 Specification [OWL 2 Specification].
• V_DT is a set containing all datatypes of D, the datatype rdfs:Literal, and possibly other datatypes; that is, N_DT ∪ { rdfs:Literal } ⊆ V_DT.
• V_LT is a set of literals LV^^DT for each datatype DT ∈ N_DT and each lexical form LV ∈ N_LS(DT).
• V_FA is the set of pairs ( F , lt ) for each constraining facet F, datatype DT ∈ N_DT, and literal lt ∈ V_LT such that ( F , ( LV , DT₁ )^LS ) ∈ N_FS(DT), where LV is the lexical form of lt and DT₁ is the datatype of lt.

Given a vocabulary V, the following conventions are used in this document to denote different syntactic parts of OWL 2 ontologies:

• OP denotes an object property;
• OPE denotes an object property expression;
• DP denotes a data property;
• DPE denotes a data property expression;
• C denotes a class;
• CE denotes a class expression;
• DT denotes a datatype;
• DR denotes a data range;
• a denotes an individual (named or anonymous);
• lt denotes a literal; and
• F denotes a constraining facet.

2.2 Interpretations

Given a datatype map D and a vocabulary V over D, an interpretation I = ( Δ_I , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) for D and V is a 9-tuple with the following structure:

• Δ_I is a nonempty set called the object domain.
• Δ_D is a nonempty set disjoint with Δ_I called the data domain such that (DT)^DT ⊆ Δ_D for each datatype DT ∈ V_DT.
• ⋅ ^C is the class interpretation function that assigns to each class C ∈ V_C a subset (C)^C ⊆ Δ_I such that
  • (owl:Thing)^C = Δ_I and
  • (owl:Nothing)^C = ∅.
• ⋅ ^OP is the object property interpretation function that assigns to each object property OP ∈ V_OP a subset (OP)^OP ⊆ Δ_I × Δ_I such that
  • (owl:topObjectProperty)^OP = Δ_I × Δ_I and
  • (owl:bottomObjectProperty)^OP = ∅.
• ⋅ ^DP is the data property interpretation function that assigns to each data property DP ∈ V_DP a subset (DP)^DP ⊆ Δ_I × Δ_D such that
  • (owl:topDataProperty)^DP = Δ_I × Δ_D and
  • (owl:bottomDataProperty)^DP = ∅.
• ⋅ ^I is the individual interpretation function that assigns to each individual a ∈ V_I an element (a)^I ∈ Δ_I.
• ⋅ ^DT is the datatype interpretation function that assigns to each datatype DT ∈ V_DT a subset (DT)^DT ⊆ Δ_D such that
  • ⋅ ^DT is the same as in D for each datatype DT ∈ N_DT, and
  • (rdfs:Literal)^DT = Δ_D.
• ⋅ ^LT is the literal interpretation function that is defined as (lt)^LT = ( LV , DT )^LS for each lt ∈ V_LT, where LV is the lexical form of lt and DT is the datatype of lt.
• ⋅ ^FA is the facet interpretation function that is defined as ( F , lt )^FA = ( F , (lt)^LT )^FS for each ( F , lt ) ∈ V_FA.

The following sections define the extensions of ⋅ ^OP, ⋅ ^DT, and ⋅ ^C to object property expressions, data ranges, and class expressions.

2.2.1 Object Property Expressions

The object property interpretation function ⋅ ^OP is extended to object property expressions as shown in Table 1.

2.2.2 Data Ranges

The datatype interpretation function ⋅ ^DT is extended to data ranges as shown in Table 3. All datatypes in OWL 2 are unary, so each datatype DT is interpreted as a unary relation over Δ_D — that is, as a set (DT)^DT ⊆ Δ_D. OWL 2 currently does not define data ranges of arity more than one; however, by allowing for n-ary data ranges, the syntax of OWL 2 provides a "hook" allowing implementations to introduce extensions such as comparisons and arithmetic. An n-ary data range DR is interpreted as an n-ary relation (DR)^DT over Δ_D — that is, as a set (DT)^DT ⊆ (Δ_D)ⁿ

2.2.3 Class Expressions

The class interpretation function ⋅ ^C is extended to class expressions as shown in Table 4. For S a set, #S denotes the number of elements in S.

2.3 Satisfaction in an Interpretation

An interpretation I = ( Δ_I , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) satisfies an axiom w.r.t. an ontology O if the axiom satisfies the relevant condition from the following sections. Satisfaction of axioms in I is defined w.r.t. O because satisfaction of key axioms uses the following function:

2.3.1 Class Expression Axioms

Satisfaction of OWL 2 class expression axioms in I w.r.t. O is defined as shown in Table 5.

2.3.2 Object Property Expression Axioms

Satisfaction of OWL 2 object property expression axioms in I w.r.t. O is defined as shown in Table 6.

2.3.3 Data Property Expression Axioms

Satisfaction of OWL 2 data property expression axioms in I w.r.t. O is defined as shown in Table 7.

2.3.4 Datatype Definitions

Satisfaction of datatype definitions in I w.r.t. O is defined as shown in Table 8.

2.3.5 Keys

Satisfaction of keys in I w.r.t. O is defined as shown in Table 9.

2.3.6 Assertions

Satisfaction of OWL 2 assertions in I w.r.t. O is defined as shown in Table 10.

2.3.7 Ontologies

An interpretation I satisfies an OWL 2 ontology O if all axioms in the axiom closure of O (with anonymous individuals standardized apart as described in Section 5.6.2 of the OWL 2 Specification [OWL 2 Specification]) are satisfied in I w.r.t. O.

2.4 Models

Given a datatype map D, an interpretation I = ( Δ_I , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) for D is a model of an OWL 2 ontology O w.r.t. D if an interpretation J = ( Δ_I , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^J , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) for D exists such that ⋅ ^J coincides with ⋅ ^I on all named individuals and J satisfies O.

Thus, an interpretation I satisfying O is also a model of O. In contrast, a model I of O may not satisfy O directly; however, by modifying the interpretation of anonymous individuals, I can always be coerced into an interpretation J that satisfies O.

2.5 Inference Problems

Let D be a datatype map and V a vocabulary over D. Furthermore, let O and O₁ be OWL 2 ontologies, CE, CE₁, and CE₂ class expressions, and a a named individual, such that all of them refer only to the vocabulary elements in V. Furthermore, variables are symbols that are not contained in V. Finally, a Boolean conjunctive query Q is a closed formula of the form

where each A_i is an atom of the form C(s), OP(s,t), or DP(s,u) with C a class, OP an object property, DP a data property, s and t individuals or some variable x_j, and u a literal or some variable y_j.

The following inference problems are often considered in practice.

Ontology Consistency: O is consistent (or satisfiable) w.r.t. D if a model of O w.r.t. D and V exists.

Ontology Entailment: O entails O₁ w.r.t. D if every model of O w.r.t. D and V is also a model of O₁ w.r.t. D and V.

Ontology Equivalence: O and O₁ are equivalent w.r.t. D if O entails O₁ w.r.t. D and O₁ entails O w.r.t. D.

Ontology Equisatisfiability: O and O₁ are equisatisfiable w.r.t. D if O is satisfiable w.r.t. D if and only if O₁ is satisfiable w.r.t D.

Class Expression Satisfiability: CE is satisfiable w.r.t. O and D if a model I = ( Δ_I , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) of O w.r.t. D and V exists such that (CE)^C ≠ ∅.

Class Expression Subsumption: CE₁ is subsumed by a class expression CE₂ w.r.t. O and D if (CE₁)^C ⊆ (CE₂)^C for each model I = ( Δ_I , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) of O w.r.t. D and V.

Instance Checking: a is an instance of CE w.r.t. O and D if (a)^I ∈ (CE)^C for each model I = ( Δ_I , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) of O w.r.t. D and V.

Boolean Conjunctive Query Answering: Q is an answer w.r.t. O and D if Q is true in each model of O w.r.t. D and V according to the standard definitions of first-order logic.

In order to ensure that ontology entailment, class expression satisfiability, class expression subsumption, and instance checking are decidable, the following restriction w.r.t. O needs to be satisfied:

For ontology equivalence to be decidable, O₁ needs to satisfy this restriction w.r.t. O and vice versa. These restrictions are analogous to the first condition from Section 11.2 of the OWL 2 Specification [OWL 2 Specification].

3 Independence of the Direct Semantics from the Datatype Map in OWL 2 DL (Informative)

OWL 2 DL has been defined so that the consequences of an OWL 2 DL ontology O do not depend on the choice of a datatype map, as long as the datatype map chosen contains all the datatypes occurring in O. This statement is made precise by the following theorem, and it has several useful consequences:

• One can apply the direct semantics to an OWL 2 DL ontology O by considering only the datatypes explicitly occurring in O.
• When referring to various reasoning problems, the datatype map D need not be given explicitly, as it is sufficient to consider an implicit datatype map containing only the datatypes from the given ontology.
• OWL 2 DL reasoners can provide datatypes not explicitly mentioned in this specification without fear that this will change the meaning of OWL 2 DL ontologies not using these datatypes.

Theorem DS1. Let O₁ and O₂ be OWL 2 DL ontologies over a vocabulary V and D = ( N_DT , N_LS , N_FS , ⋅ ^DT , ⋅ ^LS , ⋅ ^FS ) a datatype map such that each datatype mentioned in O₁ and O₂ is rdfs:Literal, a datatype defined in the respective ontology, or it occurs in N_DT. Furthermore, let D' = ( N_DT' , N_LS' , N_FS' , ⋅ ^DT ' , ⋅ ^LS ' , ⋅ ^FS ' ) be a datatype map such that N_DT ⊆ N_DT', N_LS(DT) = N_LS'(DT), and N_FS(DT) = N_FS'(DT) for each DT ∈ N_DT, and ⋅ ^DT ', ⋅ ^LS ', and ⋅ ^FS ' are extensions of ⋅ ^DT, ⋅ ^LS, and ⋅ ^FS, respectively. Then, O₁ entails O₂ w.r.t. D if and only if O₁ entails O₂ w.r.t. D'.

Proof. Without loss of generality, one can assume O₁ and O₂ to be in negation-normal form [Description Logics]. Furthermore, since datatype definitions in O₁ and O₂ are acyclic, one can assume that each defined datatype has been recursively replaced with its definition; thus, all datatypes in O₁ and O₂ are from N_DT ∪ { rdfs:Literal }. The claim of the theorem is equivalent to the following statement: an interpretation I w.r.t. D and V exists such that O₁ is and O₂ is not satisfied in I if and only if an interpretation I' w.r.t. D' and V exists such that O₁ is and O₂ is not satisfied in I'. The (⇐) direction is trivial since each interpretation I w.r.t. D' and V is also an interpretation w.r.t. D and V. For the (⇒) direction, assume that an interpretation I = ( Δ_I , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) w.r.t. D and V exists such that O₁ is and O₂ is not satisfied in I. Let I' = ( Δ_I , Δ_D' , ⋅ ^C ' , ⋅ ^OP , ⋅ ^DP ' , ⋅ ^I , ⋅ ^DT ' , ⋅ ^LT ' , ⋅ ^FA ' ) be an interpretation such that

• Δ_D' is obtained by extending Δ_D with the value space of all datatypes in N_DT' \ N_DT,
• ⋅ ^C ' coincides with ⋅ ^C on all classes, and
• ⋅ ^DP ' coincides with ⋅ ^DP on all data properties apart from owl:topDataProperty.

Clearly, DataComplementOf( DR )^DT ⊆ DataComplementOf( DR )^DT ' for each data range DR that is either a datatype, a datatype restriction, or an enumerated data range. The owl:topDataProperty property can occur in O₁ and O₂ only in tautologies. The interpretation of all other data properties is the same in I and I', so (CE)^C = (CE)^C ' for each class expression CE occurring in O₁ and O₂. Therefore, O₁ is and O₂ is not satisfied in I'. QED

5 Acknowledgments

The starting point for the development of OWL 2 was the OWL1.1 member submission, itself a result of user and developer feedback, and in particular of information gathered during the OWL Experiences and Directions (OWLED) Workshop series. The working group also considered postponed issues from the WebOnt Working Group.

This document has been produced by the OWL Working Group (see below), and its contents reflect extensive discussions within the Working Group as a whole. The editors extend special thanks to Markus Krötzsch (FZI), Michael Schneider (FZI) and Thomas Schneider (University of Manchester) for their thorough reviews.

The regular attendees at meetings of the OWL Working Group at the time of publication of this document were: Jie Bao (RPI), Diego Calvanese (Free University of Bozen-Bolzano), Bernardo Cuenca Grau (Oxford University Computing Laboratory), Martin Dzbor (Open University), Achille Fokoue (IBM Corporation), Christine Golbreich (Université de Versailles St-Quentin and LIRMM), Sandro Hawke (W3C/MIT), Ivan Herman (W3C/ERCIM), Rinke Hoekstra (University of Amsterdam), Ian Horrocks (Oxford University Computing Laboratory), Elisa Kendall (Sandpiper Software), Markus Krötzsch (FZI), Carsten Lutz (Universität Bremen), Deborah L. McGuinness (RPI), Boris Motik (Oxford University Computing Laboratory), Jeff Pan (University of Aberdeen), Bijan Parsia (University of Manchester), Peter F. Patel-Schneider (Bell Labs Research, Alcatel-Lucent), Sebastian Rudolph (FZI), Alan Ruttenberg (Science Commons), Uli Sattler (University of Manchester), Michael Schneider (FZI), Mike Smith (Clark & Parsia), Evan Wallace (NIST), Zhe Wu (Oracle Corporation), and Antoine Zimmermann (DERI Galway). We would also like to thank past members of the working group: Jeremy Carroll, Jim Hendler, Vipul Kashyap.

6 References

6.1 Normative References

[OWL 2 Specification]

OWL 2 Web Ontology Language: Structural Specification and Functional-Style Syntax Boris Motik, Peter F. Patel-Schneider, Bijan Parsia, eds. W3C Recommendation, 27 October 2009, http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/. Latest version available at http://www.w3.org/TR/owl2-syntax/.

6.2 Nonnormative References

[Description Logics]

The Description Logic Handbook: Theory, Implementation, and Applications, second edition. Franz Baader, Diego Calvanese, Deborah L. McGuinness, Daniele Nardi, and Peter F. Patel-Schneider, eds. Cambridge University Press, 2007. Also see the Description Logics Home Page.

[OWL 1 Semantics and Abstract Syntax]

OWL Web Ontology Language: Semantics and Abstract Syntax. Peter F. Patel-Schneider, Patrick Hayes, and Ian Horrocks, eds. W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-owl-semantics-20040210/. Latest version available at http://www.w3.org/TR/owl-semantics/.

[OWL 2 Profiles]

OWL 2 Web Ontology Language: Profiles Boris Motik, Bernardo Cuenca Grau, Ian Horrocks, Zhe Wu, Achille Fokoue, Carsten Lutz, eds. W3C Recommendation, 27 October 2009, http://www.w3.org/TR/2009/REC-owl2-profiles-20091027/. Latest version available at http://www.w3.org/TR/owl2-profiles/.

[SROIQ]

The Even More Irresistible SROIQ. Ian Horrocks, Oliver Kutz, and Uli Sattler. In Proc. of the 10th Int. Conf. on Principles of Knowledge Representation and Reasoning (KR 2006). AAAI Press, 2006.