Model-Theoretic Semantics

- This version:
- http://www.w3.org/TR/2008/WD-owl2-semantics-20080411/
- Latest version:
- http://www.w3.org/TR/owl2-semantics/
- Previous version:
- http://www.w3.org/TR/2008/WD-owl11-semantics-20080108/

- Authors:
- Bernardo Cuenca Grau, Oxford University
- Boris Motik, Oxford University
- Contributors:
- Ian Horrocks, Oxford University
- Bijan Parsia, The University of Manchester
- Peter F. Patel-Schneider, Bell Labs Research, Alcatel-Lucent
- Ulrike Sattler, The University of Manchester

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

OWL 2 extends the W3C OWL Web Ontology Language
with a small but useful set of features that have been requested by
users, for which effective reasoning algorithms are now available,
and that OWL tool developers are willing to support. The new
features include extra syntactic sugar, additional property and
qualified cardinality constructors, extended datatype support,
simple metamodeling, and extended annotations. This document
provides a model-theoretic semantics for OWL 2.

*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/.*

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

- Structural Specification and Functional-Style Syntax
- Model-Theoretic Semantics (this document)
- Mapping to RDF Graphs
- XML Serialization
- Profiles
- Primer

Since the previous Working Draft (dated 8 January 2008),
**the only change is the name of the
language,** from "OWL 1.1" to "OWL 2". Since the
group is publishing three new Working Drafts, and the name
has changed, it decided to publish the complete set with
consistent names.

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. You may also wish to check the Wiki Version of this document for internal-review comments and changes being drafted which may address your concerns.

*Publication as a Working Draft does not imply endorsement by the W3C Membership. This is a draft document and may be updated, replaced or obsoleted by other documents at any time. It is inappropriate to cite this document as other than work in progress.*

*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. An individual who has actual knowledge of a patent which the individual believes contains Essential Claim(s) must disclose the information in accordance with section 6 of the W3C Patent Policy.*

## Contents |

Editor's Note: See Issue-72 (Annotation Semantics).

Editor's Note: See Issue 63 (OWL Full Semantics).

Editor's Note: See Issue 69 (punning).

This document defines the formal semantics of OWL 2. The
semantics given here follows the principles for defining the
semantics of description logics [Description Logics]
and is compatible with the description logic *SROIQ* presented
in [SROIQ].
Unfortunately, the definition of *SROIQ* given in
[SROIQ] does not
provide for datatypes and metamodeling. Therefore, the semantics of
OWL 2 is defined in a direct model-theoretic way, by interpreting
the constructs of the functional-style syntax from [OWL 2 Specification].
For the constructs available in *SROIQ*, the semantics of
*SROIQ* trivially corresponds to the one defined in this
document.

OWL 2 does not have an RDF-compatible semantics. Ontologies expressed in OWL RDF are given semantics by converting then into the functional-style syntax and interpreting the result as specified in this document.

OWL 2 allows for annotations of ontologies and ontology entities (classes, properties, and individuals) and ontology axioms. Annotations, however, have no semantic meaning in OWL 2 and are ignored in this document. Definitions in OWL 2 similarly have no semantics. Constructs only used in annotations and definitions, like ObjectProperty, therefore do not show up in this document.

Since OWL 2 is an extension of OWL DL, this document also provides a formal semantics for OWL Lite and OWL DL and it is equivalent to the definition given in [OWL Abstract Syntax and Semantics].

Editor's Note: See Issue-73 (infinite universe ).

A *vocabulary* (or *signature*) *V = ( N _{C}
, N_{Po} , N_{Pd} , N_{I} , N_{D} ,
N_{V} )* is a 6-tuple where

*N*is a set of_{C}*OWL classes*,*N*is a set of_{Po}*object properties*,*N*is a set of_{Pd}*data properties*,*N*is a set of_{I}*individuals*, and*N*is a set of_{D}*datatypes*each associated with a positive integer*datatype arity*,*N*is a set of well-formed_{V}*constants*.

Since OWL 2 allows *punning* [Metamodeling] in the
signature, we do not require the sets N_{C} ,
N_{Po} , N_{Pd} , N_{I} , N_{D} ,
and N_{V} to be pair-wise disjoint. Thus, the same name can
be used in an ontology to denote a class, a datatype, a property
(object or data), an individual, and a constant. The set
*N _{D}* is defined as it is because a datatype is
defined by its name and the arity, and such a definition allows one
to reuse the same name with different arities.

The semantics of OWL 2 is defined with respect to a *concrete
domain*, which is a tuple *D = ( Δ _{D} , .^{D}
)* where

*Δ*is a fixed set called the_{D}*data domain*,*.*assigns to each constant^{D}*v ∈ N*an element_{V}*v*of^{D}*Δ*, and_{D}*.*assigns to each datatype^{D}*d ∈ N*with arity_{D}*n*an*n*-ary relation*d*over^{D}*Δ*._{D}

The set of datatypes *N _{D}* in each OWL 2
vocabulary must include a unary datatype

The set *Δ _{D}* is a fixed set that must be

Given a vocabulary *V* and a concrete domain *D*, an
interpretation *I = ( Δ _{I} , .^{Ic} ,
.^{Ipo} , .^{Ipd} , .^{Ii} )* is a
5-tuple where

*Δ*is a nonempty set disjoint with_{I}*Δ*, called the_{D}*object domain*;*.*is the^{Ic}*class interpretation*function that assigns to each OWL class*A ∈ N*a subset_{Ic}*A*of^{Ic}*Δ*;_{I}*.*is the^{Ipo}*object property interpretation*function that assigns to each object property*R ∈ N*a subset_{Po}*R*of^{Ipo}*Δ*x_{I}*Δ*;_{I}*.*is the^{Ipd}*data property interpretation*function that assigns to each data property*U ∈ N*a subset_{Pd}*U*of^{Ipd}*Δ*x_{I}*Δ*;_{D}*.*is the^{Ii}*individual interpretation*function that assigns to each individual*a ∈ N*an element_{I}*a*from^{Ii}*Δ*._{I}

We extend the object interpretation function
*. ^{Ipo}* to object property expressions as shown in
Table 1.

Object Property Expression | Interpretation |
---|---|

InverseObjectProperty(R) | { ( x , y ) | ( y , x ) ∈ R
}^{Ipo} |

We extend the interpretation function *. ^{D}* to
data ranges as shown in Table 2.

Data Range | Interpretation |
---|---|

DataOneOf(v_{1} ... v_{n}) |
{ v , ... ,
_{1}^{D}v }_{n}^{D} |

DataComplementOf(DR) | ( Δ \ _{D} )^{n}DR
where ^{D}n is the arity of DR |

DatatypeRestriction(DR f v) |
the f with value vto the data range DR as specified in [XML Schema
Datatypes] |

We extend the class interpretation function
*. ^{Ic}* to descriptions as shown in Table 3. With

Description | Interpretation |
---|---|

owl:Thing | Δ_{I} |

owl:Nothing | empty set |

ObjectComplementOf(C) | Δ \ _{I}C^{Ic} |

ObjectIntersectionOf(C_{1} ...
C_{n}) |
C ∩ ... ∩
_{1}^{Ic}C_{n}^{Ic} |

ObjectUnionOf(C_{1} ...
C_{n}) |
C ∪ ... ∪
_{1}^{Ic}C_{n}^{Ic} |

ObjectOneOf(a_{1} ... a_{n}) |
{ a , ... ,
_{1}^{Ii}a }_{n}^{Ii} |

ObjectSomeValuesFrom(R C) | { x | ∃ y : ( x, y ) ∈
R and ^{Ipo}y ∈ C }^{Ic} |

ObjectAllValuesFrom(R C) | { x | ∀ y : ( x, y ) ∈
R implies ^{Ipo}y ∈ C
}^{Ic} |

ObjectHasValue(R a) | { x | ( x, a ∈
^{Ii} )R }^{Ipo} |

ObjectExistsSelf(R) | { x | ( x, x ) ∈ R }^{Ipo} |

ObjectMinCardinality(n R C) | { x | #{ y | ( x, y ) ∈
R and ^{Ipo}y ∈ C } ≥ n
}^{Ic} |

ObjectMaxCardinality(n R C) | { x | #{ y | ( x, y ) ∈
R and ^{Ipo}y ∈ C } ≤ n
}^{Ic} |

ObjectExactCardinality(n R C) | { x | #{ y | ( x, y ) ∈
R and ^{Ipo}y ∈ C } = n
}^{Ic} |

DataSomeValuesFrom(U_{1} ... U_{n}
DR) |
{ x | ∃ y, ...,
_{1}y : _{n}( x, y ∈
_{k} )U for each 1 ≤ _{k}^{Ipd}k ≤ n
and ( y ∈
_{1}, ..., y_{n} )DR }^{D} |

DataAllValuesFrom(U_{1} ... U_{n}
DR) |
{ x | ∀ y, ...,
_{1}y : _{n}( x, y ∈
_{k} )U for each 1 ≤ _{k}^{Ipd}k ≤ n
implies ( y ∈
_{1}, ..., y_{n} )DR }^{D} |

DataHasValue(U v) | { x | ( x, v ∈
^{D} )U }^{Ipd} |

DataMinCardinality(n U DR) | { x | #{ y | ( x, y ) ∈
U and ^{Ipd}y ∈ DR } ≥ n
}^{D} |

DataMaxCardinality(n U DR) | { x | #{ y | ( x, y ) ∈
U and ^{Ipd}y ∈ DR } ≤ n
}^{D} |

DataExactCardinality(n U DR) | { x | #{ y | ( x, y ) ∈
U and ^{Ipd}y ∈ DR } = n
}^{D} |

Satisfaction of OWL 2 axioms in an interpretation *I* is
defined as shown in Table 4. With *o* we denote the
composition of binary relations.

Axiom | Condition |
---|---|

SubClassOf(C D) | C ⊆ ^{Ic}D^{Ic} |

EquivalentClasses(C_{1} ...
C_{n}) |
C =
_{j}^{Ic}C for each 1 ≤ _{k}^{Ic}j , k
≤ n |

DisjointClasses(C_{1} ...
C_{n}) |
C ∩
_{j}^{Ic}C is empty for each 1 ≤ _{k}^{Ic}j ,
k ≤ n and j ≠ k |

DisjointUnion(A C_{1} ...
C_{n}) |
A = ^{Ic}C ∪ ...
∪ _{1}^{Ic}C and
_{n}^{Ic}C ∩
_{j}^{Ic}C is empty for each 1 ≤ _{k}^{Ic}j ,
k ≤ n and j ≠ k |

SubObjectPropertyOf(R S) | R ⊆ ^{Ipo}S^{Ipo} |

SubObjectPropertyOf(SubObjectPropertyChain(R_{1} ...
R_{n}) S) |
R o ... o
_{1}^{Ipo}R ⊆ _{n}^{Ipo}S^{Ipo} |

EquivalentObjectProperties(R_{1} ...
R_{n}) |
R =
_{j}^{Ipo}R for each 1 ≤ _{k}^{Ipo}j , k
≤ n |

DisjointObjectProperties(R_{1} ...
R_{n}) |
R ∩
_{j}^{Ipo}R is empty for each 1 ≤ _{k}^{Ipo}j ,
k ≤ n and j ≠ k |

ObjectPropertyDomain(R C) | { x | ∃ y : (x , y ) ∈
R } ⊆ ^{Ipo}C^{Ic} |

ObjectPropertyRange(R C) | { y | ∃ x : (x , y ) ∈
R } ⊆ ^{Ipo}C^{Ic} |

InverseObjectProperties(R S) | R = { ^{Ipo}( x , y ) | ( y , x )
∈ S }^{Ipo} |

FunctionalObjectProperty(R) | ( x , y and _{1} ) ∈ R^{Ipo}( x ,
y imply _{2} ) ∈ R^{Ipo}y_{1} =
y_{2} |

InverseFunctionalObjectProperty(R) | ( x and _{1} , y ) ∈ R^{Ipo}(
x imply _{2} , y ) ∈ R^{Ipo}x_{1} =
x_{2} |

ReflexiveObjectProperty(R) | x ∈ Δ implies _{I}( x , x ) ∈
R^{Ipo} |

IrreflexiveObjectProperty(R) | x ∈ Δ implies _{I}( x , x ) is not in
R^{Ipo} |

SymmetricObjectProperty(R) | ( x , y ) ∈ R implies ^{Ipo}( y , x ) ∈
R^{Ipo} |

AsymmetricObjectProperty(R) | ( x , y ) ∈ R implies ^{Ipo}( y , x ) is
not in R^{Ipo} |

TransitiveObjectProperty(R) | R^{Ipo} o R^{Ipo} ⊆ R^{Ipo} |

SubDataPropertyOf(U V) | U ⊆ ^{Ipd}V^{Ipd} |

EquivalentDataProperties(U_{1} ...
U_{n}) |
U =
_{j}^{Ipd}U for each 1 ≤ _{k}^{Ipd}j , k
≤ n |

DisjointDataProperties(U_{1} ...
U_{n}) |
U ∩
_{j}^{Ipd}U is empty for each 1 ≤ _{k}^{Ipd}j ,
k ≤ n and j ≠ k |

DataPropertyDomain(U C) | { x | ∃ y : (x , y ) ∈
U } ⊆ ^{Ipd}C^{Ic} |

DataPropertyRange(U DR) | { y | ∃ x : (x , y ) ∈
U } ⊆ ^{Ipd}DR^{D} |

FunctionalDataProperty(U) | ( x , y and _{1} ) ∈ U^{Ipd}( x ,
y imply _{2} ) ∈ U^{Ipd}y_{1} =
y_{2} |

SameIndividual(a_{1} ...
a_{n}) |
a =
_{j}^{Ii}a for each 1 ≤ _{k}^{Ii}j , k
≤ n |

DifferentIndividuals(a_{1} ...
a_{n}) |
a ≠
_{j}^{Ii}a for each 1 ≤ _{k}^{Ii}j , k
≤ n and j ≠ k |

ClassAssertion(a C) | a ∈ ^{Ii}C^{Ic} |

ObjectPropertyAssertion(R a b) | ( a ∈
^{Ii} , b^{Ii} )R^{Ipo} |

NegativeObjectPropertyAssertion(R a b) | ( a is not in
^{Ii} , b^{Ii} )R^{Ipo} |

DataPropertyAssertion(U a v) | ( a ∈
^{Ii} , v^{D} )U^{Ipd} |

NegativeDataPropertyAssertion(U a v) | ( a is not in
^{Ii} , v^{D} )U^{Ipd} |

Let *O* be an OWL 2 ontology with vocabulary *V*.
*O* is *consistent* if an interpretation *I* exists
that satisfies all the axioms of the axiom closure of *O* (the
axiom closure of *O* is defined in [OWL 2
Specification]); such *I* is then called a
*model* of *O*. A description *C* is satisfiable
w.r.t. *O* if there is a model *I* of *O* such that
*C ^{Ic}* is not empty.

- [Description Logics]
- The Description Logic Handbook. Franz Baader, Diego Calvanese, Deborah McGuinness, Daniele Nardi, Peter Patel-Schneider, editors. Cambridge University Press, 2003; and Description Logics Home Page.
- [Metamodeling]
- On the Properties of Metamodeling in OWL. Boris Motik. In Proceedings of ISWC-2005
- [OWL 2 Specification]
- OWL 2 Web Ontology Language:Structural Specification and Functional-Style Syntax Boris Motik, Peter F. Patel-Schneider, Ian Horrocks. W3C Working Draft, 11 April 2008, http://www.w3.org/TR/2008/WD-owl2-syntax-20080411/. Latest version available at http://www.w3.org/TR/owl2-syntax/.
- [OWL Abstract Syntax and Semantics]
- OWL Web Ontology Language: Semantics and Abstract Syntax. Peter F. Patel-Schneider, Pat Hayes, and Ian Horrocks, Editors, W3C Recommendation, 10 February 2004.
- [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.
- [XML Schema Datatypes]
- XML Schema Part 2: Datatypes Second Edition. Paul V. Biron and Ashok Malhotra, eds. W3C Recommendation 28 October 2004.