__NUMBEREDHEADINGS__
- Document title:
- OWL 2 Web Ontology Language
Model-Theoretic Semantics (Second Edition)
- 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
- Abstract
- OWL 1.1 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 1.1.
- Status
- This document is an evolution of the OWL 1.1 Web Ontology Language: Model-Theoretic Semantics document that forms part of the OWL 1.1 Web Ontology Language W3C Member Submission.
Copyright © 2006-2007 by the Authors. This document is available under the W3C Document License. See the W3C Intellectual Rights Notice and Legal Disclaimers for additional information.
1 Introduction
This document defines the formal semantics of OWL 1.1. 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 1.1 is defined in a direct model-theoretic way, by interpreting the constructs of the functional-style syntax from [OWL 1.1 Specification]. For the constructs available in SROIQ, the semantics of SROIQ trivially corresponds to the one defined in this document.
OWL 1.1 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 1.1 allows for annotations of ontologies and ontology entities (classes, properties, and individuals) and ontology axioms. Annotations, however, have no semantic meaning in OWL 1.1 and are ignored in this document. Definitions in OWL 1.1 similarly have no semantics. Constructs only used in annotations and definitions, like ObjectProperty, therefore do not show up in this document.
Since OWL 1.1 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].
2 Model-Theoretic Semantics for OWL 1.1
2.1 Entities
A vocabulary (or signature) V = ( N_{C} , N_{Po} , N_{Pd} , N_{I} , N_{D} , N_{V} ) is a 6-tuple where
- N_{C} is a set of OWL classes,
- N_{Po} is a set of object properties,
- N_{Pd} is a set of data properties,
- N_{I} is a set of individuals, and
- N_{D} is a set of datatypes each associated with a positive integer datatype arity,
- N_{V} is a set of well-formed constants.
Since OWL 1.1 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 1.1 is defined with respect to a concrete domain, which is a tuple D = ( Δ_{D} , .^{D} ) where
- Δ_{D} is a fixed set called the data domain,
- .^{D} assigns to each constant v ∈ N_{V} an element v^{D} of Δ_{D}, and
- .^{D} assigns to each datatype d ∈ N_{D} with arity n an n-ary relation d^{D} over Δ_{D}.
The set of datatypes N_{D} in each OWL 1.1 vocabulary must include a unary datatype rdfs:Literal interpreted as Δ_{D}; furthermore, it must also include the following unary datatypes: xsd:string, xsd:boolean, xsd:decimal, xsd:float, xsd:double, xsd:dateTime, xsd:time, xsd:date, xsd:gYearMonth, xsd:gYear, xsd:gMonthDay, xsd:gDay, xsd:gMonth, xsd:hexBinary, xsd:base64Binary, xsd:anyURI, xsd:normalizedString, xsd:token, xsd:language, xsd:NMTOKEN, xsd:Name, xsd:NCName, xsd:integer, xsd:nonPositiveInteger, xsd:negativeInteger, xsd:long, xsd:int, xsd:short, xsd:byte, xsd:nonNegativeInteger, xsd:unsignedLong, xsd:unsignedInt, xsd:unsignedShort, xsd:unsignedByte, and xsd:positiveInteger. These datatypes, as well as the well-formed constants from N_{V}, are interpreted as specified in [XML Schema Datatypes]. (Note that the value spaces of primitive XML Schema Datatypes are disjoint, so "2"^^xsd:decimal^{D} is different from "2"^^xsd:float"^{D}.)
The set Δ_{D} is a fixed set that must be large enough; that is, it must contain the extension of each datatype from N_{D} and, apart from that, an infinite number of other objects. Such a definition is ambiguous, as it does not uniquely single out a particular set Δ_{D}; however, the choice of the actual set is not actually relevant for the definition of the semantics, as long as it contains the interpretations of all datatypes that one can "reasonably think of." This allows the implementations to support datatypes other than the ones mentioned in the previous paragraphs without affecting the semantics.
Given a vocabulary V and a concrete domain D, an interpretation I = ( Δ_{I} , .^{Ic} , .^{Ipo} , .^{Ipd} , .^{Ii} ) is a 5-tuple where
- Δ_{I} is a nonempty set disjoint with Δ_{D}, called the object domain;
- .^{Ic} is the class interpretation function that assigns to each OWL class A ∈ N_{Ic} a subset A^{Ic} of Δ_{I} ;
- .^{Ipo} is the object property interpretation function that assigns to each object property R ∈ N_{Po} a subset R^{Ipo} of Δ_{I} x Δ_{I} ;
- .^{Ipd} is the data property interpretation function that assigns to each data property U ∈ N_{Pd} a subset U^{Ipd} of Δ_{I} x Δ_{D} ;
- .^{Ii} is the individual interpretation function that assigns to each individual a ∈ N_{I} an element a^{Ii} from Δ_{I}.
2.2 Object Property Expressions
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} } |
2.3 Data Ranges Expressions
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^{D} where n is the arity of DR |
DatatypeRestriction(DR f v) |
the n-ary relation over Δ_{D} obtained by applying the facet f with value v |
2.4 Class Expressions
We extend the class interpretation function .^{Ic} to descriptions as shown in Table 3. With #S we denote the number of elements in a set S.
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^{Ipo} and y ∈ C^{Ic} } |
ObjectAllValuesFrom(R C) | { x | ∀ y : ( x, y ) ∈ R^{Ipo} implies 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^{Ipo} and y ∈ C^{Ic} } ≥ n } |
ObjectMaxCardinality(n R C) | { x | #{ y | ( x, y ) ∈ R^{Ipo} and y ∈ C^{Ic} } ≤ n } |
ObjectExactCardinality(n R C) | { x | #{ y | ( x, y ) ∈ R^{Ipo} and y ∈ C^{Ic} } = n } |
DataSomeValuesFrom(U_{1} ... U_{n} DR) | { x | ∃ y_{1}, ..., y_{n} : ( x, y_{k} ) ∈ U_{k}^{Ipd} for each 1 ≤ k ≤ n and ( y_{1}, ..., y_{n} ) ∈ DR^{D} } |
DataAllValuesFrom(U_{1} ... U_{n} DR) | { x | ∀ y_{1}, ..., y_{n} : ( x, y_{k} ) ∈ U_{k}^{Ipd} for each 1 ≤ 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^{Ipd} and y ∈ DR^{D} } ≥ n } |
DataMaxCardinality(n U DR) | { x | #{ y | ( x, y ) ∈ U^{Ipd} and y ∈ DR^{D} } ≤ n } |
DataExactCardinality(n U DR) | { x | #{ y | ( x, y ) ∈ U^{Ipd} and y ∈ DR^{D} } = n } |
2.5 Axioms
Satisfaction of OWL 1.1 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_{k}^{Ic} for each 1 ≤ j , k ≤ n |
DisjointClasses(C_{1} ... C_{n}) | C_{j}^{Ic} ∩ C_{k}^{Ic} is empty for each 1 ≤ j , k ≤ n and j ≠ k |
DisjointUnion(A C_{1} ... C_{n}) | A^{Ic} = C_{1}^{Ic} ∪ ... ∪ C_{n}^{Ic} and C_{j}^{Ic} ∩ C_{k}^{Ic} is empty for each 1 ≤ j , k ≤ n and j ≠ k |
SubObjectPropertyOf(R S) | R^{Ipo} ⊆ S^{Ipo} |
SubObjectPropertyOf(SubObjectPropertyChain(R_{1} ... R_{n}) S) | R_{1}^{Ipo} o ... o R_{n}^{Ipo} ⊆ S^{Ipo} |
EquivalentObjectProperties(R_{1} ... R_{n}) | R_{j}^{Ipo} = R_{k}^{Ipo} for each 1 ≤ j , k ≤ n |
DisjointObjectProperties(R_{1} ... R_{n}) | R_{j}^{Ipo} ∩ R_{k}^{Ipo} is empty for each 1 ≤ 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_{1} ) ∈ R^{Ipo} and ( x , y_{2} ) ∈ R^{Ipo} imply y_{1} = y_{2} |
InverseFunctionalObjectProperty(R) | ( x_{1} , y ) ∈ R^{Ipo} and ( x_{2} , y ) ∈ R^{Ipo} imply x_{1} = x_{2} |
ReflexiveObjectProperty(R) | x ∈ Δ_{I} implies ( x , x ) ∈ R^{Ipo} |
IrreflexiveObjectProperty(R) | x ∈ Δ_{I} implies ( x , x ) is not in R^{Ipo} |
SymmetricObjectProperty(R) | ( x , y ) ∈ R^{Ipo} implies ( y , x ) ∈ R^{Ipo} |
AsymmetricObjectProperty(R) | ( x , y ) ∈ R^{Ipo} implies ( 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_{k}^{Ipd} for each 1 ≤ j , k ≤ n |
DisjointDataProperties(U_{1} ... U_{n}) | U_{j}^{Ipd} ∩ U_{k}^{Ipd} is empty for each 1 ≤ 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_{1} ) ∈ U^{Ipd} and ( x , y_{2} ) ∈ U^{Ipd} imply y_{1} = y_{2} |
SameIndividual(a_{1} ... a_{n}) | a_{j}^{Ii} = a_{k}^{Ii} for each 1 ≤ j , k ≤ n |
DifferentIndividuals(a_{1} ... a_{n}) | a_{j}^{Ii} ≠ a_{k}^{Ii} for each 1 ≤ 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^{Ii} , b^{Ii} ) is not in R^{Ipo} |
DataPropertyAssertion(U a v) | ( a^{Ii} , v^{D} ) ∈ U^{Ipd} |
NegativeDataPropertyAssertion(U a v) | ( a^{Ii} , v^{D} ) is not in U^{Ipd} |
2.6 Ontologies
Let O be an OWL 1.1 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 1.1 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. O entails an OWL 1.1 ontology O' with vocabulary V if every model of O is also a model of O'; furthermore, O and O' are equivalent if O entails O' and O' entails O.
3 References
- [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 1.1 Specification]
- OWL 1.1 Web Ontology Language: Structural Specification and Functional-Style Syntax. Peter F. Patel-Schneider, Ian Horrocks, and Boris Motik, eds., 2006.
- [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.